Question

directions: complete each proof.

1. given: m∠4 + m∠7 = 180° prove: c || d
2. given: m∠3 = m∠8 prove: m∠3 + m∠6 = 180°

statements reasons
given
unit 3: parallel & perpendicular lines
homework 4: parallel line proofs
this is a 2 - page document!

Explanation:

Step1: Recall angle - relationships in parallel lines

When two parallel lines \(c\) and \(d\) are cut by a transversal, we use the properties of corresponding, alternate - interior, and same - side interior angles.

Step2: Analyze the given information

We know that if \(c\parallel d\), then \(\angle3\) and \(\angle7\) are corresponding angles, so \(m\angle3 = m\angle7\), and \(\angle4\) and \(\angle6\) are same - side interior angles, so \(m\angle4 + m\angle6=180^{\circ}\). Also, we are given \(m\angle3 = m\angle8\).

Step3: Substitute and prove

Since \(m\angle3 = m\angle7\) and \(m\angle3 = m\angle8\), then \(m\angle7 = m\angle8\).
We want to prove \(m\angle3 + m\angle6=180^{\circ}\).
Because \(m\angle3 = m\angle7\) and \(m\angle4 + m\angle6 = 180^{\circ}\), and \(\angle4\) and \(\angle7\) are vertical angles (\(m\angle4 = m\angle7\)).
Substituting \(m\angle3\) for \(m\angle7\) (since \(m\angle3 = m\angle7\)) in the equation \(m\angle4 + m\angle6=180^{\circ}\), we get \(m\angle3 + m\angle6 = 180^{\circ}\).

Answer:

The proof is completed as above, showing that \(m\angle3 + m\angle6 = 180^{\circ}\) using the properties of parallel lines and angle - relationships such as corresponding angles, vertical angles, and same - side interior angles.