Lesson 3.1.5

Consecutive (Same-Side) Interior Angles Theorem

The odd one out: when parallel lines are cut by a transversal, consecutive interior angles are supplementary — they add up to , not congruent.

Introduction

Corresponding, alternate interior, and alternate exterior angles with parallel lines are all congruent. But consecutive (same-side) interior angles are the exception — they're supplementary.

Past Knowledge

Alternate angles (3.1.4). Supplementary angles (1.3.3). Linear pairs (1.3.4).

Today's Goal

Apply the Consecutive Interior Angles Theorem: same-side interior angles sum to 180°.

Future Success

Proving lines parallel (3.1.6) uses the converse of this theorem.

Key Concepts

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary (sum = ).

Also Called

Same-Side Interior Angles Theorem or Co-Interior Angles Theorem. All three names mean the same thing.

Interactive Diagram — Desmos Geometry

Consecutive interior angles are on the same side and between the parallel lines. Notice the marked angle measurements add to 180°.

Quick Summary Table

Angle Pair (∥ lines)	Relationship
Corresponding	Congruent
Alternate Interior	Congruent
Alternate Exterior	Congruent
Consecutive Interior	Supplementary (180°)

Worked Examples

Basic

Finding the Supplement

. ∠4 = 115°. ∠5 is its consecutive interior angle. Find ∠5.

∠5 = 65°

Intermediate

Algebraic Setup

. Consecutive interior angles are and . Solve for .

Consecutive interior → supplementary:

. Angles are and (sum = 180° ✓)

Advanced

Multi-Step with All Theorems

. ∠1 = and ∠6 = where ∠1 and ∠6 are consecutive interior angles. Find all eight angles.

∠1 = 80°, ∠6 = 100°

∠1 = ∠3 = ∠5 = ∠7 = 80°

∠2 = ∠4 = ∠6 = ∠8 = 100°

Common Pitfalls

Setting Consecutive Interior Angles Equal

The #1 mistake! Consecutive interior angles are supplementary (add to 180°), NOT equal. Only alternate and corresponding pairs are congruent.

Real-Life Applications

Window Blinds

The slats of window blinds are parallel, and each cord acts as a transversal. Consecutive interior angles between slats are supplementary — tilt one side up, and the other side tilts down by the same amount, keeping the total at 180°.