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Alternate Exterior Angles Theorem

Master alternate exterior angles theorem with interactive lessons and practice problems! Designed for students like you! Refer to the provided diagram (alternateExteriorAnglesDiagram.png) that labels angles 1–8 around the parallel lines and transversal.

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Alternate Exterior Angles Theorem

Study Guide

Key Definition

The Alternate Exterior Angles Theorem states that, when two parallel lines are cut by a transversal, the resulting alternate exterior angles are congruent. So, if $k \parallel l$ , then $\angle 1 \cong \angle 7$ and $\angle 4 \cong \angle 6$.

Important Notes

Parallel lines are always the same distance apart.
A transversal is a line that intersects two or more lines.
Congruent angles have the same measure.
Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines.

Mathematical Notation

$\parallel$ indicates parallel lines.

$\angle$ represents an angle.

$\cong$ stands for congruence.

$k$ and $l$ are often used to represent lines.

Remember to use proper notation when solving problems

Why It Works

The theorem follows from the Corresponding Angles Postulate: when a transversal cuts two parallel lines, each pair of corresponding angles is congruent; vertical-angle congruence then yields the alternate exterior pair.

Remember

The converse of this theorem is also true; that is, if two lines $k$ and $l$ are cut by a transversal so that the alternate exterior angles are congruent, then $k \parallel l$.

Quick Reference

Alternate Exterior Angles Theorem:If $k \parallel l$, then $\angle 1 \cong \angle 7$ and $\angle 4 \cong \angle 6$

Understanding Alternate Exterior Angles Theorem

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

If two parallel lines are cut by a transversal, the alternate exterior angles are equal.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice Quiz

Beginner

If $k \parallel l$ and $t$ is a transversal, which pair of angles are alternate exterior angles? (Refer to the diagram labeling angles 1–8.)

2

Real-World Problem

Question Exercise

Intermediate

Street Grid Scenario

In a city with a grid-like street layout, two parallel streets are cut by a third street that runs diagonally. If the measure of one alternate exterior angle formed is $75^\circ$, what is the measure of the other alternate exterior angle?

3

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explain the proof of the Alternate Exterior Angles Theorem using the Corresponding Angles Postulate and the Vertical Angles Theorem. See the diagram labeling angles 1–8: angles are numbered consecutively clockwise starting at the upper-left exterior angle.

4

Challenge Quiz

Single Choice Quiz

Advanced

What does the converse of the Alternate Exterior Angles Theorem state?

Recap

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Review key concepts and takeaways