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Perpendicular Transversal Theorem

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Perpendicular Transversal Theorem

Study Guide

Key Definition

In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also. Given: $k \parallel l$, $t \perp k$. Prove: $t \perp l$.

Important Notes

If $k \parallel l$ and $t \perp k$, then $t \perp l$.
Perpendicular lines form a right angle, $90^\circ$.
Corresponding angles are congruent in parallel lines cut by a transversal.
The definition of congruent angles is used to show two angles are equal.
Substitution property allows replacing equal values.

Mathematical Notation

$\perp$ means perpendicular

$\parallel$ means parallel

$\angle$ represents an angle

$\cong$ means congruent

$=$ means equal

Remember to use proper notation when solving problems

Why It Works

If a transversal t intersects line k perpendicularly, it creates a 90° angle at their intersection. When line k is parallel to line l, the angle formed by t with l is a corresponding angle to the angle formed with k. Corresponding angles are congruent when lines are parallel, so the angle between t and l is also 90°, which means t ⟂ l.

Remember

The Perpendicular Transversal Theorem is a powerful tool for understanding the relationships between lines and angles in geometry.

Quick Reference

Perpendicular Line Property:$t \perp k \Rightarrow t \perp l$ (A line perpendicular to one of two parallel lines is perpendicular to the other.)

Understanding Perpendicular Transversal Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

When a line is perpendicular to one of two parallel lines, it forms a right angle with that line. Because parallel lines never meet and corresponding angles are congruent, the line must also form a right angle with the other parallel line and thus be perpendicular.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $k \parallel l$ and $t \perp k$, what is $t$'s relationship to $l$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine a skateboard ramp that is perpendicular to the ground which is parallel to a nearby street. How does the orientation of the street affect the ramp?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

In the diagram, lines k and l are parallel and line t is perpendicular to k at point A, creating ∠1 = 90°. When t intersects l at point B, it forms ∠2. Determine the measure of ∠2 given ∠1 = 90° and ∠1 ≅ ∠2.

Challenge Quiz

Single Choice Quiz

Advanced

In the diagram below, lines $k \parallel l$ and t is a transversal that intersects k at A and l at B, forming ∠1 at A and ∠2 at B. Prove that if $t \perp k$ then m∠2 = 90°.

Recap

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