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Chapter 6: Q12. (page 207)

Write proof in two-column form.

Given: QR炉, ST炉bisect each other. Prove: 鈭�XRT>m鈭�S.

Short Answer

Expert verified

Statement

Proof

1. QV=VR, SV=VT

Given

2. 鈭�QVS=鈭�RVT

Vertically opposite angles

3. 鈻�QSV鈮�鈻�VRT

SAS

4. 鈭�T=鈭�S

CPCT

5. 鈭�XRT=鈭�RVT+鈭�S

6. m鈭�XRT>m鈭�S

Property of Inequality If a=b+cand c>0then a>c.

Step by step solution

01

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

Two triangles QSV and VRT 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 QV=VR, SV=VT.

The vertically opposite angles are always equal.

Therefore, 鈭�QVS=鈭�RVT. Then by SAS, 鈻�QSV鈮�鈻�VRT.

02

Step 2. Property of congruent triangles

Since 鈻�QSV鈮�鈻�VRTthen by CPCT 鈭�T=鈭�S.

Use exterior angle property as follows:

鈭�XRT=鈭�RVT+鈭�T鈭�XRT=鈭�RVT+鈭�S

03

Step 3. Prove the statement

If a=b+cand c>0then a>c.

Since 鈭�XRT=鈭�RVT+鈭�Sthereforem鈭�XRT>m鈭�S

Hence proved.

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Most popular questions from this chapter

Prove the following statement by proving its contrapositive. Begin by writing what is given and what is to be proved.

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Prove: There is at most one line through P perpendicular to k.

(a) But this contradicts corollary3 of theorem 3-11: In a triangle, there can be at most one right angle or obtuse angle.

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(c) Thus, our temporary assumption is false, and there is at most one line through P perpendicular to k.

(d) Assume temporarily that there are two lines through P perpendicular to k at A, B.
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