Chapter 4: Q11. (page 131)

Given: $\bar{W X} ⊥ \bar{Y Z}; ∠ 1 ≅ ∠ 2; \bar{U X} ≅ \bar{V X}$
Which one(s) of the following statements must be true?
(1) $\bar{X W} ⊥ \bar{U V}$
(2) $\bar{U V} ∥ \bar{Y Z}$
(3) $\bar{V X} ⊥ \bar{U X}$

Short Answer

Expert verified

True statements are (1) $\bar{X W} ⊥ \bar{U V}$ and (2) $\bar{U V} ∥ \bar{Y Z}$

Step by step solution

Step 1. Show that ∠3≅∠4.

Using given statement

$m ∠ W X Y = m ∠ W X Z = 90 °$

Thus, $∠ 1 + ∠ 3 ≅ ∠ 2 + ∠ 4$

Substitute $∠ 1$ for $∠ 2$ in this equation

$\begin{matrix} ∠ 1 + ∠ 3 ≅ ∠ 1 + ∠ 4 \\ ∠ 3 ≅ ∠ 4 \end{matrix}$

Step 2. Show that ∠U+∠3=90°.

In a triangle angle opposite to congruent sides are congruent.

So in $Δ U X V$ , as $\bar{U X} ≅ \bar{V X}$ , implies $∠ U ≅ ∠ V$

Also, sum of angles of a triangle is $180 °$

$∠ U + ∠ 3 + ∠ 4 + ∠ V = 180 ° 2 ∠ U + 2 ∠ 3 = 180 ° ∠ U + ∠ 3 = 90 °$

Step 3. Show that WX¯⊥UV¯.

Since, sum of angles of a triangle is $180 °$

In $Δ U X W$

$∠ U + ∠ 3 + ∠ U W X = 180 ° 90 ° + ∠ 5 = 180 ° ∠ 5 = 90 °$

Thus, $\bar{X W} ⊥ \bar{U V}$

Step 4. Show that UV¯∥YZ¯.

Since, $\bar{X W}$ is perpendicular to both $\bar{Y Z} and \bar{U V}$

Thus, $\bar{U V} ∥ \bar{Y Z}$

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Given: $\bar{W X} ⊥ \bar{Y Z}; ∠ 1 ≅ ∠ 2; \bar{U X} ≅ \bar{V X}$
Which one(s) of the following statements must be true?
(1) $\bar{X W} ⊥ \bar{U V}$
(2) $\bar{U V} ∥ \bar{Y Z}$
(3) $\bar{V X} ⊥ \bar{U X}$

Short Answer

Step by step solution

Step 1. Show that ∠3≅∠4.

Step 2. Show that ∠U+∠3=90°.

Step 3. Show that WX¯⊥UV¯.

Step 4. Show that UV¯∥YZ¯.

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Given:WX¯⊥YZ¯;∠1≅∠2;UX¯≅VX¯Which one(s) of the following statements must be true?(1)XW¯⊥UV¯(2)UV¯∥YZ¯(3)VX¯⊥UX¯

Short Answer

Step by step solution

Step 1. Show that ∠3≅∠4.

Step 2. Show that ∠U+∠3=90°.

Step 3. Show that WX¯⊥UV¯.

Step 4. Show that UV¯∥YZ¯.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Given: $\bar{W X} ⊥ \bar{Y Z}; ∠ 1 ≅ ∠ 2; \bar{U X} ≅ \bar{V X}$
Which one(s) of the following statements must be true?
(1) $\bar{X W} ⊥ \bar{U V}$
(2) $\bar{U V} ∥ \bar{Y Z}$
(3) $\bar{V X} ⊥ \bar{U X}$