Question

complete the proof.
given: ∠1 is a complement of ∠2
∠2≅∠3
prove: ∠1 is a complement of ∠3
statements reasons
∠1 is a complement of ∠2 given
∠2≅∠3 given
m∠1 + m∠2 = 90° definition of complement
m∠2 = m∠3 definition of congruent angles
m∠1 + m∠3 = 90° substitution property of equality
∠1 is a complement of ∠3

Explanation:

Step1: Given $\angle1$ is complement of $\angle2$, so $m\angle1 + m\angle2=90^{\circ}$ by definition of complement.

Step2: Given $\angle2\cong\angle3$, so $m\angle2 = m\angle3$ by definition of congruent angles.

Step3: Substitute $m\angle2$ with $m\angle3$ in $m\angle1 + m\angle2 = 90^{\circ}$, getting $m\angle1 + m\angle3=90^{\circ}$ by substitution property of equality.

Step4: Since $m\angle1 + m\angle3 = 90^{\circ}$, $\angle1$ is a complement of $\angle3$ by definition of complementary angles.

Answer:

$\angle1$ is a complement of $\angle3$ because $m\angle1 + m\angle3 = 90^{\circ}$ by substitution and the definition of complementary - angles.