Geometry — MGU Mathematics Series
Part IV: Polygons, Circles, Area, and Volume
Polygons · Quadrilaterals · Angle Sums · Coordinate Proofs
Chapter 9 of 14
MGU Mathematics Series  ·  Geometry  ·  Library Textbook Edition

Chapter 9:
Quadrilaterals
and Polygons

Polygon angle sums, parallelograms, rectangles, rhombi, squares, trapezoids, kites, classification, and coordinate proofs — the geometry of flat figures organized by property and proven by reasoning

■ Math Level: Geometry ■

■ 14 Sections  •  Part IV of 5 ■

Follows Chapters 1–8 · Prepares students for Area, Volume, Coordinate Geometry, and Transformations

Chapter 9 — Front Matter

Purpose and Overview

Geometry is built from simple figures combined into more complex ones. Triangles are the most fundamental polygon, and you have studied them in depth. But the world of flat, closed figures extends far beyond triangles. Squares, rectangles, parallelograms, trapezoids, hexagons, octagons — these shapes appear everywhere in architecture, engineering, art, construction, design, and nature.

This chapter studies polygons: closed figures formed by straight segments joined at their endpoints. The chapter gives special attention to quadrilaterals — four-sided polygons — because they are the most varied and practically important family of polygons. You will learn how to measure their angles, identify their properties, classify their types, and prove their relationships using coordinates.

The chapter uses both geometric reasoning and algebraic calculation. Properties will be stated, explained, and proven. Unknown angle measures, side lengths, and diagonal lengths will be found using equations. Coordinate geometry will allow you to classify figures from their vertices alone. This prepares you for the area formulas in Chapter 11, the coordinate proof techniques in Chapter 13, and the modeling work in Chapter 14.

Opening Question

A contractor is designing a floor plan. She draws a four-sided room and labels the corners. Looking at her measurements, she notices that opposite sides are equal in length and that all angles appear to be right angles. Is this room a square, a rectangle, a parallelogram, or something else? How would she know for certain? What measurements or calculations would confirm the shape? By the end of this chapter, you will be able to answer this question precisely and prove your answer.

14Sections
7Objectives
20Key Terms
15+Worked Examples

Learning Objectives

  1. 1Define polygon, convex polygon, and regular polygon, and give examples of each.
  2. 2Calculate interior and exterior angle sums for any convex polygon.
  3. 3Identify and apply the properties of parallelograms, rectangles, rhombi, squares, trapezoids, and kites.
  4. 4Classify quadrilaterals correctly within the quadrilateral hierarchy.
  5. 5Use coordinate geometry to classify and prove properties of quadrilaterals.
  6. 6Recognize and explain common mistakes in polygon and quadrilateral reasoning.
  7. 7Apply polygon properties to problems in design, engineering, and measurement.
Chapter 9 — Contents

Sections at a Glance

Sections 9.1 – 9.4
Polygons and Angle Sums
What Polygons Are · Interior Angle Sums · Exterior Angle Sums · Regular Polygons
Section 9.1

What Polygons Are

A polygon is a closed plane figure formed by connecting three or more line segments end to end, with no segment crossing another. Each segment is a side, and each point where two sides meet is a vertex (plural: vertices). Polygons are named by listing their vertices in order — clockwise or counterclockwise — around the figure.

The simplest polygon is a triangle, which has three sides and three vertices. Polygons are named by the number of sides they have.

SidesNameSidesName
3Triangle8Octagon
4Quadrilateral9Nonagon
5Pentagon10Decagon
6Hexagon12Dodecagon
7Heptagonnn-gon

Convex and Concave Polygons

A polygon is convex if every interior angle measures less than 180° and every diagonal lies entirely inside the figure. A polygon is concave if at least one interior angle exceeds 180° or if a diagonal falls outside the figure. The quadrilateral families studied in this chapter — parallelograms, rectangles, rhombi, squares, trapezoids, and kites — are all convex.

Diagram Instruction

Draw two quadrilaterals side by side. The first: a convex quadrilateral with all angles less than 180° and all diagonals inside the figure. The second: a concave (arrowhead-shaped) quadrilateral where one vertex points inward. Label the inward vertex with a reflex-angle mark (greater than 180°). Label the first "Convex" and the second "Concave."

Student Warning

A polygon must be closed (the segments form a complete loop with no gaps) and simple (the sides do not cross each other). A figure whose sides intersect is called a self-intersecting polygon and is not a standard polygon studied in this course.

Practice 9.1Concept Check
  1. Name the polygon with each number of sides. Then write the number of diagonals you can draw from a single vertex: (a) 5 sides, (b) 8 sides, (c) 10 sides, (d) 12 sides.
  2. A polygon is labeled PQRST. Name all five sides and all five diagonals using correct segment notation.
  3. Explain in your own words the difference between a convex and a concave polygon. Can a triangle be concave? Explain why or why not.
Section 9.2

Interior Angle Sums

Every polygon can be divided into triangles by drawing diagonals from a single vertex. A polygon with n sides can be divided into (n − 2) triangles this way. Since each triangle has an interior angle sum of 180°, the interior angle sum of any convex n-gon is:

Interior Angle Sum Formula

S = (n − 2) · 180°

Where n is the number of sides of the polygon. For a regular polygon, each interior angle equals (n − 2) · 180° ÷ n.

