Question

∠x and ∠y are supplementary
∠z and ∠y are supplementary
∠w ≅ ∠x
∠t ≅ ∠z
prove
∠w ≅ ∠t
statement

1. ∠x and ∠y are supplementary
2. ∠z and ∠y are supplementary
3. ∠x ≅ ∠z
4. ∠w ≅ ∠x
5. ∠t ≅ ∠z
6. ∠w ≅ ∠t

reason

1. given
2. given
3. ?
4. given
5. given
6. transitive

property

what geometric reason justifies the transition
from statement 1 and 2 to statement 3?

congruent right angles theorem

congruent supplements theorem

transitive property

vertical angle theorem

Explanation:

The Congruent Supplements Theorem states that if two angles are supplementary to the same angle (or to congruent angles), then the two angles are congruent. Here, \(\angle X\) and \(\angle Z\) are both supplementary to \(\angle Y\). So, by the Congruent Supplements Theorem, \(\angle X\cong\angle Z\). The Congruent Right Angles Theorem is about right - angled congruence, the Transitive Property (\(a = b,b = c\Rightarrow a = c\)) is used for a different kind of relationship (not specifically for supplementary angles in this context), and the Vertical Angle Theorem is about angles formed by intersecting lines (not relevant here as we are dealing with supplementary angles).

Answer:

B. Congruent Supplements Theorem