Question

proof write a two - column proof.

1. given: ( mangle abc=mangle def )

( angle abc ) and ( angle def ) are supplementary.
prove: ( angle abc ) and ( angle def ) are right angles.

Explanation:

Step1: State the given information

Statements	Reasons
\(\angle ABC\) and \(\angle DEF\) are supplementary	Given

Step2: Use the definition of supplementary angles

By the definition of supplementary angles, if two angles \(\angle ABC\) and \(\angle DEF\) are supplementary, then \(m\angle ABC + m\angle DEF=180^{\circ}\)

Step3: Substitute \(m\angle ABC\) with \(m\angle DEF\)

Since \(m\angle ABC=m\angle DEF\), we can substitute \(m\angle ABC\) in the equation \(m\angle ABC + m\angle DEF=180^{\circ}\). Let \(x = m\angle ABC=m\angle DEF\). Then \(x + x=180^{\circ}\), or \(2x = 180^{\circ}\)

Step4: Solve for \(x\)

Divide both sides of the equation \(2x = 180^{\circ}\) by \(2\). We get \(x=\frac{180^{\circ}}{2}=90^{\circ}\)

Step5: Conclude the proof

Since \(x = m\angle ABC=m\angle DEF = 90^{\circ}\), by the definition of a right - angle (\(m\angle\theta=90^{\circ}\) implies \(\angle\theta\) is a right - angle), \(\angle ABC\) and \(\angle DEF\) are right angles

Answer:

Statements	Reasons
\(\angle ABC\) and \(\angle DEF\) are supplementary	Given
\(m\angle ABC + m\angle DEF=180^{\circ}\)	Definition of supplementary angles
\(2m\angle ABC=180^{\circ}\) (or \(2m\angle DEF = 180^{\circ}\))	Substitution (\(m\angle ABC=m\angle DEF\))
\(m\angle ABC = 90^{\circ}\), \(m\angle DEF=90^{\circ}\)	Division property of equality (\(x=\frac{180^{\circ}}{2}\))
\(\angle ABC\) and \(\angle DEF\) are right angles	Definition of a right - angle (\(m\angle\theta = 90^{\circ}\) implies \(\angle\theta\) is a right - angle)