Diagram Instruction

Draw a convex pentagon ABCDE. From vertex A, draw diagonals to C and to D. This divides the pentagon into three triangles: △ABC, △ACD, and △ADE. Label the three triangles and write (5 − 2) · 180° = 540° beside the figure to show where the formula comes from.

PolygonSides (n)Interior Angle Sum
Triangle3180°
Quadrilateral4360°
Pentagon5540°
Hexagon6720°
Octagon81080°
Decagon101440°
Example 9.2.1
Finding a missing interior angle
  • ProblemThe interior angles of a quadrilateral measure 90°, 115°, 83°, and x°. Find x.
  • ThinkThe interior angle sum of a quadrilateral is (4 − 2) · 180° = 360°. Add the three known angles and subtract from 360.
  • Solve90 + 115 + 83 + x = 360 288 + x = 360 x = 72
  • Check90 + 115 + 83 + 72 = 360 ✓
Answer: The missing angle measures 72°.
Example 9.2.2
Finding each angle of a regular polygon
  • ProblemA regular hexagon has six congruent interior angles. Find the measure of each angle.
  • SolveS = (6 − 2) · 180° = 4 · 180° = 720° Each angle = 720° ÷ 6 = 120°
  • Check6 × 120° = 720° ✓
Answer: Each interior angle of a regular hexagon measures 120°.
Practice 9.2Interior Angle Sums
  1. Heptagon (7 sides)
  2. Nonagon (9 sides)
  3. Dodecagon (12 sides)
  4. 15-gon
  5. Each angle of a regular pentagon
  6. Each angle of a regular octagon
  7. Quadrilateral with angles 82°, 104°, 91°, and x°
  8. Pentagon with angles 100°, 112°, 95°, 88°, and x°
Section 9.3

Exterior Angle Sums

At each vertex of a polygon, you can extend one side outward past the vertex. The angle formed between this extended side and the adjacent side of the polygon is the exterior angle at that vertex. Each interior angle and its exterior angle form a linear pair, so they are supplementary and sum to 180°.

Exterior Angle Sum Theorem

The sum of the exterior angles of any convex polygon — one at each vertex — is always 360°, regardless of the number of sides.

For a regular n-gon: each exterior angle = 360° ÷ n.

Interior vs. Exterior Angle Sums

The interior angle sum depends on n: it gets larger as n increases. The exterior angle sum is always 360°, for every convex polygon, no matter how many sides it has. These are two separate theorems; do not apply one formula to both situations.

Example 9.3.1
Finding the number of sides from an exterior angle
  • ProblemEach exterior angle of a regular polygon measures 24°. How many sides does the polygon have?
  • ThinkThe exterior angle sum is 360°. If each exterior angle equals 24°, divide 360° by 24°.
  • Solven = 360° ÷ 24° = 15
  • Check15 × 24° = 360° ✓
Answer: The polygon has 15 sides.
Practice 9.3Exterior Angle Sums
  1. Each exterior angle of a regular hexagon
  2. Each exterior angle of a regular nonagon
  3. Sides if each exterior angle = 40°
  4. Sides if each exterior angle = 30°
  5. Sides if each exterior angle = 20°
  6. Sides if each exterior angle = 45°
Section 9.4

Regular Polygons

A polygon is regular if it is both equilateral (all sides equal) and equiangular (all interior angles equal). A square is a regular quadrilateral; an equilateral triangle is the regular triangle. Regular polygons have a high degree of symmetry and appear constantly in design, architecture, and nature.

PolygonSides (n)Interior SumEach Interior ∠Each Exterior ∠
Equilateral triangle3180°60°120°
Square4360°90°90°
Regular pentagon5540°108°72°
Regular hexagon6720°120°60°
Regular octagon81080°135°45°
Regular decagon101440°144°36°
Real-World Connection

Regular polygons appear throughout design and engineering. Honeycomb cells are regular hexagons because the hexagonal shape tiles the plane perfectly and uses material efficiently. Stop signs are regular octagons. Most nuts and bolt heads are regular hexagons. Floor tile patterns often use equilateral triangles, squares, or regular hexagons — the only three regular polygons that tile the plane by themselves.

MGU Library Connection
  • MGU Geometry Reference: Polygon Angle Table — Appendix H of this textbook
  • MGU Architecture and Design Lesson: Tessellations — applications of regular polygons in construction
  • MGU Engineering Lesson: Why Triangles Are Rigid and Hexagons Are Efficient
Sections 9.5 – 9.11
Quadrilateral Families and Classification
Parallelograms · Rectangles · Rhombi · Squares · Trapezoids · Kites · Hierarchy
Section 9.5

Parallelograms

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. In parallelogram ABCD, we have AB ∥ CD and BC ∥ DA. The notation for a parallelogram uses the symbol ▱, so parallelogram ABCD may be written as ▱ABCD.

Diagram Instruction

Draw parallelogram ABCD with vertices labeled clockwise. Mark AB and CD with single arrow marks (parallel). Mark BC and DA with double arrow marks (parallel). Mark opposite sides AB = CD and BC = DA with corresponding tick marks.

