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Right Triangle Similarity

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Right Triangle Similarity

Study Guide

Key Definition

Two right triangles are similar if an acute angle of one is congruent to an acute angle of the other. Since all right angles are equal, one acute-angle equality suffices. Equivalently, their corresponding sides are proportional.

Important Notes

If $\angle M \cong \angle Y$, and $\angle N \cong \angle Z$, then $\triangle LMN \sim \triangle XYZ$
For Leg-Leg similarity in right triangles, if $\frac{AB}{PQ} = \frac{BC}{QR}$, then $\triangle ABC \sim \triangle PQR$
AA (Angle-Angle) Similarity can be used to prove triangle similarity
SAS (Side-Angle-Side) Similarity can also demonstrate triangle similarity

Mathematical Notation

$\sim$ denotes similarity between triangles

$\angle$ represents an angle

$\cong$ denotes congruence

$\frac{a}{b}$ represents a fraction

$x^2$ represents x squared

Why It Works

Right triangle similarity works because of proportional sides and equal corresponding angles, which can be mathematically proven using <a href="angle-angle-similarity">$\text{AA (Angle-Angle)}$</a> and <a href="side-angle-side-similarity">$\text{SAS (Side-Angle-Side)}$</a> theorems.

Remember

In similar triangles, corresponding angles are equal and sides are proportional: $\frac{a}{b} = \frac{c}{d}$

Quick Reference

AA Similarity:<a href="angle-angle-similarity">$\text{If two angles of one triangle are equal to two angles of another triangle, the triangles are similar}$</a>

SAS Similarity:<a href="side-angle-side-similarity">$\text{If the ratio of two corresponding sides is equal and the included angles are equal, the triangles are similar}$</a>

Understanding Right Triangle Similarity

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Video explanation of this concept

Beginner

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Beginner Explanation

Because right triangles each have a 90° angle, showing that one pair of acute angles are equal is sufficient for AA similarity. For example, if $\angle A = \angle X$, then $\triangle ABC \sim \triangle XYZ$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which condition proves the similarity of two right triangles using angle similarity?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You need to design a ramp that is similar to another with a known inclination. The ramp makes an angle of $\angle 30^\circ$ with the ground. What should be the angle of inclination for your ramp to ensure similarity?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given two right triangles, $\triangle ABC$ and $\triangle DEF$, with $\angle BAC = \angle DEF$. What else is needed to prove similarity?

Challenge Quiz

Single Choice Quiz

Advanced

For triangles $\triangle GHI$ and $\triangle JKL$, given $\frac{GH}{JK} = \frac{HI}{KL} = \frac{GI}{JL}$, what can be concluded?

Recap

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Review key concepts and takeaways