Polygons & Parallel Lines
Angles start inside polygons, move outside triangles, then travel through parallel lines and transversals. Those same relationships become the rules for parallelograms and trapezoids. The unit is really one connected idea: use structure to decide which angles or lengths must match, add to 180°, or satisfy an inequality.
Lesson Map
This follows the sequence in the Unit 3 lesson materials and emphasizes the skills that appear on the unit assessment.
| Lesson | Focus | The skill being checked |
|---|---|---|
| Lesson 1 | Sum of interior angles | Use (n − 2)180° to find an angle sum, solve for a missing angle, or work backward to find the number of sides. |
| Lesson 2 | Regular polygons | Find each interior and exterior angle of a regular polygon and solve backward from an angle measure to n. |
| Lesson 3 | Triangle inequalities | Use the exterior angle theorem, order sides and angles, test possible triangles, and find the range of a third side. |
| Lesson 4 — Day 1 | Vertical angles & angle pairs | Name corresponding, alternate interior/exterior, and same-side interior angle pairs; use vertical and linear-pair relationships. |
| Lesson 4 — Day 2 | Parallel lines & transversals | Use congruent or supplementary angle relationships created by parallel lines to solve algebraic equations. |
| Lesson 5 — Parallelograms | Properties of parallelograms | Use opposite sides/angles, consecutive angles, and bisected diagonals to find missing values. |
| Lesson 5 — Trapezoids | Isosceles trapezoids & midsegments | Use base-angle and diagonal theorems plus the trapezoid midsegment formula. |
| Performance Task | City Design | Build a street map that demonstrates parallel lines, transversals, angle-pair vocabulary, and required polygon types. |
| Assessment | Unit 3 review | Combine polygon formulas, triangle inequalities, transversal relationships, parallelograms, and trapezoids in multi-step problems. |
Content Summary
Start by identifying the structure in the figure. Once you know what kind of polygon, triangle, or angle pair you have, the equation usually follows immediately.
Interior Angles of Polygons
A convex polygon has every interior angle less than 180°. A concave polygon has at least one interior angle greater than 180°. The Unit 3 formula is for the sum of the interior angles of a convex polygon.
Why n − 2? From one vertex, a convex n-gon can be divided into exactly n − 2 non-overlapping triangles. Each contributes 180°.
A hexagon can be split into 6 − 2 = 4 triangles, so its interior angles total 720°.
Worked example
A polygon has an interior-angle sum of 1980°. How many sides?
Regular Polygons
A regular polygon has all sides congruent and all interior angles congruent. That means you can divide the total interior-angle sum evenly among the n angles.
An interior angle and its adjacent exterior angle form a linear pair, so they always add to 180°.
Worked example
Find each interior and exterior angle of a regular octagon.
Inequalities in Triangles
Exterior Angle Theorem
An exterior angle of a triangle equals the sum of its two remote interior angles.
Side–Angle Inequality
In one triangle, the larger angle is opposite the longer side. The smaller angle is opposite the shorter side. This lets you order sides from angle measures or order angles from side lengths.
Match an angle to the side directly opposite it — not to a side touching the angle.
Triangle Inequality Theorem
The sum of any two sides of a triangle must be greater than the third. When two side lengths are known, the third side x must satisfy:
Example: if two sides are 12 and 25, then 13 < x < 37. The endpoints are not included, because equality would make a straight segment rather than a triangle.
Vertical Angles, Angle Pairs & Parallel Lines
When a transversal crosses two lines, the locations of the angles give each pair a name. When the two lines are parallel, those names tell you whether the angles are congruent or supplementary.
Use position first to name the pair. Then use parallel-line rules to decide congruent or supplementary.
Vertical angles
Opposite angles at one intersection. Always congruent.
Linear pair
Adjacent angles forming a straight line. Always supplementary.
Corresponding
Same relative corner at the two intersections. Congruent if the lines are parallel.
Alternate interior
Inside the parallel lines, opposite sides of the transversal. Congruent.
Alternate exterior
Outside the parallel lines, opposite sides of the transversal. Congruent.
Same-side interior
Inside the parallel lines, same side of the transversal. Supplementary.
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Because each pair of sides acts as parallel lines cut by transversals, the angle rules from Lesson 4 generate the parallelogram theorems.
Opposite sides
Congruent.
Opposite angles
Congruent.
Consecutive angles
Supplementary.
Diagonals
Bisect each other.
Worked example
If ∠A = 56°, then ∠C = 56° because opposite angles are congruent. ∠B and ∠D are each 124° because consecutive angles are supplementary.
Trapezoids & Midsegments
A trapezoid has one pair of parallel sides, called the bases. In an isosceles trapezoid, the legs are congruent. Its base angles are congruent and its diagonals are congruent.
Worked example
Bases are 59 and 17. The midsegment is:
Vocabulary
These words tell you what relationship to use. Being precise with vocabulary is part of getting the equation right.
A closed plane figure made from straight line segments.
A polygon whose interior angles are all less than 180°.
A polygon with at least one interior angle greater than 180°.
A polygon with all sides congruent and all angles congruent.
An angle formed inside a polygon by two adjacent sides.
An angle formed outside a polygon by one side and the extension of an adjacent side.
The two triangle angles not adjacent to a chosen exterior angle.
The sum of any two side lengths of a triangle must be greater than the third.
A line that intersects two or more coplanar lines at different points.
Opposite angles formed by two intersecting lines; they are congruent.
Angles in the same relative position at two intersections.
