Question

△wxy ≅ △wvz. complete the proof that ∠xzy ≅ ∠vyz.

1 △wxy ≅ △wvz
2 \overline{vz} ≅ \overline{xy}
3 \overline{vw} ≅ \overline{wx}
4 \overline{wz} ≅ \overline{wy}
5 vy = vw + wy
6 xz = wx + wz
7 vy = wx + wz
8 vy = xz
9 \overline{yz} ≅ \overline{yz}
10 △vyz ≅ △xzy
11 ∠xzy ≅ ∠vyz

Explanation:

Step1: Given

$\triangle WXY\cong\triangle WVZ$ (Given)

Step2: Corresponding parts of congruent triangles

Since $\triangle WXY\cong\triangle WVZ$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem:

• $\overline{VZ}\cong\overline{XY}$ (CPCTC)
• $\overline{VW}\cong\overline{WX}$ (CPCTC)
• $\overline{WZ}\cong\overline{WY}$ (CPCTC)

Step3: Segment addition postulate

• $VY = VW+WY$ (Segment Addition Postulate)
• $XZ = WX + WZ$ (Segment Addition Postulate)

Step4: Substitution

Since $VW = WX$ and $WY = WZ$ (from CPCTC), then $VY=WX + WZ$ (Substitution Property of Equality)

Step5: Transitive property of equality

Since $XZ = WX + WZ$ and $VY=WX + WZ$, then $VY = XZ$ (Transitive Property of Equality)

Step6: Reflexive property

$\overline{YZ}\cong\overline{YZ}$ (Reflexive Property of Congruence)

Step7: SSS Congruence

$\triangle VYZ\cong\triangle XZY$ (SSS Congruence: $\overline{VZ}\cong\overline{XY}$, $\overline{VY}\cong\overline{XZ}$, $\overline{YZ}\cong\overline{YZ}$)

Step8: CPCTC

$\angle XZY\cong\angle VYZ$ (CPCTC)

Answer:

1. Given
2. CPCTC
3. CPCTC
4. CPCTC
5. Segment Addition Postulate
6. Segment Addition Postulate
7. Substitution Property of Equality
8. Transitive Property of Equality
9. Reflexive Property of Congruence
10. SSS Congruence
11. CPCTC