Question

1. fill in the missing information in the proof below.

given: ∠t ≅ ∠r, \\(\overline{pq} ≅ \overline{pv}\\)
prove: ∠pqt ≅ ∠pvr

statements	reasons
2) ∠tpq ≅ ∠rpv	2)
3) \\(\triangle tpq ≅ \triangle rpv\\)	3)
4) ∠pqt ≅ ∠pvr	4)

Explanation:

Step1: Identify Reason for Statement 1

The first statement is given in the problem, so the reason is "Given".

Step2: Identify Reason for Statement 2

$\angle TPQ$ and $\angle RPV$ are the same angle (they share the vertex $P$ and are the included angles between the given sides/angles), so this is the "Reflexive Property of Congruence" (or "Common Angle").

Step3: Identify Reason for Statement 3

We have two angles and a side: $\angle T\cong\angle R$, $\overline{PQ}\cong\overline{PV}$, and $\angle TPQ\cong\angle RPV$. This is the "ASA (Angle - Side - Angle) Congruence Postulate".

Step4: Identify Reason for Statement 4

Since $\triangle TPQ\cong\triangle RPV$, their corresponding angles $\angle PQT$ and $\angle PVR$ are congruent. This is "Corresponding Parts of Congruent Triangles are Congruent (CPCTC)".

Answer:

1. Given
2. Reflexive Property (Common Angle)
3. ASA Congruence Postulate
4. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)