Chapter 7, Problem 2RP

To determine

To prove : The congruency of $\overrightarrow{PR}\cong \overrightarrow{ST}$

Explanation of Solution

Given information : The following information has been given

O is the center of the circle

  $\angle W\cong \angle V$

  $\angle ORW\cong \angle OSV$

Formula used : Any triangles can be proved congruent by ASA property, if their all three angles are congruent, and any one of the corresponding sides is congruent as well.

Proof : We know that in triangles ORW and OSV,

  $\angle W\cong \angle V$

  $\angle ORW\cong \angle OSV$

Thus, if two angles are congruent, then the third one has to be congruent as well because their sum is constant.

Hence, we know that any two triangles can be proved congruent by ASA property, if their all three angles are congruent, and any one of the corresponding sides is congruent as well.

Now, we also have

  $\overrightarrow{OW}\cong \overrightarrow{OV}$ (Radius of circle)

Hence, we can say that by ASA property,

  $\Delta ORW\cong \Delta OSV$

Now, we know that corresponding sides of congruent polygons are congruent. Thus, we can say that

  $\overrightarrow{PR}\cong \overrightarrow{ST}$