Alternate Exterior Angles

Study Guide

Key Definition

Alternate exterior angles are pairs of angles on opposite sides of a transversal which lie outside a set of lines. These angles are congruent when the lines are parallel.

Important Notes

Alternate exterior angles are congruent if the lines are parallel.
The converse is also true: if alternate exterior angles are congruent, the lines are parallel.
This is a part of the study of Euclidean geometry.
Understanding this concept is fundamental for further study of geometry.
These angles can be found in many real world scenarios.

Mathematical Notation

$\angle 1 \cong \angle 7$ means angle 1 is congruent to angle 7

$\angle 4 \cong \angle 6$ means angle 4 is congruent to angle 6

Angle numbering convention: at the top intersection of the transversal with the first line, angles 1 (upper left), 2 (upper right), 3 (lower right), 4 (lower left) are labeled clockwise. At the bottom intersection, angles 5 (upper left), 6 (upper right), 7 (lower right), 8 (lower left) are labeled similarly.

Remember to use proper notation when solving problems

Why It Works

This follows from the Euclidean Parallel Postulate and the Corresponding Angles Postulate: when two parallel lines are cut by a transversal, corresponding angles are equal, and this equality extends to alternate exterior angles, proving they are congruent.

Remember

Remember that alternate exterior angles are congruent only when the lines are parallel.

Quick Reference

Alternate Exterior Angles Theorem:$\angle 1 \cong \angle 7$ and $\angle 4 \cong \angle 6$ when lines are parallel

Understanding Alternate Exterior Angles

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Consider two parallel horizontal lines and a transversal crossing them. At the top intersection, label angles clockwise as 1 (upper left), 2 (upper right), 3 (lower right), 4 (lower left). At the bottom intersection, label angles 5 (upper left), 6 (upper right), 7 (lower right), 8 (lower left). If two parallel lines are intersected by a transversal, the pairs of alternate exterior angles are congruent.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which angles are alternate exterior angles?

Real-World Problem

Question Exercise

Intermediate

City Street Layout

A city's streets are laid out in a grid pattern. The streets running north-south are parallel and the streets running east-west are also parallel. A car is driving from the east end of the city to the west end. At each intersection, the car turns right then immediately turns left at the next intersection. What can you conclude about the direction the car is heading after each pair of turns?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If two lines are cut by a transversal so that $\angle 1 \cong \angle 7$, what can you conclude about the lines?

Challenge Quiz

Single Choice Quiz

Advanced

If $\angle 1 \cong \angle 7$ and $\angle 2 \cong \angle 8$, which of the following is correct?

Recap

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Review key concepts and takeaways