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 complementary - angle definition

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: Use substitution property

From $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle2 = 90^{\circ}$, we can set $m\angle1 + m\angle2=m\angle3 + m\angle2$ (substitution property, as both sums equal 90).

Step3: Subtract $m\angle2$ from both sides

Subtract $m\angle2$ from both sides of the equation $m\angle1 + m\angle2=m\angle3 + m\angle2$. We get $m\angle1=m\angle3$.

Step4: Recall congruent - angle definition

Since $m\angle1=m\angle3$, by the definition of congruent angles (two angles are congruent if and only if their measures are equal), $\angle1\cong\angle3$.

Answer:

The blanks should be filled with 90 and 90 respectively. And the proof shows that if $\angle1$ and $\angle2$ are complementary and $\angle3$ and $\angle2$ are complementary, then $\angle1\cong\angle3$.