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Q10.

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 proofs in two-column form.
Given: $\bar{V Y} ⊥ \bar{Y Z}$ . Prove: $∠ V X Z$ is an obtuse angle.

Statement	Reason
1. $∠ Y + ∠ Z + ∠ Y X Z = 180$	Angle sum property
2. $∠ Y X Z + ∠ V X Z = 180$	Linear angles

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Angles property of a triangle

The sum of the angles of a triangle is 180. If an angle is greater than 90 then it is an obtuse angle.

Step 2. Use the angle sum property and find the angle YXZ

Since $\bar{V Y} ⊥ \bar{Y Z}$ therefore, $∠ X Y Z = ∠ Y = 90$ .

The sum of the angles of triangle will be:

role="math" localid="1637852883028" $\begin{matrix} ∠ Y + ∠ Z + ∠ Y X Z = 180 \\ ∠ Z + ∠ Y X Z = 180 - ∠ Y \\ ∠ Z + ∠ Y X Z = 180 - 90 \\ ∠ Z + ∠ Y X Z = 90 \end{matrix}$

Since $∠ Z > 90$ , therefore $∠ Y X Z < 90$ .

Step 3. Prove the statement

From the figure

Use linear angle property as follows:

$∠ Y X Z + ∠ V X Z = 180$

Since $∠ Y X Z < 90$ therefore from equation (1) $∠ V X Z > 90$ .

Therefore, $∠ V X Z$ is an obtuse angle.

Hence proved.

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