∠1 ≅ ∠3
15b.
a. Congruent Supplements Theorem
b. Congruent Complements Theorem
c. Vertical angles are congruent
The statement "∠1 ≅ ∠3" means that angle 1 is congruent (or equal) to angle 3. In geometry, there are different theorems and properties that can be used to prove the congruence of angles.
To determine which theorem applies in this case, we need to analyze the given statement and the answer choices provided.
a. Congruent Supplements Theorem: This theorem states that if two angles are supplements of the same angle (or congruent angles), then they are congruent. Since there is no information provided about angle measurements or their relationship to other angles, this theorem does not apply.
b. Congruent Complements Theorem: This theorem states that if two angles are complements of the same angle (or congruent angles), then they are congruent. Similar to the previous theorem, there is no information given about the relationships between angles as supplements or complements. Thus, this theorem does not apply either.
c. Vertical angles are congruent: Vertical angles are formed by two intersecting lines and are always congruent. This means that if angles 1 and 3 are vertical angles, then they are automatically congruent. This theorem fits the given statement, "∠1 ≅ ∠3," which states that the two angles are congruent.
Therefore, the correct answer is c. Vertical angles are congruent.