Math Problem Statement

Complete the two-column proof by filling in the blanks: Given: ∠S ≅ ∠P, ∠WSJ is a right angle Prove: ΔSJW ≅ ΔPJW

Solution

Let's work through this two-column proof and fill in the blanks step by step:

Given Information:

  • SP\angle S \cong \angle P
  • WSJ\angle WSJ is a right angle

Prove:

  • SJWPJW\triangle SJW \cong \triangle PJW

Step-by-step:

  1. Statement: SP\angle S \cong \angle P
    Reason: Given

  2. Statement: WSJ\angle WSJ is a right angle
    Reason: Given

  3. Statement: WJPWJS\angle WJP \cong \angle WJS
    Reason: The supplement is a right angle, and all right angles are congruent.

  4. Statement: WJWJ\overline{WJ} \cong \overline{WJ}
    Reason: Reflexive Property (a segment is congruent to itself).

  5. Statement: SJWPJW\triangle SJW \cong \triangle PJW
    Reason: Angle-Angle-Side (AAS) Congruence Theorem (two angles and the non-included side are congruent).

Here’s the complete proof:

StatementsReasons
SP\angle S \cong \angle P1. Given
WSJ\angle WSJ is a right angle2. Given
WJPWJS\angle WJP \cong \angle WJS3. The supplement is a right angle and right angles are congruent.
WJWJ\overline{WJ} \cong \overline{WJ}4. Reflexive Property
SJWPJW\triangle SJW \cong \triangle PJW5. AAS (Angle-Angle-Side) Congruence Theorem

Would you like further details or have any questions about the solution?

Here are 5 relative questions:

  1. How does the Angle-Angle-Side (AAS) theorem work in proving congruent triangles?
  2. What other congruence theorems exist aside from AAS?
  3. Why is the Reflexive Property important in geometry proofs?
  4. How do we identify corresponding parts in congruent triangles?
  5. What does it mean for angles to be supplements of each other?

Tip: When dealing with congruent triangles, always look for congruent angles and sides first to determine the best congruence theorem to apply.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Congruence
Angles

Formulas

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Theorems

Angle-Angle-Side (AAS) Congruence Theorem
Reflexive Property
Right Angle Congruence

Suitable Grade Level

Grades 8-10