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Isosceles Triangle Theorem

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Isosceles Triangle Theorem

Study Guide

Key Definition

Given $\triangle PQR$ with $PQ \cong PR$, if two sides of a triangle are congruent, then the angles opposite to these sides are congruent: $\angle P \cong \angle Q$

Important Notes

The sides opposite congruent angles are also congruent.
If S is the midpoint of segment AB, then $\overline{AS} \cong \overline{SB}$.
The Reflexive Property states that any segment is congruent to itself: $\overline{AB} \cong \overline{AB}$.
The SSS (Side-Side-Side) postulate helps prove triangle congruence.
Converse: If two angles are congruent, their opposite sides are congruent.

Mathematical Notation

$\angle$ represents an angle

$\cong$ means congruent

$\triangle$ represents a triangle

$\overline{AB}$ represents a line segment

Remember to use proper notation when solving problems

Why It Works

Given $\triangle PQR$ with $PQ \cong PR$, let S be the midpoint of QR. Then in $\triangle PQS$ and $\triangle PRS$, we have PQ \cong PR, QS \cong RS, and PS \cong PS (reflexive). By SSS, $\triangle PQS \cong \triangle PRS$, which implies $\angle P \cong \angle Q$.

Remember

In an isosceles triangle, the angles opposite the equal sides are always equal: $\angle A \cong \angle B$

Quick Reference

SSS Postulate:$\triangle ABC \cong \triangle DEF$ if $AB \cong DE$, $BC \cong EF$, $CA \cong FD$

Understanding Isosceles Triangle Theorem

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

In an isosceles triangle, if two sides are equal, the angles opposite those sides are equal: $\angle A \cong \angle B$

Practice Problems (No diagram needed)

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

In an isosceles triangle $\triangle ABC$, if $AB \cong AC$, which statement is true?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Alex is designing a triangular garden with two equal sides. If the equal sides are 5 meters each, how do the angles opposite compare? No diagram is needed.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a triangle where $\angle A = \angle B$. Can we conclude something about the sides? No diagram is needed.

Challenge Quiz

Single Choice Quiz

Advanced

In $\triangle DEF$, $DE \cong DF$ and $\angle D$ is bisected. What can be concluded about $\angle E$ and $\angle F$? No diagram is needed.

Recap

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Review key concepts and takeaways