Plane & Solid Geometry for the Secondary Classroom
Student Edition · Thirteen Chapters · With Worked Examples and Exercises
"There is no royal road to geometry."
— attributed to Euclid, in conversation with Ptolemy I
This textbook follows the standard sequence of an American high-school geometry course. Each section begins with the underlying concept, develops it through worked examples, and closes with a short problem set for consolidation. Cross-references throughout invite the reader to revisit foundational ideas — geometry, after all, is one of the few subjects in which everything genuinely does connect to everything else.
Tools of Geometry
Geometry begins, like most disciplines, with a vocabulary so primitive that it cannot itself be defined — only described. Point, line, and plane are the irreducible nouns. From this trio, with a handful of postulates, the entire edifice rises: angles, polygons, polyhedra, the conic sections, and ultimately the geometry of curved spaces. This chapter establishes the working tools.
Points, Lines, and Planes
A point is a location; it has no size. A line is a straight, one-dimensional figure extending infinitely in two directions; it contains infinitely many points. A plane is a flat, two-dimensional surface extending infinitely in all directions.
Points are named with capital letters: $A$, $B$, $C$. Lines are named with a lower-case letter or by any two points on them: line $\ell$, or $\overleftrightarrow{AB}$. Planes are named by a capital script letter or by three non-collinear points: plane $\mathcal{M}$, or plane $ABC$.
Two further pieces of vocabulary deserve attention. Points lying on the same line are collinear; points lying in the same plane are coplanar. (Any two points are trivially collinear; any three points are coplanar, but four may not be.)
Example 1
Naming a line and a plane.
Suppose points $A$, $B$, $C$ all lie on line $m$, and points $A$, $B$, $C$, $D$ all lie on plane $\mathcal{N}$.
The line may be named $m$, $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, or $\overleftrightarrow{AC}$. The plane may be named $\mathcal{N}$, or by any three non-collinear points such as plane $ABD$ or plane $BCD$. (Note: plane $ABC$ does not work — those three are collinear, and collinear points cannot determine a plane.)
Problem Set 1.1
Name three collinear points if line $\ell$ contains $X$, $Y$, $Z$.
How many planes contain a given line? Explain.
Sketch two planes that intersect in line $k$.
True or false: three points always determine a plane.
Can two lines intersect in more than one point? Justify briefly.
Linear Measure
A line segment is the portion of a line bounded by two endpoints, written $\overline{AB}$. Its length, denoted $AB$ (without the bar), is a positive real number. Congruent segments — written $\overline{AB} \cong \overline{CD}$ — have equal length, $AB = CD$.
If $B$ lies between $A$ and $C$ on a line, then $AB + BC = AC$.
Example 2
Segment addition.
Point $B$ lies between $A$ and $C$. If $AB = 2x + 1$, $BC = 3x - 4$, and $AC = 22$, find $x$.
By the Segment Addition Postulate, $AB + BC = AC$:
$$(2x+1) + (3x-4) = 22$$
$$5x - 3 = 22 \;\Rightarrow\; x = 5.$$
Check: $AB = 11$, $BC = 11$, total $22$. ✓
Problem Set 1.2
$Q$ lies between $P$ and $R$. $PQ = 8$, $QR = 11$. Find $PR$.
$M$ lies between $L$ and $N$. $LN = 27$, $LM = 3x$, $MN = x + 3$. Find $x$.
Are $\overline{AB}$ and $\overline{CD}$ congruent if $AB = 4.3$ cm and $CD = 43$ mm?
If $AB = 2.5$ in. and $BC = 4.75$ in. with $B$ between, what is $AC$?
Sketch and label a segment $\overline{XY}$ with $XY = 6$ and a midpoint $M$. Identify $XM$ and $MY$.
Distance and Midpoints
On a number line, the distance between points at coordinates $a$ and $b$ is $|b - a|$. In the coordinate plane, we generalize via the Pythagorean Theorem (see §8.2 for the underlying proof):
For points $A(x_1, y_1)$ and $B(x_2, y_2)$, $$d(A,B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$
The midpoint of $\overline{AB}$ is $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right).$$
Example 3
Distance and midpoint.
Find the distance and midpoint of $A(-2, 3)$ and $B(4, -5)$.
The midpoint formula is just the arithmetic mean applied coordinate-wise. The distance formula is the Pythagorean Theorem in coordinate dress. Both are worth memorizing, but their genealogies are humble.
Problem Set 1.3
Find the distance between $(1, 2)$ and $(7, 10)$.
Find the midpoint of $(-3, 8)$ and $(5, -2)$.
One endpoint of a segment is $(2, 5)$ and the midpoint is $(6, 3)$. Find the other endpoint.
Show that the points $(0,0)$, $(3,4)$, and $(-4,3)$ are all equidistant from the origin... no wait, two of them are. Which?
The distance between $(a, 0)$ and $(0, a)$ is $\sqrt{50}$. Find $a$.
Angle Measure
A ray $\overrightarrow{AB}$ is a half-line: it begins at endpoint $A$ and extends through $B$ indefinitely. An angle is formed by two rays sharing a common endpoint, the vertex. Angles are named by their vertex ($\angle B$), by a point–vertex–point sequence ($\angle ABC$), or by a number ($\angle 1$).
Angles are measured in degrees, with one full revolution equal to $360°$ — an arbitrary inheritance from Babylonian astronomy, who liked base-60. (Radians, the more mathematically honest unit, will wait until calculus.)
An angle is acute if $0° < m\angle < 90°$, right if $m\angle = 90°$, obtuse if $90° < m\angle < 180°$, and straight if $m\angle = 180°$.
If point $D$ lies in the interior of $\angle ABC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.
Example 4
Angle addition.
$\overrightarrow{BD}$ lies in the interior of $\angle ABC$. If $m\angle ABD = (3x + 5)°$, $m\angle DBC = (2x - 1)°$, and $m\angle ABC = 64°$, find $x$ and $m\angle ABD$.
$(3x+5) + (2x-1) = 64 \Rightarrow 5x + 4 = 64 \Rightarrow x = 12$. Then $m\angle ABD = 3(12)+5 = 41°$.
Problem Set 1.4
Classify: $43°$, $90°$, $137°$, $180°$.
$m\angle XYZ = 120°$ and $\overrightarrow{YW}$ bisects it. Find $m\angle XYW$.
$m\angle 1 = (4x)°$, $m\angle 2 = (2x+30)°$, and together they form a straight angle. Find $x$.
An angle's measure is three times its supplement. Find both. (Hint: supplement comes in §1.5.)
Sketch $\angle PQR = 75°$ and bisect it with $\overrightarrow{QS}$. State $m\angle PQS$.
Angle Relationships
Four named relationships pervade the rest of the course; learn them once, use them forever.
Complementary angles sum to $90°$. Supplementary angles sum to $180°$. Vertical angles are the two non-adjacent angles formed by two intersecting lines (and they are congruent). A linear pair is two adjacent angles forming a straight line (they are supplementary).
Vertical angles are congruent.
Example 5
Vertical and linear pair.
Two lines cross. One of the four angles measures $(3x+10)°$; the angle vertical to it measures $(5x-20)°$. Find $x$ and the measure.
Vertical angles are congruent: $3x + 10 = 5x - 20 \Rightarrow 2x = 30 \Rightarrow x = 15$. The angle measures $3(15)+10 = 55°$.
Problem Set 1.5
Find the complement and supplement of $37°$.
$\angle A$ and $\angle B$ are supplementary. $m\angle A = 4x$, $m\angle B = 2x + 60$. Find both.
Two angles are vertical. One is $(2x+15)°$; the other is $(x+40)°$. Find $x$.
Can two acute angles be supplementary? Justify.
Two angles form a linear pair. One is five times the other. Find both.
Two-Dimensional Figures
A polygon is a closed figure formed by three or more line segments (called sides) meeting only at endpoints (vertices). Polygons are named by side count: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10), and so on. A polygon is convex if no diagonal lies outside it; otherwise concave. It is regular if all sides and all angles are congruent.
The perimeter $P$ of a polygon is the sum of side lengths; for a circle, the analogous quantity is the circumference $C = 2\pi r$. The area is the number of unit squares the figure covers.
Find the perimeter and area of a $4$-cm by $7$-cm rectangle.
Find the circumference and area of a circle with $r = 5$ in. Round to one decimal.
A triangle has base $10$ and height $6$. Find its area.
Plot $A(-2,-1)$, $B(4,-1)$, $C(4,3)$, $D(-2,3)$. Identify the figure and find $P$ and $A$.
Three-Dimensional Figures
A polyhedron is a solid bounded by flat polygonal faces meeting at line-segment edges and point vertices. Prisms have two congruent parallel bases joined by parallelogram faces; pyramids have a single polygonal base and triangular faces meeting at an apex. Solids with curved surfaces — cylinder, cone, sphere — are not polyhedra. Naming convention: triangular prism, square pyramid, hexagonal prism, and so on, after the base.
Euler's polyhedron formula — $V - E + F = 2$ — holds for every convex polyhedron, from the humble tetrahedron to the soccer-ball-shaped truncated icosahedron. A small miracle, worth verifying on a cube.
Example 7
Identifying a solid.
A solid has 5 faces (one square base, four triangles meeting at an apex), 5 vertices, 8 edges.
It is a square pyramid. Check Euler: $5 - 8 + 5 = 2$. ✓
Problem Set 1.7
Name the solid: bases are congruent pentagons, sides are rectangles.
