Question

Given:$\angle 1\cong \angle 2;\angle PQR\cong \angle SRQ$

Prove: $\overline{PR}\cong \overline{SQ}$

Short answer

By ASA postulate it is proven that$\overline{PR}\cong \overline{SQ}$.

Step-by-step solution

- Define Congruent Triangles 

The two triangles are said to be congruent if they are copies of each other and if their vertices are superposed, then say that corresponding angles and the sides of the triangles are congruent.

Congruency of triangles can be proved by SSS, SAS and ASA, AAS and HL postulates.

- State the Congruency method used 

ASA - Angle – side – angle method means if two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, then triangles are congruent.

- State the proof 

To prove that two segments or two angles are congruent, firstly identity the two triangles in which that two segments or angles are corresponding parts. After that, prove that the identified triangles are congruent and then, state that the two parts are congruent using the reason “Corresponding parts of triangles are ”.

Consider the:$\Delta PQR\text{and}\Delta \text{SRQ}$

i. It is given that, $\angle PRQ\cong \angle SQR$

ii By reflexive property,$\overline{QR}\cong \overline{RQ}$

iii Also, it is given that$\angle PQR\cong \angle SRQ$ .

So, by ASA Postulate,$\Delta PQR\cong \Delta \text{SRQ}$.

Since two triangles are congruent, therefore, it can be concluded that$\overline{PR}\cong \overline{SQ}$ .