Properties of Parallelograms

Property 1 — Opposite sides congruent: AB = CD and BC = DA

Property 2 — Opposite angles congruent: m∠A = m∠C and m∠B = m∠D

Property 3 — Consecutive angles supplementary: m∠A + m∠B = 180°

Property 4 — Diagonals bisect each other: If diagonals AC and BD intersect at E, then AE = EC and BE = ED

Example 9.5.1
Finding angle measures in a parallelogram
  • ProblemIn parallelogram PQRS, m∠P = 67°. Find m∠Q, m∠R, and m∠S.
  • ThinkOpposite angles are equal; consecutive angles are supplementary.
  • Solvem∠R = m∠P = 67° (opposite angles) m∠Q = 180° − 67° = 113° (consecutive angles, supplementary) m∠S = m∠Q = 113° (opposite angles)
  • Check67° + 113° + 67° + 113° = 360° ✓
Answer: m∠Q = 113°, m∠R = 67°, m∠S = 113°.
Example 9.5.2
Using the diagonal bisect property algebraically
  • ProblemDiagonals of ▱ABCD intersect at E. If AE = 3x − 1 and EC = x + 7, find AC.
  • ThinkThe diagonals of a parallelogram bisect each other, so AE = EC.
  • Solve3x − 1 = x + 7 2x = 8 x = 4 AE = 3(4) − 1 = 11 EC = 4 + 7 = 11 AC = 11 + 11 = 22
  • CheckBoth segments equal 11; diagonals bisect each other. ✓
Answer: AC = 22.

Proving a Quadrilateral is a Parallelogram

Any of the following conditions, if proven, guarantees that a quadrilateral is a parallelogram: (1) both pairs of opposite sides are parallel; (2) both pairs of opposite sides are congruent; (3) both pairs of opposite angles are congruent; (4) one pair of opposite sides is both parallel and congruent; (5) the diagonals bisect each other.

Section 9.6

Rectangles

A rectangle is a parallelogram with four right angles. Because it is a parallelogram, a rectangle inherits all four parallelogram properties. The right angles add one additional property involving diagonals: the diagonals of a rectangle are congruent. In rectangle ABCD, if AC and BD are the diagonals, then AC = BD. In a general parallelogram, the diagonals bisect each other but are not necessarily equal; in a rectangle they bisect each other AND are equal.

Diagram Instruction

Draw rectangle ABCD with right-angle marks at all four corners. Draw both diagonals AC and BD. Mark the diagonals with equal (single) tick marks to show they are congruent. Mark the intersection point E and show the four equal-length segments from E with additional tick marks.

Example 9.6.1
Finding a diagonal length in a rectangle using the Pythagorean Theorem
  • ProblemRectangle WXYZ has WX = 9 and XY = 12. Find the length of diagonal WY.
  • ThinkEach diagonal is the hypotenuse of a right triangle formed with two sides of the rectangle.
  • DiagramDraw rectangle WXYZ. Draw diagonal WY, forming right △WXY with the right angle at X.
  • SolveWY² = WX² + XY² WY² = 9² + 12² = 81 + 144 = 225 WY = √225 = 15
  • Check9² + 12² = 81 + 144 = 225 = 15² ✓
Answer: WY = 15.
Section 9.7

Rhombi

A rhombus (plural: rhombi) is a parallelogram with four congruent sides. Because it is a parallelogram, a rhombus inherits all four parallelogram properties. The equal sides add two additional diagonal properties.

Additional Properties of a Rhombus

Property 1: The diagonals of a rhombus are perpendicular to each other.

Property 2: Each diagonal bisects a pair of opposite vertex angles. Diagonal AC bisects ∠A and ∠C; diagonal BD bisects ∠B and ∠D.

Diagram Instruction

Draw rhombus ABCD with all four sides marked with single tick marks. Draw both diagonals and mark a right-angle symbol at their intersection point E. Use arc marks at ∠A to show both halves of the bisected angle are equal. Do the same at ∠C.

Example 9.7.1
Using perpendicular diagonals to find a side length
  • ProblemIn rhombus ABCD, diagonals intersect at E. If AE = 5 and BE = 12, find side AB.
  • ThinkThe diagonals of a rhombus are perpendicular, so △AEB is a right triangle with the right angle at E.
  • SolveAB² = AE² + BE² AB² = 5² + 12² = 25 + 144 = 169 AB = √169 = 13
  • Check5² + 12² = 169 = 13² ✓
Answer: The side length of the rhombus is AB = 13.
Section 9.8

Squares

A square is a parallelogram that is both a rectangle and a rhombus. It has four right angles and four congruent sides. Because a square is both a rectangle and a rhombus, it inherits all properties of both figures simultaneously. The diagonals of a square bisect each other, are congruent, are perpendicular, and bisect the vertex angles.

Because a square has four right angles and four equal sides, each diagonal forms a 45°-45°-90° triangle with two sides of the square. Therefore, the diagonal of a square with side length s equals s√2.

Square Diagonal Formula

d = s√2

where s is the side length of the square and d is the diagonal length.