Identify the solid: 6 square faces, 8 vertices, 12 edges.
Is a cone a polyhedron? Why or why not?
Sketch a triangular pyramid (tetrahedron); list $V$, $E$, $F$. Verify Euler.
Geometry's distinguishing virtue, the reason it has occupied curricula since Euclid, is that it makes argument visible. We do not simply learn that the angles of a triangle sum to $180°$; we learn why, from first principles. This chapter introduces the apparatus of mathematical reasoning — inductive observation, deductive proof, and the formal scaffolding that connects them.
Inductive Reasoning and Conjecture
Inductive reasoning moves from specific observations to a general claim, called a conjecture. It is the engine of scientific discovery and the source of most mathematical questions — though, crucially, it does not constitute proof. A single counterexample is enough to demolish a conjecture; no number of confirming instances proves one.
Example 1
Pattern and conjecture.
Given the sequence $1, 4, 9, 16, 25, \ldots$, conjecture the $n$th term.
The terms are $1^2, 2^2, 3^2, 4^2, 5^2$. Conjecture: $a_n = n^2$.
Example 2
Finding a counterexample.
Claim: "All prime numbers are odd."
$2$ is prime and even. Counterexample found; the claim is false.
Problem Set 2.1
Next two terms in $2, 5, 10, 17, 26, \ldots$ and a formula for the $n$th term.
Counterexample: "If $n$ is divisible by 4, then $n$ is divisible by 8."
Conjecture about the sum of two odd integers. Then test.
Make a conjecture about the product of consecutive integers and divisibility by 2.
True or false: a conjecture supported by 100 examples is proved. Explain.
Logic
A statement (or proposition) is a sentence that is either true or false. The negation of statement $p$, written $\neg p$ or "not $p$," has the opposite truth value. Two statements may be combined:
The conjunction "$p$ and $q$" — written $p \wedge q$ — is true only when both are true.
The disjunction "$p$ or $q$" — written $p \vee q$ — is true when at least one is true. (Mathematical "or" is inclusive.)
A truth table displays every combination of input truth values and the resulting truth value of the compound:
$p$ $q$ | $p \wedge q$ | $p \vee q$
T T | T | T
T F | F | T
F T | F | T
F F | F | F
Example 3
Evaluating compound statements.
Let $p$: "$5 > 2$" (T) and $q$: "$5$ is even" (F). Evaluate $p \wedge q$, $p \vee q$, $\neg p \vee q$.
$p \wedge q$: T $\wedge$ F = F. $p \vee q$: T $\vee$ F = T. $\neg p \vee q$: F $\vee$ F = F.
Problem Set 2.2
Write the negation of "Triangle $ABC$ is equilateral."
$p$: "It is raining." $q$: "It is windy." Translate $p \wedge \neg q$.
Build a truth table for $\neg p \wedge q$.
If $p$ is T and $q$ is T, what is $\neg(p \vee q)$?
Find a statement $p$ such that $p \wedge \neg p$ is true. (Trick.)
Conditional Statements
A conditional has the form "if $p$, then $q$," written $p \to q$. Here $p$ is the hypothesis and $q$ is the conclusion. The conditional is false only when $p$ is true and $q$ is false; in every other case (including when $p$ is false), it is true. From any conditional we derive three related statements:
Critically, a conditional and its contrapositive are logically equivalent — both true together or both false together. The converse is not equivalent: "if it is raining, the ground is wet" does not imply "if the ground is wet, it is raining."
Example 4
Writing related conditionals.
Given: "If two angles are vertical, then they are congruent." Write the converse, inverse, contrapositive, and state which are true.
Converse: If two angles are congruent, they are vertical. (False — two congruent angles need not be vertical.) Inverse: If two angles are not vertical, they are not congruent. (False — same reason.) Contrapositive: If two angles are not congruent, they are not vertical. (True — equivalent to original.)
Problem Set 2.3
Identify hypothesis and conclusion: "If $x = 3$, then $2x = 6$."
Write the converse of "If a polygon is a square, then it is a rectangle." Is it true?
Write the contrapositive of "If it snows, then school is canceled."
Give an example where the conditional is true but the converse is false.
If $p \to q$ is true and $q$ is false, what can you conclude about $p$?
Deductive Reasoning
Where induction generalizes from instances, deduction derives specific conclusions from accepted general principles. Two patterns appear so often they are worth naming:
If $p \to q$ is true and $p$ is true, then $q$ is true.
If $p \to q$ and $q \to r$ are both true, then $p \to r$ is true.
Example 5
Applying the laws.
Given: (1) If a figure is a square, then it is a rectangle. (2) If a figure is a rectangle, then it has four right angles. What can be concluded?
By the Law of Syllogism: if a figure is a square, then it has four right angles.
Problem Set 2.4
Given "If a number is divisible by 6, it is divisible by 3," and "12 is divisible by 6." Conclude.
Given "If $a = b$, then $a + c = b + c$," and "$x = 5$." Conclude.
Apply syllogism: "If $p$, then $q$"; "If $q$, then $r$."
Why is the converse error ("affirming the consequent") invalid?
Construct two true conditionals chained by syllogism in everyday language.
Postulates and Paragraph Proofs
A postulate (or axiom) is a statement accepted without proof. A theorem is a statement that has been proved. A proof is a logical argument using definitions, postulates, and previously established theorems.
(1) Through any two points there is exactly one line. (2) Through any three non-collinear points there is exactly one plane. (3) If two points lie in a plane, the entire line through them lies in that plane. (4) If two planes intersect, their intersection is a line.
A paragraph proof presents reasoning in continuous prose, with each claim justified by a definition, postulate, or theorem.
Example 6
A paragraph proof.
Prove: if $M$ is the midpoint of $\overline{AB}$, then $AM = \tfrac{1}{2}AB$.
By definition of midpoint, $AM = MB$. By the Segment Addition Postulate, $AM + MB = AB$. Substituting $MB$ with $AM$: $AM + AM = AB$, so $2AM = AB$, hence $AM = \tfrac{1}{2}AB$. $\blacksquare$
Problem Set 2.5
State a postulate that justifies: "Two intersecting lines lie in exactly one plane."
Write a paragraph proof: if $\overline{AB} \cong \overline{CD}$ then $AB = CD$.
Distinguish postulate, theorem, definition.
What justifies "Through any two points, exactly one line exists"?
Prove informally: any two points determine exactly one segment.
Algebraic Proof
Before tackling geometric proofs, we rehearse the structure on familiar algebra. A two-column proof places statements on the left and justifications on the right.
Reflexive: $a = a$. Symmetric: if $a = b$ then $b = a$. Transitive: if $a = b$ and $b = c$ then $a = c$. Addition/Subtraction/Multiplication/Division properties; Substitution; Distributive: $a(b+c) = ab + ac$.
The Properties of Equality extend cleanly to segments: congruence of segments is reflexive, symmetric, and transitive. The Segment Addition Postulate (§1.2) is the workhorse of segment proofs.
$AB = CD$ and $BC = DE$ (definition of congruence). Adding: $AB + BC = CD + DE$ (addition property). By Segment Addition, $AC = AB + BC$ and $CE = CD + DE$. By substitution, $AC = CE$, so $\overline{AC} \cong \overline{CE}$. $\blacksquare$
Problem Set 2.7
Given $AB = BC$ and $BC = CD$, prove $AB = CD$.
If $M$ is the midpoint of $\overline{XY}$, prove $XM = MY$.
Justify each step of a proof that congruence is transitive.
Given $B$ between $A$ and $C$, $AB = 2x$, $BC = x+5$, $AC = 20$. Find $x$.
If $AB \cong CD$, what is justified about $BA$ and $DC$?
Proving Angle Relationships
The angle relationships from §1.5 can now be proved, not merely asserted.
If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
Example 9
Proof of the Vertical Angles Theorem.
Given: $\angle 1$ and $\angle 3$ are vertical. Prove: $\angle 1 \cong \angle 3$.
$\angle 1$ and $\angle 2$ form a linear pair, so are supplementary: $m\angle 1 + m\angle 2 = 180°$. Likewise $\angle 2$ and $\angle 3$: $m\angle 2 + m\angle 3 = 180°$. Therefore $m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3$. Subtracting $m\angle 2$ gives $m\angle 1 = m\angle 3$, i.e. $\angle 1 \cong \angle 3$. $\blacksquare$
Problem Set 2.8
If $\angle A$ and $\angle B$ are supplementary, and $m\angle A = 65°$, find $m\angle B$.
Prove: if $\angle 1 \cong \angle 2$ and $\angle 2 \cong \angle 3$, then $\angle 1 \cong \angle 3$.
Two angles form a linear pair. If one is twice the other, find both.
$\angle X$ and $\angle Y$ are both complementary to $\angle Z$. What can be concluded?
Sketch and prove informally: all right angles are congruent.
Two lines in a plane either intersect or they don't. The non-intersecting case — parallelism — is the more conceptually delicate of the two, and Euclid's notorious Fifth Postulate concerning it eventually birthed entire non-Euclidean geometries. We shall, for now, remain inside the comfortable confines of the plane.
Parallel Lines and Transversals
Two lines are parallel (notation: $\ell \parallel m$) if they are coplanar and never intersect. Two lines are skew if they are not coplanar — possible only in three-dimensional space. A line crossing two or more lines is called a transversal.
