Question

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

1. given
2. base angles theorem
3. $angle1,angle2$ and $angle3,angle4$ are linear pairs.
4. $angle1,angle2$ and $angle3,angle4$ are supplementary.
5. if two angles are a linear pair, then they are supplementary.

Explanation:

Step1: State given information

We are given $\overline{PS}\cong\overline{QS}$ and $\overline{DP}\cong\overline{LQ}$.

Step2: Apply base - angles theorem

Since $\overline{PS}\cong\overline{QS}$, by the base - angles theorem, $\angle SPQ=\angle SQP$.

Step3: Define linear pairs

We note that $\angle1,\angle2$ and $\angle3,\angle4$ are linear pairs based on their geometric arrangement.

Step4: Use linear - pair property

Because they are linear pairs, $\angle1+\angle2 = 180^{\circ}$ and $\angle3+\angle4=180^{\circ}$.

Step5: Prove angle congruences

Using the equalities from the base - angles theorem and linear - pair supplementary relationships, we can show $\angle1=\angle4$ and $\angle2=\angle3$.

Step6: Prove side congruence

By SAS congruence criterion with the given side congruences and the proven angle congruences, we get $\overline{DS}\cong\overline{LS}$.

Step7: Conclude triangle type

Since $\overline{DS}\cong\overline{LS}$, by the definition of an isosceles triangle, $\triangle DSL$ is isosceles.

Answer:

1. $\overline{PS}\cong\overline{QS},\overline{DP}\cong\overline{LQ}$ (Given)
2. $\angle SPQ=\angle SQP$ (Base - Angles Theorem, since $\overline{PS}\cong\overline{QS}$)
3. Definition of linear - pairs (To state that $\angle1,\angle2$ and $\angle3,\angle4$ are linear pairs)
4. $\angle1 + \angle2=180^{\circ},\angle3+\angle4 = 180^{\circ}$ (If two angles are a linear pair, then they are supplementary)
5. $\angle1=\angle4,\angle2=\angle3$ (By subtraction property of equality from the linear - pair supplementary relationships and the fact that $\angle SPQ=\angle SQP$)
6. $\overline{DS}\cong\overline{LS}$ (By Side - Angle - Side (SAS) congruence criterion using $\overline{DP}\cong\overline{LQ}$, $\angle1=\angle4$, and $\overline{PS}\cong\overline{QS}$)
7. $\triangle DSL$ is isosceles (Definition of an isosceles triangle: a triangle with two congruent sides)