Math Problem Statement

Complete the proof that triangle VYZ is congruent to triangle XWZ given that Z is the midpoint of VX and WY.

Solution

The proof you are working on is aimed at demonstrating that triangles VYZ\triangle VYZ and XWZ\triangle XWZ are congruent. Based on the information provided, here is how the proof can be completed step by step.

Complete Proof Steps:

  1. Statement: ZZ is the midpoint of VXVX.

    • Reason: Given.
  2. Statement: ZZ is the midpoint of WYWY.

    • Reason: Given.
  3. Statement: VZXZVZ \cong XZ.

    • Reason: Definition of midpoint (since ZZ is the midpoint of VXVX, the segments VZVZ and XZXZ are congruent).
  4. Statement: WZYZWZ \cong YZ.

    • Reason: Definition of midpoint (since ZZ is the midpoint of WYWY, the segments WZWZ and YZYZ are congruent).
  5. Statement: VZYWZX\angle VZY \cong \angle WZX.

    • Reason: Vertical Angle Theorem (the angles are vertical angles as the two lines cross at point ZZ).
  6. Statement: VYZXWZ\triangle VYZ \cong \triangle XWZ.

    • Reason: Side-Angle-Side (SAS) Congruence Theorem (using the congruent sides VZXZVZ \cong XZ, WZYZWZ \cong YZ, and the congruent angles VZYWZX\angle VZY \cong \angle WZX).

This completes the proof that VYZXWZ\triangle VYZ \cong \triangle XWZ.


Would you like more details or explanations on any step? Here are five related questions to expand on this:

  1. What is the Side-Angle-Side (SAS) Congruence Theorem?
  2. How does the Vertical Angle Theorem work in triangle congruence proofs?
  3. What is the significance of midpoints in proving triangle congruence?
  4. Could other triangle congruence postulates (like SSS or ASA) be applied to this problem?
  5. How does the congruence of these two triangles help in solving other geometric problems?

Tip: In many proofs, focusing on congruent angles and corresponding sides can often lead to a quicker resolution when using congruence theorems such as SAS or SSS.

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Math Problem Analysis

Mathematical Concepts

Congruence of Triangles
Geometric Proofs

Formulas

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Theorems

Midpoint Theorem
Vertical Angle Theorem
Side-Angle-Side (SAS) Congruence Theorem

Suitable Grade Level

Grades 8-10