Chapter 3.5, Problem 13PSC

To determine

To prove: that One of the given triangle is isosceles.

Explanation of Solution

Given information Two given line segments are congruent, i.e., $\overline{\text{NR}}\cong \overline{\text{NV}}$

Two given points (P & Q) are midpoints

Two given angles are congruent, i.e., $\angle \text{R}\cong \angle \text{V}$

Two other given line segments are congruent, i.e., $\overline{\text{PR}}\cong \overline{\text{QX}}$

Formula used Congruency means same size and same shape

Midpoint of line segment divides it in two equal parts

Proof:

  $\begin{array}{l}\text{In }\Delta \text{PRT and }\Delta \text{QSV, we have}\\ \angle \text{R}\cong \angle \text{V}\mathrm{.....}\text{given}\\ \therefore \angle \text{R=}\angle \text{V}\mathrm{.....}\text{by definition}\mathrm{........}\left(i\right)\\ \overline{\text{NR}}\cong \overline{\text{NV}}\mathrm{........}\text{given}\\ \therefore \overline{\text{NR}}=\overline{\text{NV}},\mathrm{......}given\\ \therefore \text{PR}=\text{QV}\mathrm{.......}\text{by definition of congruency}\mathrm{...}\left(ii\right)\\ \angle \text{RPT=}\angle \text{SQV}\mathrm{....}\text{ angles opposite to equal sides}\mathrm{....}\left(iii\right)\end{array}$

From (i), (ii) and (iii), we have

  $\begin{array}{l}\Delta \text{PRT }\cong \Delta \text{QSV, ASA criteria of congruency is satisfied}\\ \text{⇒}\text{PR = QX}\mathrm{.....}\text{given}\mathrm{...}\text{PR \& QX are congruent}\\ \text{⇒}\text{PR = SX}\mathrm{......}\left(a\right)\end{array}$

Further,

  $\begin{array}{l}\text{PR}=\text{QV}\\ \text{⇒}\text{QV=XT}\mathrm{......}\left(b\right)\\ \text{From }\left(a\right)\text{and}\left(b\right),\text{we have}\\ \text{SX}=\text{ST}\\ \text{⇒}\text{SX}\cong \text{ST}\end{array}$

Hence, $\Delta \text{XST }$ is an Isosceles triangle which has two congruent sides as SX and ST.