Question

1. given ∠1 is a complement of ∠2. ∠2≅∠3

prove ∠1 is a complement of ∠3.
statements

1. ∠1 is a complement of ∠2.
2. ∠2≅∠3
3. m∠1 + m∠2 = 90°
4. m∠2 = m∠3

5.

1. ∠1 is a complement of ∠3.

reasons

1. given

2.
3.

1. definition of congruent angles
2. substitution property of equality

6.

Explanation:

Step1: State given congruence

Given

Step2: Recall definition of complementary angles

If two angles are complementary, their measures sum to 90°. So if ∠1 is a complement of ∠2, then $m\angle1 + m\angle2=90^{\circ}$.

Step3: Use congruence - angle measure relation

Since ∠2≅∠3, by the definition of congruent angles, $m\angle2 = m\angle3$.

Step4: Substitute equal - angle measures

Substitute $m\angle2$ with $m\angle3$ in the equation $m\angle1 + m\angle2 = 90^{\circ}$. We get $m\angle1 + m\angle3=90^{\circ}$.

Step5: Recall definition of complementary angles again

If the sum of the measures of two angles is 90°, then the two angles are complementary. Since $m\angle1 + m\angle3 = 90^{\circ}$, ∠1 is a complement of ∠3.

Answer:

1. Given; 3. Definition of complementary angles; 5. $m\angle1 + m\angle3=90^{\circ}$; 6. Definition of complementary angles