Question

1. use the given two - column proof to write a paragraph proof. given: ∠1 and ∠2 are complementary. prove: ∠2 and ∠3 are complementary. statements: 1. ∠1 and ∠2 are complementary. 2. m∠1 + m∠2 = 90°. 3. ∠1 ≅ ∠3. 4. m∠1 = m∠3. 5. m∠3 + m∠2 = 90°. 6. ∠2 and ∠3 are complementary. reasons: 1. given. 2. def. of comp. ∠. 3. vert. ∠ thm. 4. def. of ≅ ∠. 5. subst. steps 2, 4. 6. def. of comp. ∠.

Explanation:

Step1: State the given

Given that $\angle1$ and $\angle2$ are complementary, so $m\angle1 + m\angle2=90^{\circ}$ (by the definition of complementary angles).

Step2: Use vertical - angle property

Since $\angle1\cong\angle3$ (Vertical - Angle Theorem), then $m\angle1 = m\angle3$ (by the definition of congruent angles).

Step3: Substitute

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

Step4: Apply the definition of complementary angles

Since $m\angle3 + m\angle2 = 90^{\circ}$, by the definition of complementary angles, $\angle2$ and $\angle3$ are complementary.

Answer:

The proof shows that if $\angle1$ and $\angle2$ are complementary and $\angle1\cong\angle3$, then $\angle2$ and $\angle3$ are complementary.