Question

directions: complete each proof. 1. given: m∠4 + m∠7 = 180°. prove: c || d statements reasons given

Explanation:

Step1: State the given information

Given that \(m\angle4 + m\angle7=180^{\circ}\)

Step2: Use the property of vertical - angles

Since \(\angle4\) and \(\angle6\) are vertical angles, \(m\angle4 = m\angle6\) (Vertical angles are congruent).

Step3: Substitute

Substitute \(m\angle4\) with \(m\angle6\) in the equation \(m\angle4 + m\angle7 = 180^{\circ}\), we get \(m\angle6 + m\angle7=180^{\circ}\)

Step4: Apply the same - side interior angles postulate

If the sum of same - side interior angles (\(\angle6\) and \(\angle7\)) formed by two lines \(c\) and \(d\) and a transversal is \(180^{\circ}\), then \(c\parallel d\) (If same - side interior angles are supplementary, then the lines are parallel)

Answer:

The proof is completed as follows:

Statements	Reasons
2. \(m\angle4=m\angle6\)	Vertical angles are congruent
3. \(m\angle6 + m\angle7 = 180^{\circ}\)	Substitution Property of Equality
4. \(c\parallel d\)	Same - side interior angles are supplementary, then the lines are parallel