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 = 90° and m∠3 + m∠2 = 90°. then, m∠1 + m∠2 = m∠3 + m∠2 by the transitive property of equality. subtract m∠2 from each side. by the subtraction property of equality.

Explanation:

Step1: State given information

Given $\angle1$ and $\angle2$ are complementary, $\angle3$ and $\angle2$ are complementary.

Step2: Apply definition of complementary angles

By definition, $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle2 = 90^{\circ}$.

Step3: Use transitive property of equality

Since both sums equal $90^{\circ}$, $m\angle1 + m\angle2=m\angle3 + m\angle2$.

Step4: Apply subtraction property of equality

Subtract $m\angle2$ from both sides: $m\angle1 + m\angle2 - m\angle2=m\angle3 + m\angle2 - m\angle2$, which simplifies to $m\angle1=m\angle3$.

Step5: Conclude angle - congruence

If two angles have equal measures, they are congruent. So $\angle1\cong\angle3$.

Answer:

$\angle1\cong\angle3$