Question

given
∠x and ∠y are supplementary
∠x ≅ ∠y
∠z is a right angle
prove
∠x ≅ ∠z
what geometric reason justifies the transition from statement 1 and 2 to statement 3?
congruent complements theorem
symmetric property
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 congruence of \(\angle X\) and \(\angle Y\)

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

Step3: Analyze the theorems

• Congruent Complements Theorem: Deals with angles that are complements (sum to \(90^{\circ}\)), not relevant here as we have supplementary angles.
• Symmetric Property: \(a = b\) implies \(b = a\), not relevant to the angle - measure calculation.
• Vertical Angles Theorem: Deals with angles formed by two intersecting lines, not relevant here.
• Supplemental Right Angles Theorem: If two congruent angles are supplementary, then each is a right angle. This is exactly what we have with \(\angle X\) and \(\angle Y\) (supplementary and congruent, so each is \(90^{\circ}\)).

Answer:

D. Supplemental Right Angles Theorem