Question

given
∠a and ∠b are complementary
∠c ≅ ∠b
∠d and ∠c are complementary
prove
∠a ≅ ∠d
statement

1. ∠a and ∠b are complementary
2. ∠c ≅ ∠b
3. ∠d and ∠c are complementary
4. ∠d and ∠b are complementary

reason

1. given
2. given
3. given
4. substitution

what geometric reason justifies the transition from statement 1 and 4 to statement 5?
congruent complements theorem
vertical angles theorem
supplemental right angles theorem
congruent right angles theorem

Explanation:

Step1: Understand Complementary Angles

Complementary angles sum to \(90^{\circ}\). So \(\angle A+\angle B = 90^{\circ}\) (from statement 1) and \(\angle D+\angle B=90^{\circ}\) (from statement 4).

Step2: Apply Congruent Complements Theorem

The Congruent Complements Theorem states that if two angles are complementary to the same angle (or congruent angles), then the two angles are congruent. Here, \(\angle A\) and \(\angle D\) are both complementary to \(\angle B\). So \(\angle A\cong\angle D\).

Answer:

Congruent Complements Theorem