Question

given
∠x and ∠y are supplementary
∠x ≅ ∠y
∠z is a right angle
prove
∠x ≅ ∠z
statement

1. ∠x and ∠y are supplementary
2. ∠x ≅ ∠y
3. ∠x and ∠y are right angles
4. ∠z is a right angle
5. ∠x ≅ ∠z

reason

1. given
2. given
3. ?
4. given
5. congruent right angles theorem

what geometric reason justifies the transition from statement 1 and 2 to statement 3?
symmetric property
congruent complements theorem
vertical angles theorem
supplemental right angles theorem

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles sum to \(180^{\circ}\). So, \(\angle X+\angle Y = 180^{\circ}\).

Step2: Use the congruent relationship

Since \(\angle X\cong\angle Y\), let \(\angle X=\angle Y = x\). Then \(x + x=180^{\circ}\), \(2x = 180^{\circ}\), \(x = 90^{\circ}\).

Step3: Analyze the theorems

• Symmetric Property: If \(a = b\), then \(b=a\). Not relevant here.
• Congruent Complements Theorem: If two angles are complements of the same or congruent angles, then the two angles are congruent. But we are dealing with supplements.
• Vertical Angles Theorem: Vertical angles are congruent. Not relevant.
• Supplemental Right Angles Theorem: If two congruent angles are supplementary, then each is a right angle. This matches our calculation (\(\angle X=\angle Y = 90^{\circ}\)).

Answer:

D. Supplemental Right Angles Theorem