Example 9.8.1
Finding the diagonal of a square
  • ProblemA square has side length 8. Find the length of its diagonal.
  • Solved = s√2 = 8√2
  • CheckBy the Pythagorean Theorem: d² = 8² + 8² = 64 + 64 = 128, so d = √128 = √(64 · 2) = 8√2 ✓
Answer: The diagonal of the square is 8√2 ≈ 11.31.
Section 9.9

Trapezoids

A trapezoid is a quadrilateral with exactly one pair of parallel sides. The two parallel sides are called the bases (labeled b₁ and b₂). The two non-parallel sides are called the legs. An isosceles trapezoid is a trapezoid whose legs are congruent. Isosceles trapezoids have two important symmetry properties: the base angles are congruent, and the diagonals are congruent.

Diagram Instruction

Draw trapezoid ABCD with AB as the longer base at the bottom and CD as the shorter base at the top. Mark AB and CD with arrow marks to show they are parallel. Label the legs BC and DA. Label the bases b₁ (longer) and b₂ (shorter). For the isosceles version, mark the legs with equal tick marks.

Trapezoid Midsegment Theorem

The midsegment of a trapezoid connects the midpoints of the two legs. It is parallel to both bases, and its length equals the average of the two base lengths:

Midsegment = (b₁ + b₂) ÷ 2

Example 9.9.1
Finding the midsegment length
  • ProblemA trapezoid has bases of length 14 and 22. Find the length of its midsegment.
  • SolveMidsegment = (14 + 22) ÷ 2 = 36 ÷ 2 = 18
Answer: The midsegment has length 18.
Example 9.9.2
Finding a base from the midsegment
  • ProblemThe midsegment of a trapezoid has length 20. One base has length 27. Find the other base.
  • Solve20 = (27 + b₂) ÷ 2 40 = 27 + b₂ b₂ = 13
  • Check(27 + 13) ÷ 2 = 40 ÷ 2 = 20 ✓
Answer: The other base has length 13.
Example 9.9.3
Angles in an isosceles trapezoid
  • ProblemIsosceles trapezoid ABCD has AB ∥ CD. If m∠A = 62°, find m∠B, m∠C, and m∠D.
  • ThinkBase angles of an isosceles trapezoid are congruent. The bases are parallel, so co-interior angles are supplementary.
  • Solvem∠B = m∠A = 62° (base angles at longer base, congruent) m∠D = 180° − 62° = 118° (co-interior angles with parallel bases) m∠C = m∠D = 118° (base angles at shorter base, congruent)
  • Check62° + 62° + 118° + 118° = 360° ✓
Answer: m∠B = 62°, m∠C = 118°, m∠D = 118°.
Section 9.10

Kites

A kite is a quadrilateral with two pairs of consecutive congruent sides. "Consecutive" means the equal sides share a vertex. In kite ABCD: AB = AD and CB = CD. The two pairs of equal sides are adjacent, not opposite — that is the key distinction from a parallelogram.

Diagram Instruction

Draw kite ABCD with vertex A at the top and C at the bottom, with B and D on the sides. Mark AB and AD with single tick marks. Mark CB and CD with double tick marks. Draw diagonal AC as the vertical axis of symmetry and diagonal BD as the horizontal crossing segment.

Properties of a Kite

Property 1: The diagonals of a kite are perpendicular.

Property 2: Exactly one pair of opposite angles is congruent — the angles between the unequal sides. In kite ABCD: m∠B = m∠D. The angles ∠A and ∠C are generally not equal.

Property 3: The diagonal connecting the vertices shared by equal pairs bisects the other diagonal and bisects both vertex angles at its endpoints.

Example 9.10.1
Angles in a kite
  • ProblemIn kite PQRS, PQ = PS and RQ = RS. If m∠Q = 80° and m∠P = 50°, find m∠R and m∠S.
  • ThinkThe angles between the unequal sides are congruent, so m∠Q = m∠S. All four angles sum to 360°.
  • Solvem∠S = m∠Q = 80° m∠R = 360° − 50° − 80° − 80° = 150°
  • Check50° + 80° + 150° + 80° = 360° ✓
Answer: m∠R = 150°, m∠S = 80°.
Section 9.11

Quadrilateral Classification

Quadrilaterals form a hierarchy. Some quadrilateral types are special cases of others. A square, for example, is simultaneously a rectangle, a rhombus, a parallelogram, and a quadrilateral — it inherits all properties of each. Understanding this hierarchy prevents confusion about which properties apply to which figures.

Quadrilateral— any four-sided polygon
├── Trapezoid— exactly one pair of parallel sides
│ └── Isosceles Trapezoid— legs are congruent
├── Kite— two pairs of consecutive congruent sides
└── Parallelogram— two pairs of parallel sides
├── Rectangle— parallelogram with four right angles
├── Rhombus— parallelogram with four congruent sides
└── Square— rectangle AND rhombus (most specific)
How to Classify — Step by Step

(1) Is it a quadrilateral? (2) Does it have any parallel sides? Two pairs → parallelogram; exactly one pair → trapezoid; none → kite or general. (3) If parallelogram: are angles right? Are all sides equal? (4) If trapezoid: are legs congruent? → isosceles trapezoid. Work from most general to most specific.