When a transversal cuts two lines, eight angles are formed. They are organized into pairs by position:
Corresponding angles — same position at each intersection.
Alternate interior angles — opposite sides of the transversal, between the two lines.
Alternate exterior angles — opposite sides of the transversal, outside the two lines.
Consecutive (same-side) interior angles — same side of the transversal, between the two lines.
Problem Set 3.1
Sketch two parallel lines with a transversal; label angles 1–8.
Identify a corresponding pair in your figure.
Identify a pair of alternate interior angles.
Give one pair of same-side interior angles.
True or false: skew lines can intersect.
Angles and Parallel Lines
When the two lines crossed by the transversal happen to be parallel, the angle pairs above acquire definite relationships:
If two parallel lines are cut by a transversal, then:
(a) Corresponding angles are congruent.
(b) Alternate interior angles are congruent.
(c) Alternate exterior angles are congruent.
(d) Consecutive interior angles are supplementary.
Example 1
Finding angle measures.
Lines $\ell \parallel m$ are cut by transversal $t$. One angle measures $112°$. Find the other seven.
Each of the eight angles is either $112°$ or its supplement $68°$. Four of each. Pairing them up — vertical, corresponding, and alternate — yields four angles of $112°$ and four of $68°$.
Alternate interior angles are congruent: $3x + 5 = 2x + 25 \Rightarrow x = 20$.
Problem Set 3.2
$\ell \parallel m$; one angle is $74°$. Find a same-side interior angle.
Corresponding angles measure $(5x)°$ and $(3x+40)°$. Find $x$.
Alternate exterior angles are $(2x+10)°$ and $(4x-30)°$. Find $x$.
Same-side interior angles measure $(x+30)°$ and $(2x)°$. Find $x$.
If two parallel lines are cut by a transversal at $90°$, what can be said?
Slopes of Lines
For points $(x_1, y_1)$ and $(x_2, y_2)$ with $x_1 \neq x_2$, the slope of the line through them is $$m = \frac{y_2 - y_1}{x_2 - x_1}.$$
A horizontal line has slope $0$; a vertical line has undefined slope. Two non-vertical lines are parallel iff their slopes are equal; they are perpendicular iff their slopes are negative reciprocals: $m_1 \cdot m_2 = -1$.
Example 3
Slope from two points.
Find the slope of the line through $(2, -1)$ and $(5, 8)$.
$m = \dfrac{8 - (-1)}{5 - 2} = \dfrac{9}{3} = 3$.
Example 4
Parallel and perpendicular.
A line has slope $\tfrac{2}{3}$. What is the slope of (a) a parallel line, (b) a perpendicular line?
(a) $\tfrac{2}{3}$. (b) $-\tfrac{3}{2}$.
Problem Set 3.3
Slope through $(-3, 4)$ and $(5, -2)$.
Are the lines through $(1,2), (3,8)$ and $(4,1), (5,4)$ parallel? Perpendicular?
Find slope perpendicular to one with $m = -\tfrac{4}{5}$.
What is the slope of the line $x = 7$?
Two lines have slopes $\tfrac{3}{4}$ and $-\tfrac{4}{3}$. Describe their relationship.
Equations of Lines
Slope-intercept: $y = mx + b$, where $m$ is the slope and $b$ the $y$-intercept. Point-slope: $y - y_1 = m(x - x_1)$, through $(x_1, y_1)$ with slope $m$. Standard: $Ax + By = C$, where $A, B, C$ are integers, $A \geq 0$.
Example 5
Writing an equation.
Write the equation of the line through $(2, -3)$ with slope $4$, in slope-intercept form.
Find the equation of the line through $(1, 5)$ parallel to $y = 2x - 7$.
Slope is $2$. Point-slope: $y - 5 = 2(x - 1) \Rightarrow y = 2x + 3$.
Problem Set 3.4
Equation through $(0, 4)$ with slope $-2$.
Equation through $(3, 7)$ and $(-1, -1)$.
Equation through $(2, 4)$ parallel to $y = -3x + 5$.
Equation through $(2, 4)$ perpendicular to $y = -3x + 5$.
Convert $y = \tfrac{3}{4}x - 2$ to standard form.
Proving Lines Parallel
The theorems of §3.2 all have converses — and unlike the converse of a typical conditional, these converses are true and serve as criteria for parallelism.
Given two lines cut by a transversal: if (a) corresponding angles are congruent, or (b) alternate interior angles are congruent, or (c) alternate exterior angles are congruent, or (d) consecutive interior angles are supplementary, then the two lines are parallel.
Example 7
Establishing parallelism.
Two lines are cut by a transversal forming alternate interior angles of $(2x+10)°$ and $(3x-15)°$. Find $x$ so the lines are parallel.
Set equal: $2x+10 = 3x-15 \Rightarrow x = 25$. Then both equal $60°$, confirming parallelism.
Problem Set 3.5
If corresponding angles are $(4x)°$ and $(3x+20)°$, find $x$ to make the lines parallel.
If same-side interior angles are $(3x)°$ and $(2x+40)°$, find $x$.
Two lines are perpendicular to the same line. What can you conclude?
Sketch a transversal and label one pair of angles you'd measure to test parallelism.
Justify: in a plane, if two lines are both parallel to a third, they are parallel to each other.
Perpendiculars and Distance
The distance from a point to a line is the length of the perpendicular segment from the point to the line — and only this segment, no other.
The distance between two parallel lines is constant: measure along any perpendicular segment connecting them.
To compute the distance from $(x_0, y_0)$ to the line $Ax + By + C = 0$:
The triangle is geometry's atom: every polygon can be triangulated, and every triangulation reduces a complex problem to claims about three-sided figures. Congruence — sameness of shape and size — is the relationship that lets us transfer information from one triangle to another. The four standard congruence criteria are the levers of half the proofs in this course.
Classifying Triangles
Triangles are classified by sides and by angles.
By sides: equilateral (all three congruent), isosceles (at least two congruent), scalene (no two congruent). By angles: acute (all three acute), right (exactly one $90°$), obtuse (exactly one $>90°$), equiangular (all three congruent, hence $60°$ each).
Example 1
Classification.
Triangle with sides $5, 5, 8$ and angles $52°, 52°, 76°$.
Can a triangle be both right and isosceles? Sketch one.
Can a triangle be both obtuse and equilateral? Explain.
A triangle has angles $x, 2x, 3x$. Find each.
Angles of Triangles
The sum of the interior angles of any triangle is $180°$.
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Example 2
Missing angle.
Two angles of a triangle measure $47°$ and $68°$. Find the third.
$180 - 47 - 68 = 65°$.
Example 3
Exterior angle.
An exterior angle measures $130°$; one non-adjacent interior angle is $55°$. Find the other.
$130 = 55 + x \Rightarrow x = 75°$.
Problem Set 4.2
Two angles: $34°, 87°$. Find the third.
Right triangle: one acute angle $28°$. Find the other.
Exterior angle $112°$; one non-adjacent interior $40°$. Find the other.
Equilateral triangle: each interior angle?
Angles in ratio $2 : 3 : 4$. Find each.
Congruent Triangles
Two triangles are congruent if their corresponding sides and corresponding angles are all congruent. Written $\triangle ABC \cong \triangle DEF$, the order matters: corresponding parts share positions in the name. From the statement, six congruences are implied (three pairs of sides, three pairs of angles), often abbreviated CPCTC — "corresponding parts of congruent triangles are congruent."
Example 4
Reading congruence statements.
$\triangle PQR \cong \triangle XYZ$. List all corresponding parts.
$\triangle ABC \cong \triangle DEF$. Which side corresponds to $\overline{AC}$?
$\triangle MNO \cong \triangle PQR$. Which angle corresponds to $\angle N$?
If two triangles have congruent corresponding parts, what can be concluded?
If $\triangle XYZ \cong \triangle LMN$ and $XY = 7$, find $LM$.
State CPCTC in your own words.
SSS and SAS
We need not verify all six pairs of corresponding parts to conclude congruence. Three suffice — provided we pick the right three.
If three sides of one triangle are congruent to three sides of another, the triangles are congruent.
If two sides and the included angle of one triangle are congruent to the corresponding parts of another, the triangles are congruent.
"Included" matters. SSA — two sides and a non-included angle — does not in general guarantee congruence. This is the "ambiguous case" and the source of many false proofs in student work.
Example 5
Choosing a postulate.
$\triangle ABC$ and $\triangle DEF$ share $AB \cong DE = 5$, $\angle A \cong \angle D = 40°$, $AC \cong DF = 7$. Are they congruent?
$\angle A$ is included between $\overline{AB}$ and $\overline{AC}$. By SAS, $\triangle ABC \cong \triangle DEF$.
Problem Set 4.4
State which postulate (if any) proves the triangles congruent — three sides given: $5, 7, 9$ and $5, 7, 9$.
Two sides and an included angle: $4, 6$ with $50°$ in each.
Two sides and a non-included angle: explain why congruence is not guaranteed.
Sketch two triangles with the same SSS data; what does this tell you about rigidity?
Why is the triangle the strongest engineering shape? (Hint: SSS.)
ASA and AAS
If two angles and the included side of one triangle are congruent to corresponding parts of another, the triangles are congruent.
If two angles and a non-included side of one triangle are congruent to corresponding parts of another, the triangles are congruent.
(AAS is sometimes derived as a corollary of ASA, since once two angles are fixed the third is determined; some textbooks call this the "Saccheri trick.")
