Question

prove the congruent complements theorem. if two angles are complements of the same angle, then the two angles are congruent. given: ∠1 and ∠2 are complementary ∠3 and ∠2 are complementary choose the correct statement of what is to be proved. prove: ∠1≅∠3 proof: ∠1 and ∠2 are complementary and ∠3 and ∠2 are complementary because it is given. by the definition of complementary angles, m∠1 + m∠2 = □° and m∠3 + m∠2 = □°

Explanation:

Step1: Recall definition of complementary angles

By the definition of complementary angles, if two angles are complementary, the sum of their measures is 90 degrees. So, $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle2 = 90^{\circ}$ since $\angle1$ and $\angle2$ are complementary and $\angle3$ and $\angle2$ are complementary.

Step2: Set up equations

We have the equations $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle2=90^{\circ}$. Let $m\angle1 + m\angle2=m\angle3 + m\angle2$.

Step3: Subtract $m\angle2$ from both sides

Subtracting $m\angle2$ from both sides of the equation $m\angle1 + m\angle2=m\angle3 + m\angle2$ gives $m\angle1=m\angle3$.

Step4: Use the definition of congruent angles

Since $m\angle1=m\angle3$, by the definition of congruent angles (angles with equal measures are congruent), $\angle1\cong\angle3$.

Answer:

The blanks should be filled with 90 and 90 respectively, and the proof shows that if two angles are complements of the same angle, they are congruent.