Question

$\angle t\cong\angle r$ and $\overline{qt}\cong\overline{ru}$. complete the proof that $\triangle pqt\cong\triangle pur$.

Explanation:

Step1: Identify Given Angles

Given \( \angle T \cong \angle R \), this is a given condition.

Step2: Identify Given Sides

Given \( \overline{QT} \cong \overline{RU} \), this is a given condition.

Step3: Identify Common Angle

\( \angle P \cong \angle P \) because of the Reflexive Property of Congruence (an angle is congruent to itself).

Step4: Apply Congruence Criterion

For \( \triangle PQT \) and \( \triangle PUR \), we have two angles and a non - included side? Wait, no. Wait, \( \angle T \cong \angle R \), \( \overline{QT} \cong \overline{RU} \), and \( \angle P \cong \angle P \). Wait, actually, the Angle - Angle - Side (AAS) congruence criterion states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent. Here, \( \angle T \cong \angle R \), \( \angle P \cong \angle P \), and \( \overline{QT} \cong \overline{RU} \), so by AAS, \( \triangle PQT \cong \triangle PUR \).

Now, let's fill in the reasons:

1. \( \angle T \cong \angle R \): Given
2. \( \overline{QT} \cong \overline{RU} \): Given
3. \( \angle P \cong \angle P \): Reflexive Property of Congruence
4. \( \triangle PQT \cong \triangle PUR \): AAS (Angle - Angle - Side) Congruence Criterion

Answer:

1. Reason for \( \angle T \cong \angle R \): Given
2. Reason for \( \overline{QT} \cong \overline{RU} \): Given
3. Reason for \( \angle P \cong \angle P \): Reflexive Property of Congruence
4. Reason for \( \triangle PQT \cong \triangle PUR \): AAS (Angle - Angle - Side) Congruence Criterion