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Chapter 4: Q11. (page 131)

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

Expert verified

True statements are (1) XW¯⊥UV¯ and (2)UV¯∥YZ¯

Step by step solution

01

Step 1. Show that ∠3≅∠4.

Using given statement

m∠WXY=m∠WXZ=90°

Thus, ∠1+∠3≅∠2+∠4

Substitute∠1 for∠2 in this equation

∠1+∠3≅∠1+∠4∠3≅∠4

02

Step 2. Show that ∠U+∠3=90°.

In a triangle angle opposite to congruent sides are congruent.

So in ΔUXV, as UX¯≅VX¯, 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°

03

Step 3. Show that WX¯⊥UV¯.

Since, sum of angles of a triangle is 180°

InΔUXW

∠U+∠3+∠UWX=180°90°+∠5=180°∠5=90°

Thus,XW¯⊥UV¯

04

Step 4. Show that UV¯∥YZ¯.

Since, XW¯is perpendicular to both YZ¯andUV¯

Thus,UV¯∥YZ¯

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Most popular questions from this chapter

Decide whether you can deduce by the SSS, SAS, or ASA postulate that another triangle is congruent to ΔABC. If so, write the congruence and name the postulate used. If not, write no congruence can be deduced.

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For the following figure, can the triangle be proved congruent. If so, what postulate can be used?

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