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Q12.

Expert-verified

Geometry

Found in: Page 207

Book edition Student Edition

Author(s) Ray C. Jurgensen, Richard G. Brown, John W. Jurgensen

Pages 227 pages

ISBN 9780395977279

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Short Answer

Write proof in two-column form.
Given: $\bar{Q R}$ , $\bar{S T}$ bisect each other. Prove: $∠ X R T > m ∠ S$ .

Statement	Proof
1. $Q V = V R$ , $S V = V T$	Given
2. $∠ Q V S = ∠ R V T$	Vertically opposite angles
3. $△ Q S V ≅ △ V R T$	SAS
4. $∠ T = ∠ S$	CPCT
5. $∠ X R T = ∠ R V T + ∠ S$
6. $m ∠ X R T > m ∠ S$	Property of Inequality If $a = b + c$ and $c > 0$ then $a > c$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Draw the given diagram and define the congruent property of triangles

Two triangles $Q S V$ and $V R T$ as shown below are the given triangles.

If two sides and one angle equal of two triangles are equal then from SAS corresponding triangles will be congruent triangles.

Since QR and ST bisect each other, therefore $Q V = V R$ , $S V = V T$ .

The vertically opposite angles are always equal.

Therefore, $∠ Q V S = ∠ R V T$ . Then by SAS, $△ Q S V ≅ △ V R T$ .

Step 2. Property of congruent triangles

Since $△ Q S V ≅ △ V R T$ then by CPCT $∠ T = ∠ S$ .

Use exterior angle property as follows:

$\begin{array}{l} ∠ X R T = ∠ R V T + ∠ T \\ ∠ X R T = ∠ R V T + ∠ S \end{array}$

Step 3. Prove the statement

If $a = b + c$ and $c > 0$ then $a > c$ .

Since $∠ X R T = ∠ R V T + ∠ S$ therefore $m ∠ X R T > m ∠ S$

Hence proved.

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