Example 6
Proof using ASA.
Given: $\angle A \cong \angle D$, $\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$. Prove $\triangle ABC \cong \triangle DEF$.
Two angles and the included side: $\triangle ABC \cong \triangle DEF$ by ASA. $\blacksquare$
Problem Set 4.5
Identify the postulate: $\angle A \cong \angle X$, $\angle B \cong \angle Y$, $\overline{AC} \cong \overline{XZ}$.
Why does AAA not guarantee congruence? (See §7.3.)
Given two right triangles with congruent legs, what postulate applies?
List all four congruence postulates.
Find a counterexample showing SSA does not imply congruence.
Isosceles and Equilateral Triangles
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
If two angles of a triangle are congruent, then the sides opposite are congruent.
Corollary. An equilateral triangle is equiangular (each angle $60°$), and vice versa.
Example 7
Base-angle theorem.
$\triangle XYZ$ is isosceles with $XY = XZ$. $m\angle X = 40°$. Find $m\angle Y$ and $m\angle Z$.
Base angles $\angle Y$ and $\angle Z$ are congruent. Their sum is $180 - 40 = 140°$, so each is $70°$.
Problem Set 4.6
Isosceles triangle: vertex angle $80°$. Find each base angle.
Two base angles each $55°$. Find vertex angle.
Equilateral triangle: side $6$. Find perimeter and each angle.
Two angles of a triangle measure $50°$ and $80°$. Is it isosceles? Which sides are equal?
If $\triangle ABC$ is isosceles with $AB = AC$ and $m\angle B = 65°$, find $m\angle A$.
Congruence Transformations
A rigid motion (or isometry) is a transformation preserving distance: reflections, translations, rotations, and any composition thereof. Two figures are congruent if and only if one can be mapped onto the other by a rigid motion. (We will study these transformations formally in Chapter 9.)
Example 8
Identifying a congruence transformation.
$\triangle ABC$ has vertices $(1,1), (4,1), (1,3)$. After a translation 2 units right and 3 units up, name the image vertices.
$A'(3,4), B'(6,4), C'(3,6)$. The triangles are congruent.
Are two figures related by a rigid motion always congruent? Justify.
Triangles and Coordinate Proof
A coordinate proof places figures on the coordinate plane, then uses algebra — the distance formula, slope, midpoint — to establish geometric claims. The art lies in choosing coordinates that simplify calculations: place a vertex at the origin, align a side with an axis.
Example 9
Coordinate proof.
Prove that the triangle with vertices $A(0,0), B(6,0), C(3,4)$ is isosceles.
$AC = \sqrt{9 + 16} = 5$. $BC = \sqrt{9 + 16} = 5$. Since $AC = BC$, the triangle is isosceles. $\blacksquare$
Problem Set 4.8
Show $(0,0), (4,0), (2, 2\sqrt{3})$ is equilateral.
Classify the triangle with vertices $(0,0), (3,0), (3,4)$.
Prove $(0,0), (a,0), (a,a)$ is right isosceles.
Find the midpoints of the sides of $(-2,1), (4,3), (0,5)$.
Why does choosing one vertex at the origin simplify coordinate proofs?
Beyond the question of congruence, triangles harbor a rich internal anatomy: special segments, points of concurrency, and inequalities that govern which side-and-angle combinations are even possible. This chapter surveys that anatomy.
Bisectors of Triangles
A perpendicular bisector of a side passes through its midpoint at right angles. An angle bisector divides an angle into two congruent angles.
A point is on the perpendicular bisector of a segment iff it is equidistant from the endpoints.
A point is on the bisector of an angle iff it is equidistant from the two sides of the angle.
The three perpendicular bisectors of a triangle concur at a single point — the circumcenter — equidistant from all three vertices and thus the center of the circumscribed circle. The three angle bisectors concur at the incenter, equidistant from all three sides and the center of the inscribed circle.
Example 1
Using the bisector theorem.
Point $P$ is on the perpendicular bisector of $\overline{AB}$. $PA = 3x + 2$, $PB = 5x - 8$. Find $x$.
$PA = PB$: $3x + 2 = 5x - 8 \Rightarrow x = 5$.
Problem Set 5.1
$P$ on perpendicular bisector of $\overline{AB}$; $PA = 7$. Find $PB$.
$Q$ on bisector of $\angle XYZ$; distance to $\overline{YX}$ is $4$. Find distance to $\overline{YZ}$.
Where is the circumcenter of a right triangle? (Hypotenuse midpoint.)
Where is the incenter of an equilateral triangle? (Centroid; they coincide.)
Find equidistant point from $(0,0)$ and $(6,0)$: describe its locus.
Medians and Altitudes
A median of a triangle joins a vertex to the midpoint of the opposite side. An altitude is a perpendicular segment from a vertex to the line containing the opposite side.
The three medians concur at the centroid, which divides each median in a $2:1$ ratio from the vertex. The three altitudes (or their extensions) concur at the orthocenter. (Fun fact: in any triangle, the circumcenter, centroid, and orthocenter are collinear — they lie on the Euler line. Quite the cocktail-party tidbit.)
Example 2
Centroid ratio.
$G$ is the centroid of $\triangle ABC$. Median from $A$ has length $12$. Find $AG$ and $GM$ (where $M$ is the midpoint of $\overline{BC}$).
The centroid divides the median $2:1$ from the vertex: $AG = 8$, $GM = 4$.
Problem Set 5.2
Median from $A$ has length $15$. $G$ is centroid. Find $AG$.
Sketch the orthocenter of an obtuse triangle. (Outside the triangle.)
Where is the centroid of an equilateral triangle relative to incenter and circumcenter?
Find the centroid of the triangle with vertices $(0,0), (6,0), (0,6)$. (Average the coordinates.)
Inequalities in One Triangle
In any triangle, the longer side is opposite the larger angle; the larger angle is opposite the longer side.
Example 3
Ordering sides by angles.
$\triangle ABC$ has $m\angle A = 80°, m\angle B = 60°, m\angle C = 40°$. Order the sides.
Opposite to $\angle A$ (largest) is $\overline{BC}$; opposite to $\angle C$ (smallest) is $\overline{AB}$. So $AB < AC < BC$.
Problem Set 5.3
Angles: $50°, 60°, 70°$. Order the sides.
Sides: $5, 7, 9$. Which angle is largest?
True or false: in a right triangle, the hypotenuse is the longest side.
$\triangle ABC$: $AB = 3$, $BC = 4$, $AC = 5$. Order the angles.
Can a triangle have two obtuse angles? Justify.
Indirect Proof
An indirect proof (or proof by contradiction) assumes the opposite of what is to be shown, derives a logical contradiction, and concludes the original claim. This is the same maneuver Euclid used to prove $\sqrt{2}$ is irrational and there are infinitely many primes.
Example 4
Indirect proof.
Prove: a triangle cannot have two right angles.
Suppose, for contradiction, that a triangle $\triangle ABC$ has two right angles, say $\angle A$ and $\angle B$. Then $m\angle A + m\angle B = 180°$, leaving $m\angle C = 0°$ — impossible for a triangle. Contradiction; the assumption is false. $\blacksquare$
Problem Set 5.4
Prove indirectly: a triangle cannot have two obtuse angles.
Prove indirectly: if $n^2$ is even, then $n$ is even.
First step in proving "$\ell$ is parallel to $m$" indirectly?
Prove indirectly: in a triangle, at most one angle is obtuse.
What is the contradiction sought in proving $\sqrt{2}$ irrational?
The Triangle Inequality
For any triangle with sides $a, b, c$: $a + b > c$, $a + c > b$, $b + c > a$. Equivalently, the sum of any two sides exceeds the third.
Example 5
Possible triangle?
Can a triangle have sides $4, 6, 11$?
$4 + 6 = 10 < 11$. Triangle Inequality fails — no such triangle exists.
Example 6
Range of third side.
Two sides are $7$ and $10$. What are the possible lengths of the third side $x$?
$|10 - 7| < x < 10 + 7$, so $3 < x < 17$.
Problem Set 5.5
Possible triangle? $5, 5, 9$.
Possible triangle? $2, 3, 5$.
Two sides $8, 13$. Range for third side?
Two sides $4, 4$. Range for third?
If three sticks have lengths $a, b, c$ with $a+b = c$, what do they form?
Inequalities in Two Triangles
If two sides of one triangle are congruent to two sides of another, but the included angle of the first is larger, then the third side of the first is longer.
Same setup, but with third sides given: the larger third side is opposite the larger included angle.
Imagine a door hinge: the wider you open it, the farther the door's edge is from the jamb. (The metaphor is the theorem's name.)
Example 7
Hinge in action.
$\triangle ABC$ and $\triangle DEF$: $AB = DE$, $AC = DF$. $m\angle A = 50°$, $m\angle D = 65°$. Compare $BC$ and $EF$.
Larger included angle ($\angle D$) gives the longer third side: $EF > BC$.
Problem Set 5.6
Two triangles share two sides. Included angles: $40°, 80°$. Which third side is longer?
Two triangles share two sides. Third sides: $5, 8$. Which included angle is larger?
Sketch a door at $30°$ and $60°$ — which configuration has the farther knob?
State the contrapositive of the Hinge Theorem.
Apply Hinge: $\triangle ABC$ with $AB = AC$. Compare $BC$ to itself if $\angle A$ doubles. (Not exact — describe qualitatively.)
