Chapter 4.3, Problem 13WE

To determine

To prove: $\Delta RTV$ is isosceles.

Explanation of Solution

Giveninformation:In the figure shown, $\overline{RS}\perp \text{plane}Y$ ; $\angle TRS\cong \angle VRS$

  

Concept/Formula used:In an isosceles triangle, any two sides are always congruent.

Also in two triangles, if two pairs of corresponding angles are congruent and their included side is common, then by ASA property, both are congruent triangles.

So, first show that $\Delta TRS\cong \Delta VRS$ ,and then by CPCT, their other corresponding sides will also be congruent.

Calculation: In $\Delta TRS\cong \Delta VRS$ ,

  $\begin{array}{l}(i)\angle TRS\cong \angle VRS(\text{Given)}\\ \text{(}ii)RS\cong RS(\text{Reflexive property)}\\ \text{(}iii)\angle RST=\angle RSV={90}^0\end{array}$

Thus, by ASA( Angle-Side-Angle) congruency, $\Delta TRS\cong \Delta VRS$

Now, by CPCT (Corresponding parts of corresponding triangles),

  $\overline{RT}\cong \overline{RV}$

Conclusion: Thus,as two sides of triangles $\Delta RTV$ are congruent, so it is an isosceles triangle.