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Congruent Triangles

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Congruent Triangles

Study Guide

Key Definition

Congruent triangles are those with three pairs of corresponding sides and angles. If one triangle, $\triangle ABC$, and another triangle, $\triangle DEF$, are congruent, we write $\triangle ABC \cong \triangle DEF$.

Important Notes

Congruent triangles can be obtained by translating, rotating, and/or flipping one triangle to match the other.
It's not necessary to measure all three sides and all three angles to determine if two triangles are congruent. There are several shortcuts such as SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and RHS (Right angle-Hypotenuse-Side).
The corresponding vertices must come in the correct order when indicating congruence.

Mathematical Notation

$\triangle$ represents a triangle.

$\cong$ represents congruence.

$\angle$ represents an angle.

Remember to use proper notation when solving problems

Why It Works

The concept of congruent triangles works because it is based on the idea that a shape's size and position do not affect its fundamental properties. By translating, rotating, or reflecting a triangle, we can see that it's still the same triangle - it's just in a different position or orientation.

Remember

Remember the shortcuts for proving congruence in triangles: SSS, SAS, ASA, AAS, and RHS.

Quick Reference

SSS:Side-Side-Side

SAS:Side-Angle-Side

ASA:Angle-Side-Angle

AAS:Angle-Angle-Side

RHS:Right angle-Hypotenuse-Side

Understanding Congruent Triangles

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Congruent triangles are triangles that have the same size and shape. This means that all corresponding sides and angles are equal.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following conditions would not be sufficient to prove that triangles are congruent?

Real-World Problem

Question Exercise

Intermediate

Field Day Challenge

Your school is having a field day and you're in charge of setting up the three-legged race. You need to mark a triangular path for the race. You have a rope of 50 feet and you want all sides of the triangle to be equal. How long should each side be?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider two right triangles, $\triangle ABC$ and $\triangle DEF$, with $\angle ABC$ and $\angle DEF$ as the right angles. If $AB = DE, BC = EF$, and $AC \neq DF$, are the two triangles congruent?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following is not a pair of congruent triangles?

Recap

Watch & Learn

Review key concepts and takeaways