Interior angles on opposite sides of a transversal.
Exterior angles on opposite sides of a transversal.
Interior angles on the same side of a transversal; supplementary when the lines are parallel.
A quadrilateral with both pairs of opposite sides parallel.
A quadrilateral with one pair of parallel sides.
A trapezoid with congruent legs; each pair of base angles is congruent.
The segment joining the midpoints of the legs; its length is the average of the two bases.
Sample Problems
Twelve problems in homework and assessment format, grouped by lesson. Work each one on paper first — the solutions are one click away, which makes them very easy to read too early.
Lesson 1 — Interior Angles of Polygons
Polygon angle sum
Find the sum of the interior angles of a 20-sided polygon.
Show solution
Work backward from the sum
The interior angles of a polygon total 4860°. How many sides does it have?
Show solution
Lesson 2 — Regular Polygons
Regular polygon
Find each interior angle and each exterior angle of a regular 12-gon.
Show solution
Regular polygon from an exterior angle
Each exterior angle of a regular polygon measures 40°. How many sides?
Show solution
Lesson 3 — Inequalities in Triangles
Exterior Angle Theorem
A triangle has remote interior angles of 48° and 67°. Find the exterior angle.
Show solution
Triangle inequality range
Two sides of a triangle are 11 and 20. Between what two values must the third side x fall?
Show solution
Order the sides
In △ABC, ∠A = 41°, ∠B = 63°, and ∠C = 76°. List the sides from shortest to longest.
Show solution
The smallest angle A is opposite BC; B is opposite AC; the largest angle C is opposite AB.
Lesson 4 — Angle Pairs, Parallel Lines & Transversals
Vertical angles
Two vertical angles are labeled 6x − 21 and 105°. Find x.
Show solution
Parallel lines: same-side interior
Two same-side interior angles are 4x − 12 and 120°. The lines are parallel. Find x.
Show solution
Same-side interior angles are supplementary, so the two expressions add to 180° rather than being set equal.
Lesson 5 — Parallelograms, Trapezoids & Midsegments
Parallelogram angles
In parallelogram ABCD, m∠A = 56°. Find B, C, and D.
Show solution
Opposite angles are congruent; consecutive angles are supplementary.
Parallelogram diagonals
The diagonals bisect each other. If one diagonal is split into segments 4x − 11 and x + 10, find x.
Show solution
Trapezoid midsegment
A trapezoid has bases 12 and 26. Find its midsegment.
Show solution
Equations & Rules
Every formula and rule for the unit on one printable page.
Polygon interior sum
Regular polygon interior angle
Regular polygon exterior angle
Interior/exterior pair
Triangle exterior angle
Third side range
Parallel-line congruent pairs
Corresponding, alternate interior, and alternate exterior angles.
Parallel-line supplementary pair
Same-side interior angles sum to 180°.
Parallelogram
Opposite sides congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other.
Trapezoid midsegment
Video Library
Walkthroughs matched to the lessons in this unit, mostly Khan Academy. Use them after a first attempt, not instead of one. Videos do not appear on the printed copy.
Lesson 1 — Interior Angles of Polygons
Lesson 2 — Regular Polygons
Lesson 3 — Inequalities in Triangles
Lesson 4 Day 1 — Vertical Angles & Angle Pairs
Lesson 4 Day 2 — Parallel Lines & Transversals
Lesson 5 — Parallelograms
Lesson 5 — Trapezoids & Midsegments
Unit 3 Review Sheet
The review moves through the same progression as the assessment: polygon sums and equations, triangle side/angle inequalities, regular polygons, parallel lines and transversals, parallelograms, and trapezoids with midsegments.
Polygons & triangles
Solve forward and backward with the polygon formulas, find regular-polygon angles, use exterior angles, order sides and angles, and find the interval for a third side.
Lines & quadrilaterals
Name angle pairs, decide congruent vs. supplementary, solve the resulting equations, and apply the parallelogram and trapezoid theorems.
Before you turn in the assessment
- Every polygon answer checked for the n − 2 step — did you add 2 back at the end?
- Each angle-pair problem started with the name of the relationship, written down, before any algebra.
- Same-side interior angles summed to 180° rather than set equal.
- Every variable substituted back into both expressions it came from.
- Side and angle orderings matched to the side opposite each angle, not a side touching it.
- Third-side answers written as an interval with strict inequalities, endpoints excluded.
- Parallelogram answers using opposite angles congruent and consecutive angles supplementary as a check.
- Midsegment answers averaged, not summed.
Performance Task: City Design
Parallel lines and transversals, placed in a context where the labels have to convince somebody else.
The build
Design a city map with three parallel streets and two non-parallel transversal streets. Place locations so their intersections demonstrate specific angle relationships. The task also requires a scalene-triangle gas station, a regular-pentagon police station, and a right-triangle courthouse.
Required angle relationships
- Gas station & restaurant — alternate exterior
- Home & school — same-side interior
- Post office & bank — vertical
- Fire department & police station — corresponding
- Library & park — alternate interior
Design goal
The point is not just to draw a city. Your labels should make another student able to inspect the map and verify every required angle pair from the street geometry alone.
Where to Go Next
Unit 3 leans on the triangle work from Unit 2 and sets up the congruence proofs in Unit 4.
Unit 2: Congruence & Similarity
Triangle classification, isosceles triangles, and similar figures — the triangle facts Unit 3 assumes you already have.
Open Unit 2 →All Geometry Units
The full six-unit course map, with every content summary, review sheet, and inquiry project in one place.
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