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SAS Inequality

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SAS Inequality

Study Guide

Key Definition

If one $\triangle$ has two sides that are $\cong$ to two sides of another $\triangle$ and an included angle greater than the other, then the third side of the first $\triangle$ is longer.

Important Notes

$\triangle$ ABC and $\triangle$ PQR with $\overline{AB} \cong \overline{PQ}$ and $\overline{AC} \cong \overline{PR}$
If $\angle A > \angle P$, then $\overline{BC} > \overline{QR}$
Used to compare side lengths in triangles
Relies on congruency and angle comparison
Helpful in triangle inequality proofs

Mathematical Notation

$\cong$ means congruent

$\angle$ represents an angle

$\triangle$ represents a triangle

$\overline{AB}$ represents line segment AB

$>$ represents greater than

Remember to use proper notation when solving problems

Why It Works

SAS Inequality relies on two triangles having two pairs of congruent sides and then comparing the included angles. If the included angle in one triangle is larger than in the other, the third side opposite that angle is longer.

Remember

Use SAS Inequality when two sides and the included angle of one triangle are known relative to another.

Quick Reference

SAS Inequality:If $\overline{AB} \cong \overline{PQ}$, $\overline{AC} \cong \overline{PR}$, and $\angle A > \angle P$, then $\overline{BC} > \overline{QR}$

Understanding SAS Inequality

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

If two triangles have $AB≅PQ$ and $AC≅PR$, and $∠A>∠P$, then by SAS Inequality $BC>QR$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $\overline{AB} \cong \overline{PQ}$ and $\overline{AC} \cong \overline{PR}$, which condition implies $\overline{BC} > \overline{QR}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Triangle ABC has $AB=5cm$ and $AC=7cm$ with $∠A=60°$; triangle PQR has $PQ=5cm$ and $PR=7cm$ with $∠P=45°$. Determine which base, $BC$ or $QR$, is longer.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Triangle ABC has $AB=4cm$ and $AC=6cm$ with $∠A=50°$; triangle PQR has $PQ=4cm$ and $PR=6cm$ with $∠P=70°$. Determine which side, $BC$ or $QR$, is longer.

Challenge Quiz

Single Choice Quiz

Advanced

Given two triangles, $\triangle ABC$ and $\triangle DEF$, with sides $\overline{AB} \cong \overline{DE}$ and $\overline{AC} \cong \overline{DF}$, if $\angle A > \angle D$, what can be said about $\overline{BC}$ and $\overline{EF}$?