Question

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

Explanation:

Step1: Use the property 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 \(m\angle M = m\angle N\), we substitute \(m\angle M\) into \(m\angle M + m\angle N = 180\). We get \(m\angle M+m\angle M = 180\), or \(2m\angle M=180\).

Step3: Solve for \(m\angle M\)

Using the Division Property of Equality, divide both sides of \(2m\angle M = 180\) by \(2\). So \(m\angle M=\frac{180}{2}=90\).

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

Since \(m\angle M = m\angle N\), when \(m\angle M = 90\), then \(m\angle N=90\)

Answer:

180, 180, 180, 90, 90