Question

given: $overline{ps}congoverline{qs},overline{dp}congoverline{lq}
prove: \triangle dsl is isosceles.

1. \triangle dpscong\triangle lqs
2. corresponding parts of congruent tri - angles are congruent (c.p.c.t.c.)

Explanation:

Step1: Identify congruence criterion

In $\triangle DPS$ and $\triangle LQS$, we have two pairs of given congruent sides and a common side, so by SSS $\triangle DPS\cong\triangle LQS$.

Step2: Use C.P.C.T.C.

From the congruence of $\triangle DPS$ and $\triangle LQS$, we get $\overline{DS}\cong\overline{LS}$ as corresponding parts of congruent triangles.

Step3: Define isosceles triangle

A triangle with two congruent sides is isosceles. Since $\overline{DS}\cong\overline{LS}$ in $\triangle DSL$, $\triangle DSL$ is isosceles.

Answer:

1. By Side - Side - Side (SSS) congruence criterion since $\overline{PS}\cong\overline{QS}$, $\overline{DP}\cong\overline{LQ}$ and $\overline{DS}$ is common to both $\triangle DPS$ and $\triangle LQS$.
2. Since $\triangle DPS\cong\triangle LQS$, then $\overline{DS}\cong\overline{LS}$ (by C.P.C.T.C.).
3. $\triangle DSL$ is isosceles because it has two congruent sides $\overline{DS}$ and $\overline{LS}$.