Quadrilaterals are the next rung up from triangles in the polygon hierarchy, and they admit a richer taxonomy: parallelograms (with their various special cases), trapezoids, kites. Much of the interest lies in the criteria — what minimal information suffices to declare a quadrilateral one type rather than another.
Angles of Polygons
The sum of the interior angles of an $n$-gon is $(n-2) \cdot 180°$.
The reason is satisfying: any convex $n$-gon can be triangulated from a single vertex into $n - 2$ triangles, each contributing $180°$.
The sum of the exterior angles of any convex polygon, taking one at each vertex, is $360°$.
Example 1
Interior angle of a regular hexagon.
$n = 6$: sum $= 4 \cdot 180 = 720°$; each angle $720/6 = 120°$.
Problem Set 6.1
Sum of interior angles of a $12$-gon.
Each interior angle of a regular pentagon.
Each exterior angle of a regular octagon.
How many sides has a polygon whose interior angles sum to $1440°$?
How many sides has a regular polygon whose interior angle is $150°$?
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
In any parallelogram: (a) opposite sides are congruent; (b) opposite angles are congruent; (c) consecutive angles are supplementary; (d) the diagonals bisect each other.
$ABCD$ parallelogram: $m\angle A = (3x)°$, $m\angle C = (x + 50)°$. Find $x$.
True or false: every rectangle is a parallelogram.
Tests for Parallelograms
To conclude that a quadrilateral is a parallelogram, any one of these suffices:
(a) Both pairs of opposite sides parallel. (Definition.)
(b) Both pairs of opposite sides congruent.
(c) Both pairs of opposite angles congruent.
(d) One pair of opposite sides both parallel and congruent.
(e) Diagonals bisect each other.
Example 3
Verifying.
Quadrilateral $ABCD$ with $A(0,0), B(4,0), C(5,3), D(1,3)$. Parallelogram?
$\overline{AB}$ horizontal length $4$; $\overline{DC}$ from $(1,3)$ to $(5,3)$, also horizontal, length $4$. One pair of opposite sides parallel and congruent ⇒ parallelogram. ✓
Rectangle: one diagonal is $13$, one side $5$. Find the other side.
Diagonals of rectangle $WXYZ$ meet at $P$. $WP = 3x - 2$, $YP = x + 6$. Find $x$ and $WY$.
True or false: every parallelogram with one right angle is a rectangle.
Find the area of a $7 \times 9$ rectangle.
Rhombi and Squares
A rhombus is a parallelogram with four congruent sides. A square is both a rectangle and a rhombus.
The diagonals of a rhombus (a) are perpendicular and (b) bisect opposite angles.
Example 5
Rhombus diagonals.
Rhombus $ABCD$, diagonals $\overline{AC}$ and $\overline{BD}$ meet at $E$. $m\angle BAC = 35°$. Find $m\angle BAD$.
$\overline{AC}$ bisects $\angle A$: $m\angle BAD = 2(35) = 70°$.
Problem Set 6.5
Rhombus side $10$. Find perimeter.
Rhombus diagonals $6$ and $8$. Find side length. (Half-diagonals $3, 4$; side $5$.)
Square side $7$. Find diagonal.
State the relationship: parallelogram, rectangle, rhombus, square.
Find $x$ if a rhombus has $\angle 1 = (3x + 6)°$ where one diagonal bisects it into $24°$ halves.
Kites and Trapezoids
A trapezoid has exactly one pair of parallel sides — the bases. An isosceles trapezoid has congruent legs. A kite has two pairs of consecutive congruent sides (no opposite sides parallel).
The midsegment of a trapezoid (joining the midpoints of the legs) is parallel to each base and has length $\tfrac{1}{2}(b_1 + b_2)$.
Example 6
Midsegment of trapezoid.
Trapezoid with parallel bases $8$ and $14$. Find the midsegment.
$\tfrac{1}{2}(8+14) = 11$.
Problem Set 6.6
Bases $6, 10$. Midsegment?
Midsegment $9$, one base $5$. Find the other base.
Isosceles trapezoid: list congruent angles.
Kite: diagonals — what is true about them?
Is a parallelogram a trapezoid? (Some texts say yes, modern inclusive definition; Glencoe says no, exclusive.)
Where congruence demands sameness in both shape and size, similarity requires only sameness of shape. The two concepts are siblings: similar figures are congruent up to a scale factor. The notion of proportional reasoning that emerges here also drives trigonometry, scale modeling, and (later) the calculus of related rates.
Ratios and Proportions
A ratio compares two quantities by division: $a:b$ or $a/b$. A proportion equates two ratios: $\dfrac{a}{b} = \dfrac{c}{d}$.
The ratio of cats to dogs is $3:5$ and there are 24 cats. How many dogs?
Solve $\dfrac{x+1}{4} = \dfrac{x-2}{2}$.
If $\dfrac{a}{b} = \dfrac{c}{d}$, prove $\dfrac{a+b}{b} = \dfrac{c+d}{d}$.
A scale on a map reads $1\text{ cm} : 50\text{ km}$. How far in reality is $4.5$ cm on the map?
Similar Polygons
Two polygons are similar (written $\sim$) if (a) corresponding angles are congruent and (b) corresponding sides are proportional. The constant ratio of corresponding sides is the scale factor.
If $\overline{DE}$ cuts $\overline{AB}$ and $\overline{AC}$ at midpoints, what is the relationship to $\overline{BC}$? (Midsegment.)
Midsegment $\overline{MN}$ of $\triangle ABC$ with $BC = 14$. Find $MN$.
Three parallel lines cut transversals into segments $4, 6$ and $x, 9$. Find $x$.
State the converse of the Triangle Proportionality Theorem.
Parts of Similar Triangles
If $\triangle ABC \sim \triangle DEF$ with scale factor $k$, then any pair of corresponding linear measures — sides, perimeters, medians, altitudes, angle bisectors — also satisfy ratio $k$. Areas, however, scale by $k^2$; volumes by $k^3$ (see §11.5 and §12.8).
Example 5
Scaling perimeters and altitudes.
$\triangle ABC \sim \triangle DEF$, scale factor $\tfrac{1}{3}$. Perimeter of $\triangle ABC = 15$, altitude of $\triangle DEF = 12$. Find perimeter of $\triangle DEF$ and altitude of $\triangle ABC$.
Similar triangles, scale factor $2:5$. Perimeter of smaller $18$. Find larger.
Altitude ratio $3:7$. Scale factor?
Two similar triangles have angle bisectors of $5$ and $8$. Scale factor?
If scale factor is $1:2$, what is ratio of areas?
If areas are $9:16$, what is scale factor?
Similarity Transformations
A dilation is a transformation that scales every distance from a fixed center by a constant factor. Dilations are similarity transformations: image and pre-image are similar, with scale factor equal to the dilation factor $k$. When $|k| > 1$, an enlargement; when $0 < |k| < 1$, a reduction. (More in §9.6.)
Example 6
Dilating about the origin.
Dilate $A(2, -3)$ about the origin by factor $4$.
$A' = (4 \cdot 2, 4 \cdot -3) = (8, -12)$.
Problem Set 7.6
Dilate $(3, 4)$ about origin by factor $2$.
Dilate $(-6, 9)$ by factor $\tfrac{1}{3}$.
If a dilation maps $(2, 5)$ to $(8, 20)$, find the scale factor.
Distinguish dilation from translation, rotation, reflection.
Is a dilation an isometry? Why or why not?
Scale Drawings and Models
A scale drawing represents an object proportional to its actual size. The scale states the ratio: $1\text{ in.}:10\text{ ft}$, $1:48$, etc. Once you have the scale, every linear measurement is a single proportion away.
Example 7
Reading a scale.
Floor plan scale $1\text{ in.}:4\text{ ft}$. A room measures $3.5$ in. on the plan. Find its actual length.
$3.5 \times 4 = 14$ ft.
Problem Set 7.7
Scale $1\text{ cm}:25\text{ km}$. Map shows $6.4$ cm. Actual distance?
A model car at scale $1:24$ measures $7$ in. Actual car length?
A blueprint shows a $2.5$ in. door at scale $1\text{ in.}:3\text{ ft}$. Actual door height?
If the area of a scale model is $1/100$ of actual, what is the linear scale?
Is the relationship between scale model and actual a similarity? Justify.
Right triangles have their own geometry — a thicket of theorems flowering from a single seed, the Pythagorean Theorem. Trigonometry, despite its later association with calculus and waves, begins life here as ratios of right-triangle sides.
Geometric Mean
The geometric mean of two positive numbers $a$ and $b$ is $\sqrt{ab}$.
The altitude from the right angle of a right triangle to its hypotenuse creates two smaller triangles, each similar to the original. Consequently, the altitude is the geometric mean of the two segments it creates on the hypotenuse, and each leg is the geometric mean of the hypotenuse and the segment adjacent to that leg.
Example 1
Altitude on hypotenuse.
In a right triangle, the altitude from the right angle splits the hypotenuse into segments of length $4$ and $9$. Find the altitude.
$h = \sqrt{4 \cdot 9} = 6$.
Problem Set 8.1
Geometric mean of $5$ and $20$.
Geometric mean of $8$ and $18$.
Altitude to hypotenuse splits it into $2$ and $8$. Find altitude.
Same setup: find each leg.
Verify: $\sqrt{ab}$ is at most the arithmetic mean $(a+b)/2$ (AM–GM inequality).
