Substitution Congruent Supplements Theorem Law of Syllogism

Substitution Congruent Supplements Theorem Law of Syllogism

Linear Pairs Theorem Angle Addition Postulate Definition of Perpendicular Lines Definition of Linear Pair Vertical Angles Theorem Transitive Property of Equality

                    Statement

                   Reason

1

WZX=WZV

Given

2

WZX and WZV are a linear pair

3

MWZX + mWZV = 180°

4

MWZX +mWZX 180°

5

MWZX-90°

Substraction property of equality

6

WY 1 VX

The Correct Answer and Explanation is :

iturn0image0To complete the given proof, we’ll provide the missing statements and reasons, ensuring each step logically follows from the previous one.

Proof:

Statement	Reason
1. ∠WZX ≅ ∠WZV	Given
2. ∠WZX and ∠WZV are a linear pair.	Definition of Linear Pair
3. m∠WZX + m∠WZV = 180°	Linear Pair Theorem
4. m∠WZX + m∠WZX = 180°	Substitution (since ∠WZX ≅ ∠WZV)
5. 2 × m∠WZX = 180°	Simplification
6. m∠WZX = 90°	Division Property of Equality
7. WY ⊥ VX	Definition of Perpendicular Lines (since ∠WZX is 90°)

Explanation:

1. Given: We start with the information that ∠WZX is congruent to ∠WZV.
2. Definition of Linear Pair: The angles ∠WZX and ∠WZV are stated to form a linear pair. By definition, a linear pair consists of two adjacent angles whose non-common sides form a straight line.
3. Linear Pair Theorem: Angles that form a linear pair are supplementary, meaning their measures add up to 180 degrees. Therefore, m∠WZX + m∠WZV = 180°.
4. Substitution: Since ∠WZX is congruent to ∠WZV, their measures are equal. We can substitute m∠WZV with m∠WZX in the equation from step 3, resulting in m∠WZX + m∠WZX = 180°.
5. Simplification: Combining like terms, we get 2 × m∠WZX = 180°.
6. Division Property of Equality: To isolate m∠WZX, we divide both sides of the equation by 2, yielding m∠WZX = 90°.
7. Definition of Perpendicular Lines: An angle measuring 90 degrees is a right angle. If ∠WZX is a right angle, then by definition, the lines WY and VX are perpendicular to each other.

This proof demonstrates that if two congruent angles form a linear pair, each must be a right angle, leading to the conclusion that the lines forming these angles are perpendicular.