Question

activity 2: what do we know for sure about isosceles triangles?
draw _______________
segment pa is congruent to segment pb because of the definition of isosceles triangle.
angle _____________ is congruent to angle ___________ because _____________
segment pq is congruent to itself.
therefore, triangle apq is congruent to triangle bqp by the side-angle-side triangle congruence theorem.
therefore, _______________

Explanation:

Step1: Identify missing segment

Draw segment $PQ$ where $Q$ is the midpoint of $\overline{BA}$.

Step2: Identify congruent angles

Angle $\angle PAQ$ is congruent to angle $\angle PBQ$ because base angles of isosceles triangles are congruent.

Step3: State final congruence conclusion

Corresponding parts of congruent triangles are congruent, so segment $AQ$ is congruent to segment $BQ$ (or angle $\angle APQ \cong \angle BPQ$, or segment $AQ \cong BQ$).

Answer:

1. Draw: $\boldsymbol{\overline{PQ}}$ (where $Q$ is on $\overline{BA}$)
2. Angle $\boldsymbol{\angle PAQ}$ is congruent to angle $\boldsymbol{\angle PBQ}$ because base angles of isosceles triangles are congruent
3. Therefore, $\boldsymbol{\overline{AQ} \cong \overline{BQ}}$ (or $\boldsymbol{\angle APQ \cong \angle BPQ}$, or $\boldsymbol{\angle AQP \cong \angle BQP}$)