Question

In Exercises 9 and 10 you are given more information than you need. For each exercise state one piece of given information that you do not need for the proof. Then give a two–column proof that does not use that piece of information.

Given:$\overline{PQ}\cong \overline{PS};\overline{QR}\cong \overline{SR};\angle 1\cong \angle 2$

Prove:$\angle 3\cong \angle 4$

Short answer

$\angle 1\cong \angle 2$is not needed in given information to give proof.

Statement	Reason
1.$\overline{PQ}\cong \overline{PS}$	Given
2.$\overline{PR}\cong \overline{PR}$	Given
3.$\overline{QR}\cong \overline{SR}$	Given
4.$\mathrm{\Delta }PQR\cong \mathrm{\Delta }PSR$	SSS congruency criteria
5.$\angle 3\cong \angle 4$	corresponding parts of congruent triangle are congruent

Step-by-step solution

Step 1. Show that PR¯≅PR¯.

Since, line segment$\overline{PR}$ is common to both triangles, by relexive property it is congruent to itself

So,$\overline{PR}\cong \overline{PR}$

Step 2. Show that ΔPQR≅ΔPSR.

From step 1 using given statement$\mathrm{\Delta }PQR\cong \mathrm{\Delta }PSR$ by SSS (Side-Side-Side) congruency criteria

Step 3. Show that ∠3≅∠4.

Since, corresponding parts of congruent triangle are congruent

Thus,$\angle 3\cong \angle 4$

Also proof is provided without using $\angle 1\cong \angle 2$, so it is not needed in given information.