Question

△xyz and △vwz are equilateral. complete the proof that ∠vyz ≅ ∠wxz.
1 △xyz is equilateral
2 △vwz is equilateral
3 overline{xz} ≅ overline{yz}
4 overline{vz} ≅ overline{wz}
5 ∠vzy ≅ ∠wzx
6 △vyz ≅ △wxz
7 ∠vyz ≅ ∠wxz

Explanation:

Step1: Given

Statement 1: $\triangle XYZ$ is equilateral. Reason: Given.
Statement 2: $\triangle VWZ$ is equilateral. Reason: Given.

Step2: Properties of equilateral triangle

Statement 3: $\overline{XZ}\cong\overline{YZ}$. Reason: All sides of an equilateral triangle are congruent ($\triangle XYZ$ is equilateral).
Statement 4: $\overline{VZ}\cong\overline{WZ}$. Reason: All sides of an equilateral triangle are congruent ($\triangle VWZ$ is equilateral).

Step3: Vertical angles

Statement 5: $\angle VZY\cong\angle WZX$. Reason: Vertical angles are congruent.

Step4: SAS (Side - Angle - Side) congruence

Statement 6: $\triangle VYZ\cong\triangle WXZ$. Reason: SAS ( $\overline{VZ}\cong\overline{WZ}$, $\angle VZY\cong\angle WZX$, $\overline{YZ}\cong\overline{XZ}$).

Step5: Corresponding parts of congruent triangles

Statement 7: $\angle VYZ\cong\angle WXZ$. Reason: Corresponding parts of congruent triangles are congruent (CPCTC).

Answer:

1. Given
2. Given
3. All sides of an equilateral triangle are congruent
4. All sides of an equilateral triangle are congruent
5. Vertical angles are congruent
6. SAS (Side - Angle - Side)
7. Corresponding parts of congruent triangles are congruent (CPCTC)