Question

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

Explanation:

Step1: Identify Midpoint Segments

Since \( R \) is the midpoint of \( \overline{QT} \), we have \( QR = TR \) (Definition of midpoint: a midpoint divides a segment into two equal parts). Similarly, as \( R \) is the midpoint of \( \overline{PS} \), \( PR = SR \).

Step2: Identify Vertical Angles

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

Step3: Apply SAS Congruence

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

• \( QR = TR \) (from Step 1, midpoint definition)
• \( \angle QRP = \angle TRS \) (from Step 2, vertical angles theorem)
• \( PR = SR \) (from Step 1, midpoint definition)

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 triangles are congruent. So, \( \triangle QRP \cong \triangle TRS \) by SAS.

Answer:

To prove \( \triangle QRP\cong\triangle TRS \):

Statement	Reason
2. \( PR = SR \)	\( R \) is the midpoint of \( \overline{PS} \) (Definition of midpoint)
3. \( \angle QRP=\angle TRS \)	Vertical Angles Theorem (Vertical angles are congruent)
4. \( \triangle QRP\cong\triangle TRS \)	SAS Congruence Postulate (from 1, 3, 2)

The triangles \( \triangle QRP \) and \( \triangle TRS \) are congruent by the SAS Congruence Postulate.