Question

∠a and ∠b form a linear pair
∠a ≅ ∠b
∠c is a right angle
prove
∠a ≅ ∠c
statement

1. ∠a and ∠b form a linear pair
2. ∠a and ∠b are supplementary
3. ∠a ≅ ∠b
4. ∠a is a right angle
5. ∠c is a right angle
6. ∠a ≅ ∠c

reason

1. given
2. definition of a linear pair
3. given
4. ?
5. given
6. congruent right angles theorem

what geometric reason justifies the transition from statement 2 and 3 to statement 4?
congruent complements theorem
distributive property
congruent right angles theorem
vertical angles theorem

Explanation:

Step1: Recall the definition of supplementary angles

Supplementary angles sum to \(180^{\circ}\). Since \(\angle A\) and \(\angle B\) are supplementary (\(\angle A+\angle B = 180^{\circ}\)) and \(\angle A\cong\angle B\) (so \(\angle A=\angle B\)), we can substitute.

Step2: Solve for \(\angle A\)

Let \(\angle A = x\) and \(\angle B=x\). Then \(x + x=180^{\circ}\), \(2x = 180^{\circ}\), \(x = 90^{\circ}\). This is an application of the Congruent Complements Theorem (if two angles are congruent and supplementary, then each is a right angle). The Distributive Property (\(a(b + c)=ab+ac\)) is an algebraic property not relevant here. The Congruent Right Angles Theorem is used in step 6 (to show \(\angle A\cong\angle C\) given they are both right angles). The Vertical Angles Theorem (vertical angles are congruent) is about angles formed by two intersecting lines and not relevant to this pair - of - supplementary - and - congruent angles situation.

Answer:

Congruent Complements Theorem