Question

write a paragraph proof of the following theorem. if ∠n and ∠m are congruent and supplementary, then ∠n and ∠m are right angles. because ∠n and ∠m are congruent, m∠m = m∠n by the definition of congruent angles. because ∠n and ∠m are supplementary, m∠m + m∠n = □. substitute m∠m for m∠n to get m∠m + m∠m = □, or 2m∠m = □. the division property of equality then yields m∠m = □. substitute m∠n for m∠m to get m∠n = □. by the definition of right angles, ∠n and ∠m are right angles. (do not include the degree symbols in your answer.)

Explanation:

Step1: Use the definition of supplementary angles

By the definition of supplementary angles, if two angles are supplementary, the sum of their measures is \(180\). So \(m\angle M + m\angle N=180\).

Step2: Substitute \(m\angle M\) for \(m\angle N\)

Since \(m\angle M = m\angle N\), substituting gives \(m\angle M+m\angle M = 180\), or \(2m\angle M=180\).

Step3: Use the Division Property of Equality

Dividing both sides of \(2m\angle M = 180\) by \(2\), we get \(m\angle M=\frac{180}{2}=90\).

Step4: Substitute \(m\angle N\) for \(m\angle M\)

Since \(m\angle M = m\angle N\), substituting \(m\angle N\) for \(m\angle M\) gives \(m\angle N = 90\).

Answer:

\(180\), \(180\), \(180\), \(90\), \(90\)