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Consecutive Interior Angles Theorem

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Consecutive Interior Angles Theorem

Study Guide

Key Definition

Consider two parallel lines k and l cut by a transversal t as shown in the diagram. When k ∥ l and t intersects them, the consecutive interior angles (for example, ∠3 and ∠5, which lie between the lines on the same side of t) are supplementary.

Important Notes

Consecutive interior angles are on the same side of the transversal.
If lines are not parallel, consecutive interior angles need not be supplementary.
The sum of supplementary angles is $180^\circ$.
Understanding the theorem helps in proving lines are parallel.
This concept is crucial for solving many geometric proofs.

Mathematical Notation

$\angle$ represents an angle

$\parallel$ means parallel lines

$+$ represents addition

$=$ represents equality

$180^\circ$ is the sum of supplementary angles

Remember to use proper notation when solving problems

Why It Works

By the Parallel Postulate, alternate interior angles formed by a transversal with two parallel lines are congruent. Each pair of consecutive interior angles then forms a linear pair, and so their measures add up to $180^\circ$.

Remember

Consecutive interior angles theorem is a fundamental concept in geometry that helps in determining the properties of parallel lines cut by a transversal.

Quick Reference

Supplementary Angles:$\angle 3 + \angle 5 = 180^\circ$

Understanding Consecutive Interior Angles Theorem

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When two parallel lines are intersected by a transversal, the consecutive interior angles (such as ∠3 and ∠5 in the diagram) lie between the lines on the same side of the transversal and always add to 180°.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

In the diagram where lines k and l are parallel and cut by transversal t with angles labeled as shown, what is the measure of $\angle 5$ if $\angle 3 = 110^\circ$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are decorating a room. Two parallel shelves are cut by a diagonal piece. If one angle is $60^\circ$, find the measure of its consecutive interior angle.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If two lines are not parallel and cut by a transversal, can the consecutive interior angles still be supplementary? Explain your reasoning.

Challenge Quiz

Single Choice Quiz

Advanced

Given $\angle 3 = x + 20^\circ$ and $\angle 5 = 2x - 10^\circ$ in the diagram where lines k and l are parallel and cut by transversal t with angles numbered accordingly, find $x$.

Recap

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