Question

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

1. $angle1$, $angle2$ and $angle3$, $angle4$ are supplementary.
2. if two angles are a linear pair, then they are supplementary.
3. click here to insert
4. click here to insert
5. $overline{dp}congoverline{lq}$
6. click here to insert

Explanation:

Step1: Identify congruent parts

We know \(\overline{PS}\cong\overline{QS}\) and \(\overline{DP}\cong\overline{LQ}\) from the given. Also, linear - pair properties and vertical - angle property are used to find equal angles.

Step2: Prove triangle congruence

With \(\angle 1=\angle 4\), \(\overline{PS}\cong\overline{QS}\), \(\overline{DP}\cong\overline{LQ}\), by SAS, \(\triangle DPS\cong\triangle LQS\).

Step3: Prove the triangle is isosceles

Since \(\triangle DPS\cong\triangle LQS\), corresponding parts of congruent triangles are congruent, so \(\overline{DS}\cong\overline{LS}\), making \(\triangle DSL\) isosceles.

Step4: State the given fact

\(\overline{DP}\cong\overline{LQ}\) is given information.

Answer:

1. In \(\triangle DPS\) and \(\triangle LQS\), since \(\overline{PS}\cong\overline{QS}\), \(\overline{DP}\cong\overline{LQ}\) and \(\angle 1 + \angle 2=\angle 3+\angle 4 = 180^{\circ}\) (linear - pair supplementary), and \(\angle 2\) and \(\angle 3\) are vertical angles so \(\angle 2=\angle 3\), then \(\angle 1=\angle 4\). By Side - Angle - Side (SAS) congruence criterion, \(\triangle DPS\cong\triangle LQS\). So \(\overline{DS}\cong\overline{LS}\) and \(\triangle DSL\) is isosceles.
2. Given.