The Pythagorean Theorem
In a right triangle with legs $a, b$ and hypotenuse $c$: $a^2 + b^2 = c^2$.
If $a^2 + b^2 = c^2$, then the triangle with these sides is right (with $c$ the hypotenuse).
The theorem has hundreds of proofs — Loomis's 1927 compendium catalogues over 370. We omit the proofs here but note the theorem is the geometric statement equivalent to "Euclidean space is flat."
Example 2
Finding the hypotenuse.
Legs $9$ and $12$. Find hypotenuse.
$c = \sqrt{81 + 144} = \sqrt{225} = 15$.
Example 3
Pythagorean triple identification.
Is $(7, 24, 25)$ a right triangle?
$49 + 576 = 625 = 25^2$. Yes.
Problem Set 8.2
Legs $5, 12$. Hypotenuse?
One leg $8$, hypotenuse $17$. Other leg?
Sides $9, 40, 41$: right triangle?
Sides $4, 5, 7$: right? If not, acute or obtuse? (Compare $a^2 + b^2$ to $c^2$.)
Diagonal of a unit square: simplify.
Special Right Triangles
In an isosceles right triangle, legs are equal length $x$; hypotenuse is $x\sqrt{2}$.
Short leg $x$, long leg $x\sqrt{3}$, hypotenuse $2x$.
Example 4
$45–45–90$.
Leg $5$. Find hypotenuse.
$5\sqrt{2}$.
Example 5
$30–60–90$.
Short leg $4$. Find long leg and hypotenuse.
Long leg $4\sqrt{3}$; hypotenuse $8$.
Problem Set 8.3
$45–45–90$: leg $7$. Hypotenuse?
$30–60–90$: hypotenuse $14$. Find both legs.
$30–60–90$: long leg $6\sqrt{3}$. Find short leg and hypotenuse.
Diagonal of a square with side $8$.
Altitude of an equilateral triangle with side $10$.
Trigonometry
For acute angle $\theta$ in a right triangle:
$\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}.$
The mnemonic SOH–CAH–TOA survives because it works. The ratios are well-defined precisely because all right triangles with a given acute angle are similar (AA), so their side ratios are constant.
Example 6
Finding a side.
Right triangle, $\theta = 35°$, hypotenuse $20$. Find the side opposite $\theta$.
Right triangle: hypotenuse $10$, $\theta = 40°$. Find both legs.
Right triangle: legs $5, 12$. Find both acute angles.
$\sin 30°$, $\cos 60°$, $\tan 45°$: exact values.
If $\sin\theta = \tfrac{3}{5}$, find $\cos\theta$ and $\tan\theta$.
Why does the ratio depend only on the angle, not the size of the triangle?
Angles of Elevation and Depression
The angle of elevation is measured upward from horizontal; angle of depression downward from horizontal. The two are equal when measured from the two endpoints of the same line of sight (alternate interior angles between parallel horizontals).
Example 8
Height of a tree.
Standing $50$ ft from the base of a tree, the angle of elevation to the top is $58°$. Find the tree's height.
$\tan 58° = h/50 \Rightarrow h = 50 \tan 58° \approx 80.04$ ft.
Problem Set 8.5
Angle of elevation $32°$, horizontal distance $60$ ft to a tower. Find tower height.
From a $200$ ft cliff, angle of depression to a boat is $18°$. Find horizontal distance.
A ladder leans against a wall at $70°$ elevation, reaching $15$ ft up. Find ladder length.
The angle of elevation from $30$ ft away to a flagpole top is $50°$. Find pole height.
Explain why angle of elevation equals angle of depression for the same sight line.
Laws of Sines and Cosines
For triangles that are not right, two laws extend trigonometric reasoning to general triangles. In each, sides $a, b, c$ are opposite angles $A, B, C$.
State conditions under which Law of Sines applies; same for Law of Cosines.
Sides $4, 5, 6$. Find the largest angle. (Law of Cosines.)
Explain why Pythagoras is a special case of Law of Cosines.
Vectors
A vector has both magnitude and direction; geometrically, a directed segment. Written $\vec{v} = \langle a, b \rangle$ in component form. Magnitude: $|\vec{v}| = \sqrt{a^2 + b^2}$. Direction angle $\theta$ satisfies $\tan\theta = b/a$.
Vectors add component-wise: $\langle a_1, b_1 \rangle + \langle a_2, b_2 \rangle = \langle a_1 + a_2, b_1 + b_2 \rangle$. They scale: $k\langle a, b \rangle = \langle ka, kb \rangle$.
Example 11
Magnitude and direction.
$\vec{v} = \langle 3, 4 \rangle$. Find magnitude and direction.
A transformation is a function mapping points of the plane to points of the plane. The four canonical ones — reflection, translation, rotation, dilation — generate, by composition, every motion that interests classical geometry. The Erlangen program of Felix Klein (1872) recast geometry itself as the study of properties invariant under particular groups of transformations: rigid motions for Euclidean geometry, projective transformations for projective geometry, and so on.
Reflections
A reflection across a line $\ell$ maps each point $P$ to the point $P'$ such that $\ell$ is the perpendicular bisector of $\overline{PP'}$. Reflections preserve distance (isometries) but reverse orientation.
Across $x$-axis: $(x, y) \to (x, -y)$. Across $y$-axis: $(x, y) \to (-x, y)$. Across $y = x$: $(x, y) \to (y, x)$. Across origin (point reflection): $(x, y) \to (-x, -y)$.
Example 1
Reflecting across an axis.
Reflect $A(3, -2)$ across the $x$-axis.
$A' = (3, 2)$.
Problem Set 9.1
Reflect $(-4, 5)$ across the $y$-axis.
Reflect $(2, 7)$ across $y = x$.
Reflect $(-3, -6)$ across origin.
The image of $(a, b)$ across $x$-axis equals the pre-image. What is $b$?
Reflect triangle $(0,0), (4,0), (4,3)$ across the $y$-axis.
Translations
A translation shifts every point by a fixed vector $\langle a, b \rangle$: $(x, y) \to (x + a, y + b)$. Translations are isometries that preserve orientation.
Example 2
Translating a triangle.
Translate $\triangle ABC$ with vertices $(1, 2), (4, 2), (4, 5)$ by $\langle -3, 1 \rangle$.
$A'(-2, 3), B'(1, 3), C'(1, 6)$.
Problem Set 9.2
Translate $(5, -2)$ by $\langle -4, 7 \rangle$.
Find the translation vector if $(2, 3) \to (5, -1)$.
Translate $\triangle(0,0), (3,0), (0,4)$ by $\langle 2, 2 \rangle$.
Is a translation an isometry? Justify.
What translation undoes $\langle 6, -3 \rangle$?
Rotations
A rotation about a center point $O$ by angle $\theta$ (counter-clockwise positive) maps each point $P$ to $P'$ such that $OP = OP'$ and $m\angle POP' = \theta$.
Rotate $(3, 4)$ by $90°$ counter-clockwise about origin.
$(3, 4) \to (-4, 3)$.
Problem Set 9.3
Rotate $(2, -5)$ by $180°$.
Rotate $(-1, 3)$ by $90°$ CCW.
Rotate $(4, 2)$ by $270°$ CCW.
Two consecutive $90°$ rotations yield what total rotation?
Is rotation an isometry? An orientation-preserving one?
Compositions of Transformations
A composition applies one transformation, then another. Order matters in general: a reflection followed by a translation need not equal the translation followed by the reflection.
The composition of two reflections across parallel lines is a translation (perpendicular to those lines, of magnitude twice the distance between them). The composition of two reflections across intersecting lines is a rotation about the point of intersection (by twice the angle between the lines).
Example 4
Composition.
Reflect $(2, 3)$ across the $x$-axis, then translate by $\langle 4, -1 \rangle$.
$(2, 3) \to (2, -3) \to (6, -4)$.
Problem Set 9.4
Reflect $(1, 5)$ across $y$-axis, then translate $\langle 3, 2 \rangle$.
Translate $(1, 5)$ by $\langle 3, 2 \rangle$, then reflect across $y$-axis. Compare.
Rotate $(2, 0)$ by $90°$, then reflect across $x$-axis.
What single transformation is equivalent to reflection across $x$-axis followed by reflection across $y$-axis?
Two reflections across the same line equal what?
Symmetry
A figure has line symmetry (or reflectional symmetry) if reflection across some line maps the figure onto itself. It has rotational symmetry of order $n$ if rotation by $360°/n$ maps it onto itself.
Example 5
Symmetries.
A regular hexagon has how many lines of symmetry? Rotational symmetry of what order?
$6$ lines of symmetry; rotational symmetry of order $6$ (smallest rotation $60°$).
Problem Set 9.5
Lines of symmetry of an equilateral triangle? Rotational order?
Lines of symmetry of a square? Order?
A regular $n$-gon has how many lines of symmetry? Rotational order?
Letter "H" — symmetries?
Snowflake symmetries: typically what order?
Dilations
A dilation centered at $O$ with scale factor $k$ maps each point $P$ to $P'$ on ray $\overrightarrow{OP}$ with $OP' = |k| \cdot OP$ (and on the opposite ray if $k < 0$).
Dilations preserve angles and ratios but not distances. They are similarity transformations but not isometries.
Example 6
Dilation about origin.
Dilate $\triangle(1, 1), (4, 1), (4, 3)$ by factor $2$ about origin.
$(2, 2), (8, 2), (8, 6)$.
