Question

given: r is the midpoint of \\(\overline{qt}\\) and \\(\overline{ps}\\)
prove : \\(\triangle qrp \cong \triangle trs\\)?

Explanation:

Step1: Identify Midpoint Properties

Since \( R \) is the midpoint of \( \overline{QT} \) and \( \overline{PS} \), we have \( QR = RT \) and \( PR = RS \) (Definition of midpoint: a midpoint divides a segment into two congruent segments).

Step2: Identify Vertical Angles

\( \angle QRP \) and \( \angle TRS \) are vertical angles. Vertical angles are congruent, so \( \angle QRP \cong \angle TRS \).

Step3: Apply SAS Congruence

In \( \triangle QRP \) and \( \triangle TRS \):

• \( QR = RT \) (from Step 1)
• \( \angle QRP = \angle TRS \) (from Step 2)
• \( PR = RS \) (from Step 1)

By the Side - Angle - Side (SAS) Congruence Postulate, 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 \( \triangle QRP \cong \triangle TRS \) by SAS.

Answer:

\( \triangle QRP \cong \triangle TRS \) (proven by SAS congruence postulate using midpoint properties and vertical angles)