Property▱Rect.Rhom.SquareTrap.Kite
Opposite sides parallel✓✓✓✓one pair✗
Opposite sides congruent✓✓✓✓✗✗
All sides congruent✗✗✓✓✗✗
All angles right angles✗✓✗✓✗✗
Diagonals bisect each other✓✓✓✓✗✗
Diagonals congruent✗✓✗✓isosc. only✗
Diagonals perpendicular✗✗✓✓✗✓
Example 9.11.1
Classifying from properties
  • ProblemA quadrilateral has four congruent sides and two pairs of parallel sides, but its angles are not right angles. Classify it as specifically as possible.
  • ThinkFour congruent sides and two pairs of parallel sides → parallelogram with equal sides → rhombus. Angles are not right angles → not a square.
Answer: The figure is a rhombus (but not a square).
Practice 9.5 – 9.11Quadrilateral Property Practice

Parallelograms

  1. ▱ABCD, m∠A = 74°. Find m∠B, m∠C, m∠D.
  2. ▱EFGH, EF = 2x + 3 and GH = x + 9. Find EF.
  3. Diagonals of ▱JKLM meet at P. JP = 4y − 1, PL = 2y + 7. Find JL.
  4. ▱ABCD, m∠B = 3x + 15° and m∠C = x + 45°. Find m∠B.

Rectangles, Rhombi, Squares

  1. Rectangle PQRS, PQ = 5, QR = 12. Find diagonal PR.
  2. Rectangle WXYZ, diagonal = 20, one side = 12. Find other side.
  3. Rhombus MNOP, diagonals meet at Q. MQ = 6, NQ = 8. Find side length.
  4. Rhombus ABCD, m∠A = 70°. Find all other angles.
  5. Square side = 10. Find the diagonal.
  6. Square diagonal = 18. Find the side length.

Trapezoids and Kites

  1. Trapezoid bases 18 and 30. Find midsegment.
  2. Midsegment = 22, one base = 28. Find other base.
  3. Isosceles trapezoid, m∠E = 55°. Find other three angles.
  4. Kite PQRS, PQ = PS, RQ = RS, m∠P = 40°, m∠Q = 100°. Find m∠R and m∠S.
Sections 9.12 – 9.13
Coordinate Proofs and Real-World Applications
Distance · Midpoint · Slope · Classification · Design and Engineering
Section 9.12

Coordinate Proofs with Quadrilaterals

Coordinate geometry gives us a powerful tool for proving properties of quadrilaterals. By placing a quadrilateral on the coordinate plane and calculating slopes, distances, and midpoints, we can determine exactly what type of quadrilateral it is — or prove that specific properties hold — without relying only on angle marks and congruence arguments.

The Three Coordinate Tools

Distance Formula — finds side lengths and diagonal lengths:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Midpoint Formula — verifies that diagonals bisect each other:

M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Slope Formula — determines parallel (equal slopes) or perpendicular (negative reciprocal slopes) lines:

m = (y₂ − y₁) / (x₂ − x₁)

To prove it is a…Check…
ParallelogramBoth pairs of opposite sides have equal slopes (parallel), OR opposite sides have equal lengths, OR diagonals share a midpoint
RectangleParallelogram PLUS two adjacent sides have slopes that are negative reciprocals (perpendicular), OR diagonals are equal in length
RhombusParallelogram PLUS all four sides have equal length
SquareRectangle properties PLUS all four sides have equal length
TrapezoidExactly one pair of opposite sides has equal slopes
Isosceles TrapezoidTrapezoid properties PLUS the two legs have equal length
Example 9.12.1
Proving a parallelogram using midpoints of diagonals
  • ProblemQuadrilateral ABCD has vertices A(1, 2), B(5, 4), C(7, 0), D(3, −2). Prove ABCD is a parallelogram.
  • ThinkIf the diagonals bisect each other, then ABCD is a parallelogram. Find the midpoints of AC and BD and check whether they are the same point.
  • DiagramPlot A(1,2), B(5,4), C(7,0), D(3,−2). Connect in order and draw diagonals AC and BD.
  • SolveMidpoint of AC: M_AC = ((1+7)/2, (2+0)/2) = (4, 1) Midpoint of BD: M_BD = ((5+3)/2, (4+(−2))/2) = (4, 1) Both diagonals share midpoint (4, 1).
  • ConclusionBecause the diagonals of ABCD bisect each other, ABCD is a parallelogram.
Answer: ABCD is a parallelogram. The diagonals share the midpoint (4, 1).
Example 9.12.2
Classifying a quadrilateral completely from its vertices
  • ProblemQuadrilateral JKLM has vertices J(0, 0), K(6, 0), L(8, 4), M(2, 4). Classify JKLM as specifically as possible.
  • Step 1Find slopes of all four sides to identify parallel pairs.Slope of JK: (0−0)/(6−0) = 0 Slope of ML: (4−4)/(8−2) = 0 → JK ∥ ML Slope of JM: (4−0)/(2−0) = 2 Slope of KL: (4−0)/(8−6) = 2 → JM ∥ KLBoth pairs of opposite sides are parallel → parallelogram.
  • Step 2Check side lengths.JK = √(6² + 0²) = 6 ML = √(6² + 0²) = 6 ✓ equal JM = √(2² + 4²) = √20 = 2√5 KL = √(2² + 4²) = √20 = 2√5 ✓ equal But 6 ≠ 2√5, so not all sides equal → not a rhombus.
  • Step 3Check for right angles: slope of JK = 0, slope of JM = 2. These are not negative reciprocals, so ∠J is not 90°. Not a rectangle.
Answer: JKLM is a parallelogram but is not a rectangle, rhombus, or square.
Example 9.12.3
Proving an isosceles trapezoid
  • ProblemQuadrilateral PQRS has vertices P(1, 0), Q(7, 0), R(5, 4), S(3, 4). Prove PQRS is an isosceles trapezoid.
  • Step 1Slopes:Slope PQ = 0, Slope SR = 0 → PQ ∥ SR (one pair) Slope PS = (4−0)/(3−1) = 2 Slope QR = (4−0)/(5−7) = −2 → PS not parallel to QRExactly one pair of parallel sides → trapezoid.
  • Step 2Leg lengths:PS = √(2² + 4²) = √20 = 2√5 QR = √(2² + 4²) = √20 = 2√5 → legs are congruent
  • ConclusionOne pair of parallel sides and congruent legs → isosceles trapezoid.
Answer: PQRS is an isosceles trapezoid.
Practice 9.12Coordinate Proof Practice

