Question

write a two - column proof.
given: $\angle 1\cong\angle 3$
$m\angle 3 = 90^{circ}$
prove: $\angle 2$ and $\angle 4$ are supplementary.

Explanation:

Step1: State the given information

Given: \( \angle 1\cong\angle 3\) and \(m\angle 3 = 90^{\circ}\)

Step2: Use the congruence property

Since \( \angle 1\cong\angle 3\), by the definition of congruent angles, \(m\angle 1=m\angle 3\). Substituting \(m\angle 3 = 90^{\circ}\), we get \(m\angle 1=90^{\circ}\)

Step3: Use the linear - pair postulate

\(\angle 1\) and \(\angle 2\) form a linear pair. By the linear - pair postulate, \(m\angle 1 + m\angle 2=180^{\circ}\). Substituting \(m\angle 1 = 90^{\circ}\), we have \(90^{\circ}+m\angle 2 = 180^{\circ}\), so \(m\angle 2=90^{\circ}\)

Step4: Use the linear - pair postulate again

\(\angle 3\) and \(\angle 4\) form a linear pair. By the linear - pair postulate, \(m\angle 3 + m\angle 4=180^{\circ}\). Substituting \(m\angle 3 = 90^{\circ}\), we get \(90^{\circ}+m\angle 4 = 180^{\circ}\), so \(m\angle 4=90^{\circ}\)

Step5: Calculate \(m\angle 2 + m\angle 4\)

\(m\angle 2+m\angle 4=90^{\circ}+90^{\circ}=180^{\circ}\)

Step6: Use the definition of supplementary angles

By the definition of supplementary angles (if \(m\angle A + m\angle B=180^{\circ}\), then \(\angle A\) and \(\angle B\) are supplementary), since \(m\angle 2 + m\angle 4 = 180^{\circ}\), \(\angle 2\) and \(\angle 4\) are supplementary

Answer:

Statements	Reasons
\(m\angle 1=m\angle 3\)	Definition of congruent angles
\(m\angle 1 = 90^{\circ}\)	Substitution property (\(m\angle 3 = 90^{\circ}\))
\(m\angle 1 + m\angle 2=180^{\circ}\)	Linear - pair postulate (\(\angle 1\) and \(\angle 2\) form a linear pair)
\(90^{\circ}+m\angle 2 = 180^{\circ}\), \(m\angle 2=90^{\circ}\)	Substitution (\(m\angle 1 = 90^{\circ}\)) and subtraction property of equality
\(m\angle 3 + m\angle 4=180^{\circ}\)	Linear - pair postulate (\(\angle 3\) and \(\angle 4\) form a linear pair)
\(90^{\circ}+m\angle 4 = 180^{\circ}\), \(m\angle 4=90^{\circ}\)	Substitution (\(m\angle 3 = 90^{\circ}\)) and subtraction property of equality
\(m\angle 2+m\angle 4=180^{\circ}\)	Substitution (\(m\angle 2 = 90^{\circ}\), \(m\angle 4 = 90^{\circ}\))
\(\angle 2\) and \(\angle 4\) are supplementary	Definition of supplementary angles (\(m\angle 2 + m\angle 4=180^{\circ}\))