Question

∠s and ∠q are complementary and ∠q and ∠r are complementary. complete the proof that ∠r ≅ ∠s.

	statement	reason
1	∠s and ∠q are complementary	given
2	∠q and ∠r are complementary	given
3	m∠s + m∠q = 90°
4	m∠q + m∠r = 90°	definition of complementary angles
5	m∠s + m∠q = m∠q + m∠r	transitive property of equality
6	m∠r = m∠s	properties of addition, subtraction, multiplication, and division
7	∠r ≅ ∠s

Explanation:

Step1: Determine Reason for Step 3

The definition of complementary angles states that if two angles are complementary, the sum of their measures is \(90^\circ\). Since \(\angle S\) and \(\angle Q\) are complementary (given in Step 1), the reason for \(m\angle S + m\angle Q = 90^\circ\) is the "Definition of complementary angles".

Step2: Determine Reason for Step 7

If two angles have equal measures (\(m\angle R = m\angle S\) from Step 6), then the angles are congruent. The reason for \(\angle R \cong \angle S\) is "Definition of congruent angles" (angles with equal measures are congruent).

Answer:

• Step 3 Reason: Definition of complementary angles
• Step 7 Reason: Definition of congruent angles