Question

given
∠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 right angles theorem
vertical angles theorem
distributive property
congruent complements theorem

Explanation:

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 \(\angle B\) with \(\angle A\) in the supplementary - angle equation: \(\angle A+\angle A=180^{\circ}\), \(2\angle A = 180^{\circ}\), \(\angle A = 90^{\circ}\). This is based on the definition of supplementary angles (\(\angle A+\angle B = 180^{\circ}\)) and substitution (because \(\angle A=\angle B\)). The Distributive Property \(a(b + c)=ab+ac\) (in the form \(1\times\angle A+1\times\angle B=(1 + 1)\times\angle A\) when \(\angle A=\angle B\)) is used to simplify \(\angle A+\angle B = 180^{\circ}\) to \(2\angle A=180^{\circ}\).

Answer:

Distributive Property