Chapter 7, Problem 3PE

To determine

To prove: $\Delta QRS\cong \Delta QTS$

Explanation of Solution

Given information: $\Delta QRU\cong \Delta QTV,\overline{US}\cong \overline{VS}$

Proof:

  

  $\Delta QRU\cong \Delta QTV$

  $\overline{QR}\cong \overline{QT}(i)$ (CPCT)

Here, CPCT denotes ‘’Corresponding parts of congruent triangles are congruent’’

  $\overline{RU}\cong \overline{TV}(ii)$ (CPCT)

Also,

  $\overline{US}\cong \overline{VS}(iii)$

Add both sides of $(ii)$ and $(iii)$

  $\begin{array}{l}\overline{RU}+\overline{US}\cong \overline{TV}+\overline{VS}\\ \Rightarrow \overline{RS}\cong \overline{TS}(iv)\end{array}$

  $\overline{QS}\cong \overline{QS}(v)$

(According to reflexive property of congruence, a line segment is congruent to itself)

From $(i),(iv),(v)$ ,

  $\Delta QRS\cong \Delta QTS$ (SSS congruence criteria)

(according to SSS congruence criteria, if all the sides of one triangle are congruent to the corresponding sides of the other triangle then the triangles are congruent)