Problem Set 9.6
Dilate $(3, -2)$ by factor $\tfrac{1}{2}$ about origin.
Dilate $(6, 0)$ by factor $-1$. (Equivalent to what?)
The circle is geometry's tidiest object — every point equidistant from a centre — and yet it underwrites the deepest results: $\pi$ itself, the inscribed-angle theorem, the power of a point. The Greeks built much of their cosmology on it, and we still measure angles in fractions of a circle's full revolution.
Circles and Circumference
A circle is the locus of points in a plane equidistant from a fixed point (the centre). The common distance is the radius $r$; twice the radius is the diameter $d$.
$C = 2\pi r = \pi d$. The ratio $C/d = \pi$ is the same for every circle — a fact so unobvious that proving it rigorously took the better part of two millennia.
Example 1
Circumference from radius.
Find $C$ for a circle with $r = 7$ cm. Leave the answer in terms of $\pi$.
$C = 2\pi(7) = 14\pi$ cm.
Problem Set 10.1
Find $C$ if $r = 10$.
Find $r$ if $C = 30\pi$.
Find $d$ if $C = 18\pi$.
A bicycle wheel has diameter $26$ in. How far does it travel in one revolution?
If $C = 50$ cm exactly, find $r$ to two decimal places.
Angles and Arcs
A central angle has its vertex at the centre of the circle. Its sides cut off two arcs: a minor arc (less than a semicircle) and a major arc. The measure of the minor arc equals the measure of the central angle.
For a central angle of $\theta$ degrees in a circle of radius $r$: $\text{arc length} = \dfrac{\theta}{360} \cdot 2\pi r$.
Example 2
Arc length.
Find the length of an arc subtended by a $60°$ central angle in a circle of radius $12$.
Find the arc length for a $90°$ central angle, $r = 8$.
Find the arc length for a $120°$ central angle, $r = 9$.
A central angle of $45°$ subtends an arc of length $\pi$. Find $r$.
The measures of two arcs of a circle sum to what?
Convert: a $270°$ arc is what fraction of the circle?
Arcs and Chords
A chord is a segment whose endpoints lie on the circle. A diameter is a chord through the centre.
(i) In the same circle, congruent chords have congruent arcs (and vice versa). (ii) A diameter perpendicular to a chord bisects the chord and its arc. (iii) The perpendicular bisector of a chord passes through the centre.
Example 3
Finding chord length via perpendicular from centre.
A chord lies $5$ cm from the centre of a circle of radius $13$ cm. Find the chord's length.
The perpendicular from centre to chord bisects the chord. Half-chord $= \sqrt{13^2 - 5^2} = \sqrt{144} = 12$. Full chord $= 24$ cm.
Problem Set 10.3
A chord of length $16$ lies in a circle of radius $10$. Find the distance from chord to centre.
A chord lies $6$ cm from the centre of a circle of radius $10$. Find the chord's length.
Two chords in the same circle have equal length. What can you say about their distances from centre?
True or false: every diameter is a chord.
If a chord bisects another chord, must one of them be a diameter? Justify.
Inscribed Angles
An inscribed angle has its vertex on the circle and its sides as chords.
The measure of an inscribed angle is half the measure of its intercepted arc. Corollaries: (i) inscribed angles intercepting the same arc are congruent; (ii) an angle inscribed in a semicircle is a right angle (Thales).
Thales of Miletus (c. 624–546 BCE) is credited with the first abstract geometric theorem in recorded history — that any angle inscribed in a semicircle is right. He is also said to have predicted a solar eclipse and to have fallen into a well while contemplating the stars, which Plato found amusing.
Example 4
Inscribed angle.
An inscribed angle intercepts an arc of $80°$. Find the angle.
$\frac{1}{2}(80°) = 40°$.
Problem Set 10.4
Find the inscribed angle intercepting an arc of $130°$.
An inscribed angle measures $35°$. Find the arc.
An angle inscribed in a semicircle measures what?
If two inscribed angles intercept the same arc, what can you conclude?
A quadrilateral inscribed in a circle has opposite angles. Show that they sum to $180°$.
Tangents
A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point — the point of tangency.
A tangent line is perpendicular to the radius drawn to the point of tangency. Conversely, a line perpendicular to a radius at its endpoint on the circle is tangent to the circle.
If two tangent segments are drawn to a circle from the same external point, they are congruent.
Example 5
Tangent length from external point.
From a point $P$ outside a circle, a tangent of length $\ell$ is drawn to the circle of radius $r$. If $OP = 13$ and $r = 5$, find $\ell$.
Triangle $OTP$ is right-angled at $T$ (point of tangency). $\ell = \sqrt{13^2 - 5^2} = 12$.
Problem Set 10.5
Tangent length $\ell = ?$ if $r = 6$ and $OP = 10$.
From external point $P$, two tangents touch the circle at $A$ and $B$. If $PA = 8$, find $PB$.
True or false: a tangent and a radius drawn to the point of tangency are parallel.
How many tangent lines can be drawn to a circle from a point on the circle?
How many from an external point?
Secants and Tangents: Angles
A secant is a line that intersects a circle in two points. Two secants, two tangents, or a secant and a tangent meeting outside the circle form an angle whose measure depends on the intercepted arcs.
If two secants meet outside the circle, the angle they form equals half the positive difference of the intercepted arcs: $m\angle = \frac{1}{2}|arc_{\text{far}} - arc_{\text{near}}|$.
If two chords intersect inside the circle, the angle formed equals half the sum of the intercepted arcs.
Example 6
Angle from two secants meeting outside.
Two secants from an external point intercept arcs of $100°$ and $40°$. Find the angle.
$m\angle = \tfrac{1}{2}(100 - 40) = 30°$.
Problem Set 10.6
Two chords intersect inside a circle, intercepting arcs of $70°$ and $30°$. Find the angle.
Two secants from outside intercept arcs of $150°$ and $50°$. Find the angle.
A tangent and a secant from an external point intercept arcs of $120°$ and $40°$. Find the angle.
An inscribed angle is a special case of which theorem (chord meets chord at the circle)?
If two secants make a $25°$ angle and intercept a far arc of $90°$, find the near arc.
Special Segments in a Circle
If two chords intersect inside a circle at point $P$, dividing the chords into segments of lengths $a, b$ and $c, d$, then $ab = cd$.
From external point $P$, two secants reach the circle. For each secant, let $a$ be the distance from $P$ to the near intersection and $b$ to the far. Then $a_1 b_1 = a_2 b_2$, where $a_i b_i$ means $a_i \cdot b_i$ (whole secant times external segment, in either of two equivalent formulations).
If a tangent of length $t$ and a secant with external part $a$ and full length $b$ are drawn from an external point, $t^2 = a \cdot b$.
Example 7
Two chords intersect.
Two chords intersect inside a circle. One chord is divided into parts $3$ and $8$; the other into parts $4$ and $x$. Find $x$.
$3 \cdot 8 = 4 \cdot x \Rightarrow x = 6$.
Problem Set 10.7
Two chords meet inside. Parts $5, 6$ and $3, x$. Find $x$.
Tangent length $t = 6$ from external point; secant from same point has external part $4$. Find the whole secant length.
Two secants from outside: secant 1 has external $a = 3$, whole $b = 12$; secant 2 has external $4$. Find the whole secant 2.
State the power of a point as a single equation valid for chord, secant, and tangent cases.
If a tangent and secant from external $P$ give $t = 8$ and external $a = 4$, find the whole secant.
Equations of Circles
A circle with centre $(h, k)$ and radius $r$ has equation $(x - h)^2 + (y - k)^2 = r^2$.
Example 8
From general to standard form.
Write $x^2 + y^2 - 6x + 4y - 12 = 0$ in standard form.
Group and complete the square: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4 = 25$. So $(x - 3)^2 + (y + 2)^2 = 25$: centre $(3, -2)$, radius $5$.
Problem Set 10.8
Write the equation of a circle with centre $(2, -5)$ and radius $4$.
Find centre and radius of $(x + 1)^2 + (y - 3)^2 = 49$.
Convert $x^2 + y^2 + 8x - 2y + 8 = 0$ to standard form.
Does $(5, 1)$ lie on the circle $(x - 2)^2 + (y + 3)^2 = 25$?
Find the equation of the circle with diameter endpoints $(0, 0)$ and $(6, 8)$.
If perimeter measures the boundary, area measures the interior — and the move from one dimension to two is where geometry first betrays its scale-sensitivity. Doubling the side of a square does not double its area; it quadruples it. The squared-versus-linear distinction underpins everything from biological allometry to gravitational flux.
Parallelograms and Triangles
Parallelogram: $A = bh$, where $b$ is the base and $h$ the perpendicular height. Triangle: $A = \tfrac{1}{2} bh$. The triangle is half of a parallelogram built on the same base and height — the easiest dissection proof in the curriculum.
Example 1
Triangle area.
A triangle has base $10$ and height $7$. Find its area.
$A = \tfrac{1}{2}(10)(7) = 35$ square units.
Problem Set 11.1
Parallelogram: $b = 12$, $h = 5$. Find $A$.
Triangle: $b = 9$, $h = 8$. Find $A$.
A triangle has area $48$ and base $12$. Find its height.
A parallelogram has sides $6$ and $10$ and an included angle of $30°$. Find its area. (Hint: $h = 10 \sin 30°$.)
True or false: two parallelograms with the same base and same height have equal area.
