Question

Decide whether the two triangles must be congruent. If so, write the congruence and name the postulate used. If not, write no congruence can be deduced.

Short answer

$\Delta TRP\cong \Delta TRS$ by ASA (Angle-Side-Angle) Postulate

Step-by-step solution

Step 1. First observation from figure.

Line segment$\overline{TR}$ is common in both triangles.

So, by reflexive property of congruency

$\overline{TR}\cong \overline{TR}$

Step 2. Find ∠TRP.

Since sum of angles of a triangle is $180°$

$\begin{array}{c}\angle TRP+\angle RPT+\angle PTR=180°\\ \angle TRP+60°+50°=180°\\ \angle TRP=180°-110°\\ \angle TRP=70°\end{array}$

Step 3. Find ∠RTS.

Since sum of angles of a triangle is$180°$

$$\begin{gathered} \angle RTS+\angle TSR+\angle SRT=180° \\ \angle RTS+60°+70°=180° \\ \angle RTS=180°-130° \\ \angle RTS=50° \end{gathered}$$

Step 4. Observations from figure and above steps.

$\angle TRP$and $\angle TRS$are congruent

So, $\angle TRP\cong \angle TRS$

Also, $\angle PTR$and$\angle STR$ are congruent

So,$\angle PTR\cong \angle STR$

Step 5. Conclude given two triangles are congruent or not.

From step 1 and 4

Two angles and there including side is congruent

Thus,$\Delta TRP\cong \Delta TRS$ by ASA (Angle-Side-Angle) Postulate