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. 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, m∠1 = m∠3. angles with the same measure are

Explanation:

Step1: State given information

Given $\angle1$ and $\angle2$ are complementary, so $m\angle1 + m\angle2=90^{\circ}$. Also $\angle3$ and $\angle2$ are complementary, so $m\angle3 + m\angle2 = 90^{\circ}$.

Step2: Apply transitive property

Since $m\angle1 + m\angle2=90^{\circ}$ and $m\angle3 + m\angle2 = 90^{\circ}$, by the transitive property of equality, $m\angle1 + m\angle2=m\angle3 + m\angle2$.

Step3: Use subtraction property

Subtract $m\angle2$ from both sides of the equation $m\angle1 + m\angle2=m\angle3 + m\angle2$. By the subtraction - property of equality, $m\angle1=m\angle3$.

Step4: Recall congruent - angle definition

Angles with the same measure are congruent. Since $m\angle1 = m\angle3$, $\angle1\cong\angle3$.

Answer:

congruent