Classify each quadrilateral as specifically as possible. Justify using distance, midpoint, and slope.

  1. A(0, 0), B(6, 0), C(6, 4), D(0, 4)
  2. A(1, 1), B(5, 1), C(7, 5), D(3, 5)
  3. A(0, 0), B(4, 0), C(5, 3), D(1, 3)
  4. A(0, 0), B(5, 0), C(4, 3), D(1, 3)
  5. A(−2, 0), B(2, 0), C(3, 4), D(−3, 4)
  6. Given ▱ABCD with A(0,0), B(4,2), C(6,6), D(2,4): Prove it is a parallelogram by showing the diagonals bisect each other.
  7. Given EFGH with E(0,0), F(5,0), G(6,3), H(1,3): Prove or disprove that EFGH is a rectangle.
  8. Given JKLM with J(−3,0), K(0,3), L(3,0), M(0,−3): Prove that JKLM is a rhombus.
Section 9.13

Polygons in Design and Engineering

Polygons are the foundation of nearly every built structure. Floor plans are composed of rectangles and quadrilaterals. Rooflines use triangles and trapezoids. Facades use hexagons, pentagons, and octagons. Bridges use parallelograms in their truss frameworks. The shape of a room is not merely decorative — it determines structural load paths, space efficiency, and material requirements.

The triangle, with its rigid angles under load, is the strongest polygon for frameworks. This is why trusses and steel bridges are built from triangular patterns. The rectangle is preferred for rooms because right angles create stable, stackable structures. The hexagon is preferred for tiling because it fills area with minimum perimeter — the principle behind honeycomb cells.

Regular polygons that tessellate the plane alone are the equilateral triangle (60° angles, six at each vertex), the square (90° angles, four at each vertex), and the regular hexagon (120° angles, three at each vertex). These are the only three. Any other regular polygon leaves gaps or requires combinations of shapes.

MGU Library Connection
  • MGU Architecture Lesson: Structural Polygons in Building Design
  • MGU Engineering Lesson: Truss Frameworks and Triangle Rigidity
  • MGU Art and Design Lesson: Tessellations and Pattern Design
  • Appendix H of this textbook: Quadrilateral Properties Chart
Section 9.14
Common Polygon and Quadrilateral Mistakes
Eight recurring errors — with explanations and corrections
Section 9.14

Common Polygon and Quadrilateral Mistakes

Mistake 1 — Equal Sides Do Not Mean Equal Angles
"Four equal sides means it must be a square."

A rhombus has four equal sides but its angles are generally not right angles. A square requires both equal sides AND right angles. Always check angle properties separately from side-length properties.

Mistake 2 — Reversing the Hierarchy
"All squares are rectangles AND all rectangles are squares."

Every square is a rectangle, but not every rectangle is a square. The hierarchy goes only one direction: a square is the more specific figure (more restrictions). A rectangle with unequal adjacent sides is not a square.

Mistake 3 — Using the Interior Formula for Exterior Angles
Exterior angle sum of a hexagon = (6 − 2) · 180° = 720°

The formula (n − 2) · 180° gives the interior angle sum. The exterior angle sum is always 360°, for any convex polygon, no formula needed.

Mistake 4 — Applying Parallelogram Properties to Opposite Angles as Supplementary
"Opposite angles in a parallelogram are supplementary."

In a parallelogram, opposite angles are congruent. Consecutive angles (sharing a side) are supplementary. Draw the figure and label clearly before applying a property.

Mistake 5 — Assuming Properties from Diagrams
"The diagram looks like a rectangle, so the angles must be 90°."

In geometry, properties must be stated (given) or proven — not assumed from appearance. Do not conclude right angles, equal sides, or parallel sides from how a figure looks unless they are marked or stated.

Mistake 6 — Midsegment = Sum of Bases
Midsegment of a trapezoid with bases 10 and 18 = 10 + 18 = 28

The midsegment equals the average of the two bases: (10 + 18) ÷ 2 = 14. It is always between the two bases in length — never equal to their sum.

Mistake 7 — Applying Parallelogram Theorems to Trapezoids
"Since ABCD is a trapezoid, opposite sides are congruent and diagonals bisect each other."

Trapezoids have exactly one pair of parallel sides. They do not have opposite sides equal (unless isosceles), diagonals that bisect each other, or supplementary consecutive angles at every pair. Apply parallelogram theorems only to parallelograms.

