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Chapter 3: Q32 (page 99)

Given: PRbisects 鈭�SPQand PS鈯�SQ;RQ鈯�PQ.

Which numbered angles must be congruent?

Short Answer

Expert verified

The congruent angles are 鈭�7鈮�鈭�8and 鈭�11鈮�鈭�12.

Step by step solution

01

Step 1. Apply the definition of angle bisector.

Angle bisector is a line which divides an angle into two equal parts.

From the figure, it can be observed that PR炉bisect 鈭�SPQ. Therefore,

鈭�7鈮�鈭�8

02

Step 2. Description of step.

PS炉鈯�SQ炉and RQ炉鈯�PQ炉, therefore,

鈭�Q鈮�鈭�Sm鈭�Q=90

03

Step 3. Description of step.

Consider 螖PQRand螖PST, in which 鈭�Q鈮�鈭�S,鈭�RPQ鈮�鈭�SPTimplies that 鈭�PTS鈮�鈭�11.

04

Step 4. Apply definition of vertically opposite angles.

Vertically opposite angles are the opposite angles formed due to intersection of two lines.

Consider 螖PQRand螖PST,

鈭�11and鈭�12are vertically opposite angles, therefore, 鈭�11鈮�鈭�12.

Therefore, the congruent angles are 鈭�7鈮�鈭�8and 鈭�11鈮�鈭�12.

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