Questions: PQRS is a parallelogram with diagonals PR and QS intersecting at point T. Given PQ RS Definition of parallelogram triangle PQT congruent to triangle RST ASA congruency criteria QT / PT = ST / RT Corresponding parts of congruent triangles are congruent. QS bisects PR Definition of segment bisector Which statement and reason complete the proof? A. PS congruent to QR; Opposite sides of a parallelogram are congruent. B. angle QTR congruent to angle PTS; Vertical angles congruence theorem C. angle STR congruent to angle QTP : Vertical angles congruence theorem D. PQ approximately equal to RS; Opposite sides of a parallelogram are congruent.

PQRS is a parallelogram with diagonals PR and QS intersecting at point T. Given

PQ RS Definition of parallelogram

triangle PQT congruent to triangle RST ASA congruency criteria

QT / PT = ST / RT Corresponding parts of congruent triangles are congruent.

QS bisects PR Definition of segment bisector

Which statement and reason complete the proof?
A. PS congruent to QR; Opposite sides of a parallelogram are congruent.
B. angle QTR congruent to angle PTS; Vertical angles congruence theorem
C. angle STR congruent to angle QTP : Vertical angles congruence theorem
D. PQ approximately equal to RS; Opposite sides of a parallelogram are congruent.

Transcript text: PQRS is a parallelogram with diagonals $\overline{P R}$ and $\overline{\mathrm{QS}}$ intersecting at point T. Given $\overline{\mathrm{PQ}} \| \overline{\mathrm{RS}}$ Definition of parallelogram $\triangle \mathrm{PQT} \cong \triangle \mathrm{RST}$ ASA congruency criteria $\frac{\overline{Q T}}{\overline{P T}} = \frac{\overline{S T}}{R T}$ Corresponding parts of congruent triangles are congruent. $\overline{\mathrm{QS}}$ bisects $\overline{\mathrm{PR}}$ Definition of segment bisector Which statement and reason complete the proof? A. $\overline{P S} \cong \overline{Q R}$; Opposite sides of a parallelogram are congruent. B. $\angle \mathrm{QTR} \cong \angle \mathrm{PTS}$; Vertical angles congruence theorem C. $\angle S T R \cong \angle Q T P$ : Vertical angles congruence theorem D. $\overline{\mathrm{PQ}} \approx \overline{\mathrm{RS}}$; Opposite sides of a parallelogram are congruent.

Solution

Solution Steps

To complete the proof, we need to identify the missing statement and reason that logically follows from the given information and leads to the congruence of triangles $\triangle \mathrm{PQT}$ and $\triangle \mathrm{RST}$. Given that the diagonals of a parallelogram bisect each other, we can use the property of vertical angles to establish the congruence of the triangles.

Solution Approach

Identify the missing statement and reason that would help in proving the congruence of the triangles.
Use the property of vertical angles to establish the congruence.

Step 1: Identify the Missing Statement

To complete the proof, we need to find a statement that logically follows from the properties of the parallelogram and leads to the congruence of triangles $ \triangle PQT $ and $ \triangle RST $.

Step 2: Use Vertical Angles

Since $ \overline{PR} $ and $ \overline{QS} $ are diagonals of the parallelogram $ PQRS $ and intersect at point $ T $, the angles $ \angle QTR $ and $ \angle PTS $ are vertical angles. By the Vertical Angles Congruence Theorem, we have: \[ \angle QTR \cong \angle PTS \]

Step 3: Establish Triangle Congruence

With the information that $ \angle QTR \cong \angle PTS $ and the fact that $ \overline{PQ} \parallel \overline{RS} $ (which gives us alternate interior angles), we can apply the Angle-Side-Angle (ASA) congruency criterion. Thus, we conclude that: \[ \triangle PQT \cong \triangle RST \]