Mistake 8 — Confusing Consecutive and Opposite Sides in the Kite Definition
"A kite has two pairs of equal sides, just like a parallelogram."

A kite has two pairs of consecutive (adjacent) congruent sides. A parallelogram has two pairs of opposite congruent sides. This is a critical difference — the equal sides are in completely different positions relative to the vertices.

Chapter 9 — Back Matter

Chapter Summary

Key Terms Review

polygon
A closed plane figure formed by three or more straight, non-crossing segments joined at their endpoints.
convex polygon
A polygon in which every interior angle is less than 180° and every diagonal lies inside the figure.
regular polygon
A polygon that is both equilateral (all sides equal) and equiangular (all angles equal).
diagonal
A segment connecting two non-adjacent vertices of a polygon.
parallelogram
A quadrilateral with two pairs of parallel sides. Opposite sides and angles are congruent; diagonals bisect each other.
rectangle
A parallelogram with four right angles. Its diagonals are congruent.
rhombus
A parallelogram with four congruent sides. Its diagonals are perpendicular and bisect the vertex angles.
square
A parallelogram with four right angles and four congruent sides — simultaneously a rectangle and a rhombus.
trapezoid
A quadrilateral with exactly one pair of parallel sides (the bases). The non-parallel sides are the legs.
isosceles trapezoid
A trapezoid with congruent legs. Its base angles are congruent and its diagonals are congruent.
midsegment
The segment connecting the midpoints of the legs of a trapezoid. Its length equals the average of the two bases.
kite
A quadrilateral with two pairs of consecutive congruent sides. Its diagonals are perpendicular; one pair of opposite angles is congruent.
consecutive sides
Two sides of a polygon that share a vertex.
opposite sides
Two sides of a quadrilateral that do not share a vertex.
tessellation
A tiling of the plane with one or more polygon shapes, with no gaps and no overlaps. Angles at every vertex must sum to exactly 360°.

Concept Review Questions

  1. What is the difference between a convex and a concave polygon? Give an example of each.
  2. Where does the formula (n − 2) · 180° come from? Explain the reasoning, not just the formula.
  3. Why is the exterior angle sum of every convex polygon 360°, no matter how many sides it has?
  4. What is the difference between a rhombus and a square? What properties do they share, and what property distinguishes them?
  5. A student says, "All squares are rectangles, and all rectangles are squares." Is this correct? Explain.
  6. In what way is a kite different from a parallelogram, even though both can have perpendicular diagonals?
  7. What is the midsegment of a trapezoid, and what theorem describes its length?
  8. Explain why you cannot look at a diagram alone to determine that a figure is a parallelogram. What must you verify?

Angle Sum Practice

Part AInterior Angle Sums

Find the interior angle sum for each polygon.

  1. Heptagon
  2. Nonagon
  3. Dodecagon
  4. 15-gon
  5. 20-gon
Part BEach Interior Angle of a Regular Polygon
  1. Regular pentagon
  2. Regular octagon
  3. Regular 12-gon
  4. Regular 18-gon
Part CEach Exterior Angle and Sides from Exterior Angle
  1. Each exterior angle, regular hexagon
  2. Each exterior angle, regular nonagon
  3. Sides, if each exterior angle = 40°
  4. Sides, if each exterior angle = 30°
  5. Sides, if each exterior angle = 20°
  6. Sides, if each exterior angle = 45°
Part DFinding Missing Angles Algebraically
  1. Quadrilateral: 90°, 115°, 83°, x°
  2. Pentagon: 100°, 112°, 95°, 88°, x°
  3. Hexagon: five angles each 115°, find the sixth
  4. Quadrilateral: 3x°, 2x°, (x+20)°, (x+40)°. Find x and each angle.

Word Problems and Applications

For each problem, define all quantities, draw or describe a diagram, write the key relationship or formula, solve, and check.

  1. Flooring: A rectangular room is 14 feet long and 11 feet wide. A diagonal support beam must be installed from one corner to the opposite corner. How long is the beam, to the nearest tenth of a foot?
  2. Engineering: A trapezoidal steel plate has parallel sides of 40 cm and 60 cm. A midsegment brace is welded across the plate at the midpoints of both legs. How long is the brace?
  3. Tile Design: A designer is planning a floor using regular hexagonal tiles, each with a side length of 4 inches. What is the interior angle of each tile? Do these tiles tessellate the floor completely? Explain.
  4. Construction: A rhombus-shaped roof panel has diagonals of length 8 feet and 6 feet. What is the length of each edge of the panel?
  5. Land Survey: A surveyor plots a quadrilateral lot at A(0, 0), B(120, 0), C(130, 80), D(10, 80), with coordinates in feet. Classify the shape of the lot. Find the length of the midsegment that would divide the lot equally in height.
  6. Robotics: A robotic arm moves along a parallelogram linkage with vertices A(0,0), B(8,0), C(10,5), D(2,5). Verify the figure is a parallelogram by showing the diagonals bisect each other.
  7. Stained Glass: A kite-shaped piece of stained glass has sides AB = AD = 10 cm and CB = CD = 6 cm. The longer diagonal AC has length 14 cm. Find the length of the shorter diagonal BD.
  8. Architecture: A regular hexagonal window has a perimeter of 72 inches. Describe how a brace running from one vertex to the opposite vertex could be computed. (Hint: a regular hexagon can be divided into equilateral triangles.)