$A = \tfrac{1}{2}(b_1 + b_2)h$. The two parallel sides are the bases; $h$ is the perpendicular distance between them.
For both rhombus and kite: $A = \tfrac{1}{2} d_1 d_2$, where $d_1, d_2$ are the diagonals. (For a rhombus the diagonals bisect each other at right angles; for a kite at least one diagonal is bisected by the other.)
If two figures are similar with linear scale factor $k$, their areas are in ratio $k^2$. (Volumes, when applicable, scale by $k^3$ — see §12.8.)
This is why a kitten cannot simply be scaled up into a tiger: the mass (volume) grows as the cube of length, but bone cross-section (area) grows only as the square. Eventually the legs cannot support the body. Galileo worked this out in 1638, and it still governs why ants can carry many times their weight while elephants cannot.
Example 5
Area ratio.
Two similar triangles have corresponding sides in ratio $2:5$. Find the ratio of their areas.
$\left(\dfrac{2}{5}\right)^2 = \dfrac{4}{25}$.
Problem Set 11.5
Similar pentagons have sides in ratio $3:4$. Area ratio?
Similar circles have radii $5$ and $8$. Area ratio?
The areas of two similar figures are $9$ and $25$. Find the ratio of their corresponding sides.
If the side of a square is tripled, by what factor does the area grow?
Two similar triangles have areas $18$ and $50$. The smaller has perimeter $12$. Find the larger triangle's perimeter.
Three dimensions complicate matters in ways students rarely anticipate. Surface area scales with the square of linear measure; volume with the cube. This is the lever by which physicists explain why mice scurry and elephants lumber, why ice cubes shaped like spheres melt slower than cubes (lower surface-to-volume ratio), and why no land animal larger than an elephant exists.
Representations of Three-Dimensional Figures
A polyhedron is a solid bounded by polygonal faces. A prism has two parallel congruent bases; a pyramid has one polygonal base and triangular faces meeting at an apex. A cylinder is the curved analogue of a prism; a cone, of a pyramid; a sphere is the locus of points in space equidistant from a centre.
For any convex polyhedron with $V$ vertices, $E$ edges, $F$ faces: $V - E + F = 2$.
Euler discovered this in 1750 but did not prove it rigorously; the first complete proof came from Cauchy in 1813. The number $2$ on the right is a topological invariant of the sphere — for a torus (a doughnut), the corresponding number is $0$. Topology was waiting in the wings.
Prism: $V = Bh$. Cylinder: $V = \pi r^2 h$. Cavalieri's principle: two solids of equal height whose cross-sections at every level have equal area have equal volume.
Pyramid: $V = \tfrac{1}{3} Bh$. Cone: $V = \tfrac{1}{3} \pi r^2 h$. The factor of $\tfrac{1}{3}$ is not coincidence: Cavalieri's principle and an integration argument both give the same result.
Example 5
Cone volume.
$r = 6$, $h = 9$. Find $V$.
$V = \tfrac{1}{3}\pi(36)(9) = 108\pi$.
Problem Set 12.5
Square pyramid: base $4 \times 4$, height $9$. Find $V$.
$SA = 4\pi r^2$. $V = \tfrac{4}{3} \pi r^3$. Archimedes proved both, and is said to have requested a sphere inscribed in a cylinder carved on his tombstone — the ratio of their volumes ($2:3$) being his favourite result.
Example 6
Sphere volume and surface.
$r = 3$. Find $SA$ and $V$.
$SA = 4\pi(9) = 36\pi$. $V = \tfrac{4}{3}\pi(27) = 36\pi$. (A coincidence that holds only for $r = 3$.)
Problem Set 12.6
Sphere: $r = 6$. Find $SA$ and $V$.
Sphere: $V = \tfrac{32}{3}\pi$. Find $r$.
Sphere: $SA = 100\pi$. Find $r$.
Earth's radius $\approx 6{,}370$ km. Surface area (in $\pi$)?
If a sphere's radius doubles, by what factor does its volume grow?
On the surface of a sphere, a great circle is the intersection of the sphere with a plane through its centre. Great circles are the spherical analogues of straight lines.
A triangle drawn with arcs of great circles has angle sum strictly greater than $180°$. This is the first hint that Euclid's parallel postulate is not universally true — spherical geometry is a non-Euclidean geometry in which through a point outside a "line" (great circle), no parallel line exists.
Spherical geometry is the working tool of navigators, astronomers, and crystallographers. Aviation routes appear curved on a Mercator projection precisely because the shortest path between two points on a sphere is an arc of a great circle, not the straight line of flat-paper intuition.
Problem Set 12.7
True or false: on a sphere, the shortest distance between two points is along a great circle.
How many great circles can be drawn through two non-antipodal points?
What is the angle sum of a triangle whose three vertices are the North Pole, $0°$ longitude on the equator, and $90°$ E longitude on the equator?
Are lines of latitude (other than the equator) great circles?
Name two parallels of Euclidean geometry that fail on a sphere.
Congruent solids have all corresponding measures equal. Similar solids have corresponding linear measures in ratio $k$. Then surface areas are in ratio $k^2$ and volumes in ratio $k^3$.
Example 8
Scaling solids.
Two similar cylinders have radii in ratio $1:3$. Find the ratio of (a) surface areas, (b) volumes.
(a) $1:9$. (b) $1:27$.
Problem Set 12.8
Two similar cubes have sides $2$ and $5$. Ratio of volumes?
Two similar cones have surface areas in ratio $4:25$. Ratio of radii? Of volumes?
If a sphere's radius is tripled, by what factor does volume change?
Two similar prisms have volumes $27$ and $125$. Ratio of corresponding edges?
A scale model of a building uses ratio $1:50$. If model volume is $0.4$ m³, find actual volume.
Probability is the geometry of uncertainty — areas of possibility carved out of a total sample space. It joins counting (combinatorics) to geometric measure, and underwrites everything from gambling to gene expression to weather forecasting. The Geometry textbook treatment is necessarily brief, but the principles below scale all the way up.
Representing Sample Spaces
A sample space is the set of all possible outcomes of an experiment. An event is a subset of the sample space. The probability of an event $E$ in an equally-likely sample space $S$ is $P(E) = \dfrac{|E|}{|S|}$.
Example 1
Sample space.
Flip two coins. List the sample space and find $P(\text{exactly one head})$.
If one event can occur in $m$ ways and an independent second event in $n$ ways, the two together occur in $mn$ ways.
A permutation is an ordered arrangement: $P(n, r) = \dfrac{n!}{(n - r)!}$. A combination is an unordered selection: $C(n, r) = \dbinom{n}{r} = \dfrac{n!}{r!(n - r)!}$.
Example 2
Combination.
From $10$ books, how many ways to choose $3$?
$\binom{10}{3} = \dfrac{10!}{3!\,7!} = 120$.
Problem Set 13.2
How many ways to arrange the letters of WORD?
$P(8, 3) = ?$
$C(7, 4) = ?$
From $5$ men and $4$ women, choose a committee of $3$ men and $2$ women. How many committees?
Pizza menu: $4$ crusts, $3$ sauces, $8$ toppings (choose one of each). How many combinations?
A simulation uses random trials to estimate a probability that may be hard to compute analytically. Each trial is a model of one performance of the experiment.
Example 4
Coin-flip simulation.
To estimate $P(\text{exactly 2 heads in 4 flips})$, generate $1{,}000$ groups of $4$ random bits and count those with exactly two $1$s. The proportion approximates the true probability $\binom{4}{2}/16 = 6/16 = 0.375$.
Problem Set 13.4
Describe a simulation using a die to estimate $P(\text{roll a 6})$.
Why might a simulation be preferred to analytic calculation?
If a simulation of $1{,}000$ trials gives a relative frequency of $0.38$ for an event whose true $P = 0.4$, is this reasonable?
Design a simulation using coins to model a $50\%$-shooter's $5$ free throws.
True or false: a simulation always converges to the true probability.
If $A$ and $B$ are independent (occurrence of one does not affect the other), $P(A \text{ and } B) = P(A) \cdot P(B)$. If dependent, $P(A \text{ and } B) = P(A) \cdot P(B \mid A)$, where $P(B \mid A)$ is the conditional probability of $B$ given $A$.
Example 5
With and without replacement.
A bag has $3$ red and $7$ blue marbles. Draw two. $P(\text{both red})$ (a) with replacement, (b) without?
If $A$ and $B$ are mutually exclusive (cannot both happen), $P(A \text{ or } B) = P(A) + P(B)$. Otherwise, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.
The general formula is a special case of the inclusion–exclusion principle, the combinatorial counterpart to integrating across overlapping regions. It generalises to three or more sets — and is precisely what stops a careless student from double-counting the kings that are also hearts.
Example 6
Card draw.
Draw one card. $P(\text{king or heart})$?
Not mutually exclusive (the king of hearts is both). $P = \tfrac{4}{52} + \tfrac{13}{52} - \tfrac{1}{52} = \tfrac{16}{52} = \tfrac{4}{13}$.
Problem Set 13.6
Roll a die. $P(\text{even or } 5)$?
Draw one card. $P(\text{ace or face card})$?
$P(A) = 0.3$, $P(B) = 0.4$, mutually exclusive. $P(A \text{ or } B) = ?$
$P(A) = 0.5$, $P(B) = 0.6$, $P(A \text{ and } B) = 0.2$. $P(A \text{ or } B) = ?$
If two events are mutually exclusive, can they be independent? (Trick — think carefully.)