Question

$$overline{xy} cong overline{wy}$$ and $$overline{uy} cong overline{vy}$$. complete the proof that $$angle u cong angle v$$.

	statement	reason
2	$$overline{uy} cong overline{vy}$$
3	$$angle uyx cong angle vyw$$
4	$$\triangle uxy cong \triangle vwy$$
5	$$angle u cong angle v$$

Explanation:

Step1: Identify the given information

We are given that \(\overline{XY}\cong\overline{WY}\) (Statement 1) and \(\overline{UY}\cong\overline{VY}\) (Statement 2). These are the side - side equalities.

Step2: Use the Vertical Angles Theorem

\(\angle UYX\) and \(\angle VYW\) are vertical angles. By the Vertical Angles Theorem, \(\angle UYX\cong\angle VYW\) (Statement 3).

Step3: Prove triangle congruence

We have two sides and the included angle (SAS - Side - Angle - Side congruence criterion) for \(\triangle UXY\) and \(\triangle VWY\). Since \(\overline{XY}\cong\overline{WY}\), \(\angle UYX\cong\angle VYW\), and \(\overline{UY}\cong\overline{VY}\), by the SAS congruence criterion, \(\triangle UXY\cong\triangle VWY\) (Statement 4).

Step4: Use the CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle UXY\cong\triangle VWY\), their corresponding angles \(\angle U\) and \(\angle V\) are congruent. By CPCTC, \(\angle U\cong\angle V\) (Statement 5).

Answer:

1. Given
2. Given
3. Vertical Angles Theorem
4. SAS (Side - Angle - Side) Congruence Criterion
5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)