Error Analysis

Each problem below shows a student's work or statement. Identify the error and write the correct reasoning.

Problem 1
"Quadrilateral ABCD has AB = CD = 6 and BC = DA = 8, so it must be a rectangle."

What error did the student make? What shape can definitely be concluded, and what additional information would confirm a rectangle?

Problem 2
"The exterior angle sum of a hexagon is (6 − 2) · 180° = 720°."

What error did the student make? What are the correct interior and exterior angle sums for a hexagon?

Problem 3
"Quadrilateral PQRS has PQ ∥ SR, so it must be a parallelogram."

What error did the student make? What can actually be concluded from this single condition?

Problem 4
"A square is a rhombus, so all rhombi are squares."

What error did the student make? How does the quadrilateral hierarchy actually work?

Problem 5
"The midsegment of a trapezoid with bases 10 and 18 has length 10 + 18 = 28."

What error did the student make? What is the correct midsegment length?

Chapter 9 Checkpoint

Chapter 9 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 9
Part A — Angle Sums
  1. Find the interior angle sum of a polygon with 11 sides.
  2. Each interior angle of a regular polygon measures 150°. How many sides does the polygon have?
  3. Each exterior angle of a regular polygon measures 18°. How many sides does the polygon have?
  4. A quadrilateral has angles 3x°, 2x°, (x + 20)°, and (x + 40)°. Find the value of x and the measure of each angle.
Part B — Quadrilateral Properties
  1. In parallelogram ABCD, m∠A = 5x + 10° and m∠C = 3x + 30°. Find m∠A.
  2. In rectangle EFGH, the diagonal has length 26 and one side is 10. Find the other side.
  3. In rhombus JKLM, m∠J = 60°. Find all other angle measures.
  4. Trapezoid PQRS has bases PQ = 24 and RS = 14. Find the length of the midsegment.
  5. Isosceles trapezoid WXYZ has m∠W = 70°. Find m∠X, m∠Y, and m∠Z.
  6. In kite ABCD with AB = AD and CB = CD, m∠A = 90° and m∠C = 40°. Find m∠B and m∠D.
Part C — Classification
  1. A quadrilateral has diagonals that bisect each other and are perpendicular, but are not congruent. Classify it as specifically as possible.
  2. A quadrilateral has diagonals that are congruent and bisect each other. Classify it as specifically as possible.
  3. A quadrilateral has exactly one pair of parallel sides and two pairs of congruent base angles. Classify it.
Part D — Coordinate Geometry
  1. Quadrilateral ABCD has vertices A(0, 0), B(8, 0), C(9, 4), D(1, 4). Classify ABCD as specifically as possible and justify your answer.
  2. Given EFGH with E(0,0), F(6,0), G(6,5), H(0,5): Classify it as specifically as possible.
  3. Show that the diagonals of the quadrilateral with vertices P(2, 5), Q(8, 5), R(8, −1), S(2, −1) are congruent and bisect each other. Classify the figure.
Part E — Applications
  1. A rectangular garden is 9 meters by 40 meters. A fence must be built diagonally across it. How long is the diagonal, to the nearest tenth of a meter?
  2. A trapezoidal plot of land has parallel sides of 80 feet and 120 feet. A fence is to be built along the midsegment. How long is the fence?
  3. A rhombus has perpendicular diagonals of length 12 and 16. Find the perimeter of the rhombus.
  4. A contractor claims a room with vertices A(0, 0), B(10, 0), C(10, 8), D(0, 8) is a square. Do you agree? Justify your answer using coordinate geometry.

Answers to the Chapter 9 Checkpoint may be placed in the answer key.

Looking Ahead
From Polygons to Circles, Area, and Beyond

Chapter 9 has built your understanding of polygon families and their properties. You have seen that shape families are organized by properties, not just appearance, and that both geometric reasoning and algebraic calculation are needed to work with these figures.

Chapter 10: Circles introduces a fundamentally different type of figure — one defined by a center and radius rather than by vertices and sides. You will study arcs, chords, tangents, secants, inscribed angles, and the equation of a circle. The reasoning skills practiced here transfer directly.

Chapter 11: Area and Surface Area returns to polygons and uses the properties from Chapter 9 to derive area formulas. Each quadrilateral family has its own area formula, and understanding why requires knowing the shape's defining properties.

Chapters 13 and 14 deepen the coordinate proof work from Section 9.12 and extend polygon classification to transformations and modeling. The quadrilateral hierarchy, angle sum formulas, and midsegment theorems from this chapter will appear again and again throughout the rest of the course.

MGU Library Connection
  • Appendix H of this textbook: Quadrilateral Properties Chart
  • Appendix J: Area Formula Sheet — Chapter 11 area formulas for all quadrilateral families
  • Appendix M: Coordinate Geometry Formula Sheet — distance, midpoint, and slope tools used in Section 9.12
  • MGU Dictionary entries: polygon, parallelogram, trapezoid, midsegment, tessellation
  • Algebra I, Chapter 6: Proportions — background for the midsegment theorem reasoning
  • Geometry Chapter 13: Coordinate Geometry and Algebraic Proof — extends all coordinate proof work from this chapter