Math Problem Statement

Given: \( WZ \parallel XY \) & \( P \) is the midpoint of \( WY \). Prove: \( \triangle WZP \cong \triangle YXP \).

Solution

To prove that WZPYXP\triangle WZP \cong \triangle YXP given that WZXYWZ \parallel XY and PP is the midpoint of WYWY, let’s go through the steps and reasoning behind this congruency.

Solution

  1. Given Information:

    • WZXYWZ \parallel XY
    • PP is the midpoint of WYWY
  2. Goal:

    • Prove that WZPYXP\triangle WZP \cong \triangle YXP
  3. Congruency Strategy:

    • We can use the SAS (Side-Angle-Side) criterion to prove the triangles congruent.
  4. Step-by-Step Proof:

    • Step 1: Identify Corresponding Sides: Since PP is the midpoint of WYWY, we know that: WP=PYWP = PY This gives us one pair of equal sides.

    • Step 2: Identify Corresponding Angles: Since WZXYWZ \parallel XY and WPWP and PYPY serve as transversals, the alternate interior angles at PP are equal. So: WPZ=YPX\angle WPZ = \angle YPX This gives us one pair of equal angles.

    • Step 3: Identify Another Pair of Equal Sides: By the given information, WZXYWZ \parallel XY. Given that these two segments are opposite sides of the shape and both are parallel, they are also equal in length: WZ=XYWZ = XY

    • Conclusion: Using the SAS criterion (Side-Angle-Side), we have established:

      • WP=PYWP = PY (side),
      • WPZ=YPX\angle WPZ = \angle YPX (angle),
      • WZ=XYWZ = XY (side).

      Therefore, WZPYXP\triangle WZP \cong \triangle YXP by SAS.

Summary

We have proven that WZPYXP\triangle WZP \cong \triangle YXP by the SAS criterion, based on the information that WZXYWZ \parallel XY, PP is the midpoint of WYWY, and corresponding sides and angles are congruent.

Would you like further details or have any questions on this proof?


Here are some related questions for further exploration:

  1. What are other methods besides SAS that could potentially prove two triangles congruent?
  2. How does the parallel nature of WZWZ and XYXY contribute to the congruency of these triangles?
  3. Can congruency be established if PP was not the midpoint of WYWY?
  4. What is the significance of the midpoint in creating congruent triangles in geometry?
  5. How does the alternate interior angle theorem apply to this scenario?

Tip: In geometry, when dealing with parallel lines and transversals, always consider the relationships between angles they create, such as alternate interior or corresponding angles, which are often equal.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Congruence

Formulas

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Theorems

SAS (Side-Angle-Side) Congruence Theorem
Alternate Interior Angle Theorem

Suitable Grade Level

Grades 9-10