Question

t is the midpoint of pr and qs. complete the proof that ∠p ≅ ∠r.

(image of two triangles intersecting at t, with vertices q, r, t and p, s, t)

	statement	reason
1	t is the midpoint of \\(\overline{pr}\\)	given
2	t is the midpoint of \\(\overline{qs}\\)	given
3	\\(\overline{pt} \cong \overline{rt}\\)	definition of midpoint
4	\\(\overline{qt} \cong \overline{st}\\)	definition of midpoint
5	\\(\angle pts \cong \angle qtr\\)	vertical angle theorem
6	blank	dropdown

Explanation:

Step1: Identify Congruent Triangles

We have \( \overline{PT} \cong \overline{RT} \) (step 3), \( \overline{QT} \cong \overline{ST} \) (step 4), and \( \angle PTS \cong \angle QTR \) (step 5, vertical angles). By the SAS (Side - Angle - Side) congruence criterion, \( \triangle PTS \cong \triangle RTQ \).

Step2: Corresponding Angles of Congruent Triangles

In congruent triangles, corresponding angles are congruent. Since \( \triangle PTS \cong \triangle RTQ \), the corresponding angles \( \angle P \) (in \( \triangle PTS \)) and \( \angle R \) (in \( \triangle RTQ \)) are congruent. So the statement for step 6 should be \( \triangle PTS \cong \triangle RTQ \) (by SAS), and then we can conclude \( \angle P \cong \angle R \) from the congruence of triangles. But for the statement in step 6 (to lead to \( \angle P \cong \angle R \)), the key triangle congruence statement is \( \triangle PTS \cong \triangle RTQ \).

Answer:

\( \triangle PTS \cong \triangle RTQ \) (with reason "SAS Congruence Criterion" for the next step, but the statement here is the triangle congruence)