Question

t is the midpoint of \\(\overline{pr}\\). complete the proof that \\(\triangle pst \cong \triangle rqt\\).

	statement	reason
1	t is the midpoint of \\(\overline{pr}\\)	given
2	\\(\angle p \cong \angle r\\)	given
3	\\(\overline{pt} \cong \overline{rt}\\)
4	\\(\angle pts \cong \angle qtr\\)
5	\\(\triangle pst \cong \triangle rqt\\)

Explanation:

Step1: Recall Midpoint Definition

A midpoint of a segment divides it into two congruent segments.
Since \( T \) is the midpoint of \( \overline{PR} \), by the definition of a midpoint, \( \overline{PT} \cong \overline{RT} \).

Step2: Identify Reason for Step 3

For statement 3 (\( \overline{PT} \cong \overline{RT} \)), the reason is the definition of a midpoint (a midpoint divides a segment into two congruent segments).

Step3: Reason for Step 4

\( \angle PTS \) and \( \angle QTR \) are vertical angles. By the Vertical Angles Theorem, vertical angles are congruent. So \( \angle PTS \cong \angle QTR \) because vertical angles are congruent.

Step4: Reason for Step 5

We have \( \angle P \cong \angle R \) (given), \( \overline{PT} \cong \overline{RT} \) (midpoint definition), and \( \angle PTS \cong \angle QTR \) (vertical angles theorem). By the Angle - Side - Angle (ASA) Congruence Postulate, \( \triangle PST \cong \triangle RQT \).

Answer:

• For step 3: Reason is "Definition of a midpoint"
• For step 4: Reason is "Vertical Angles Theorem"
• For step 5: Reason is "ASA Congruence Postulate"