Chapter 7, Problem 5RP

To determine

To prove : The congruency of $\overrightarrow{TS}\cong \overrightarrow{TX}$

Explanation of Solution

Given information : The following information has been given

SV lies in the plane m

VX lies in the plan m

  $\angle S\cong \angle X$

TV is perpendicular to plane m

Formula used : Any two triangles are congruent if they have all equal angles, and any one of the corresponding sides is also congruent

Proof : We know that as SV lies in the plane m and VX lies in the plan m, also that TV is perpendicular to plane m, we can say that

  $\angle TVS\cong \angle TVX={90}^0$

Now, $\angle S\cong \angle X$ is given.

Thus, if two angles of any triangles are congruent, then the third one also has to be because their sum is constant.

Now we know that if any two triangles are congruent if they have all equal angles, and any one of the corresponding sides is also congruent. So, we know that

TV is a common side

Hence, we can say that $\Delta TVS\cong \Delta TVX$

Now, we that corresponding sides of congruent polygons are congruent, thus we can say that

  $\overrightarrow{TS}\cong \overrightarrow{TX}$ (by the virtue of congruency)