Question

technology nevaeh has developed a new timer app. the icon for the app contains an hourglass that can be modeled by two triangles. if r is the midpoint of \\(\overline{qs}\\) and \\(\overline{pt}\\), complete the paragraph proof to prove \\(\triangle prq \cong \triangle trs\\).
proof:
because r is the midpoint of \\(\overline{qs}\\) and \\(\overline{pt}\\), \\(pr \cong rt\\) and \\(rq \cong rs\\) by definition of a midpoint. \\(\angle prq \cong \angle trs\\) by the \\(\boldsymbol{\text{select choice}}\\). so, \\(\triangle prq \cong \triangle trs\\) by \\(\boldsymbol{\text{select choice}}\\).

Explanation:

Step1: Identify the angle relationship

Angles ∠PRQ and ∠TRS are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So ∠PRQ ≅ ∠TRS because they are vertical angles.

Step2: Identify the triangle congruence criterion

We have PR ≅ RT (given by midpoint of PT), RQ ≅ RS (given by midpoint of QS), and ∠PRQ ≅ ∠TRS (vertical angles). This satisfies the Side - Angle - Side (SAS) congruence criterion, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. So △PRQ ≅ △TRS by SAS.

Answer:

First blank: Vertical Angles Theorem (∠PRQ and ∠TRS are vertical angles, so they are congruent)
Second blank: SAS (Side - Angle - Side) Congruence Criterion