Question

letters a, b, c, and d are angles measures. which equation is enough information to prove that lines m and n are parallel lines cut by transversal p? select three options. a = c a = d c = d b + c = 180° b + d = 180°

Explanation:

Step1: Recall parallel - line angle relationships

If two parallel lines are cut by a transversal:

• Alternate exterior angles: If \(m\parallel n\), then \(a = d\) (alternate exterior angles).
• Alternate interior angles: If \(m\parallel n\), then \(c = d\) (alternate interior angles).
• Consecutive interior angles: If \(m\parallel n\), then \(b + c=180^{\circ}\) (consecutive interior angles are supplementary).

Step2: Analyze each option

• For \(a = c\): \(a\) and \(c\) are vertical angles. \(a = c\) is always true regardless of whether \(m\) and \(n\) are parallel.
• For \(a = d\): By the alternate - exterior - angle theorem, if \(a = d\), then \(m\parallel n\).
• For \(c = d\): By the alternate - interior - angle theorem, if \(c = d\), then \(m\parallel n\).
• For \(b + c=180^{\circ}\): By the consecutive - interior - angle theorem, if \(b + c = 180^{\circ}\), then \(m\parallel n\).
• For \(b + d=180^{\circ}\): There is no parallel - line angle relationship that directly gives \(b + d = 180^{\circ}\) when \(m\parallel n\).

Answer:

\(a = d\), \(c = d\), \(b + c=180^{\circ}\)