Question

oving lines parallel
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? choose three correct answers.
b + c = 180°
a = c
c = d
a = d
b + d = 180°

Explanation:

Step1: Recall parallel line theorems

When two lines are cut by a transversal, corresponding angles, alternate interior angles, and consecutive interior angles (supplementary) can prove parallelism.

Step2: Analyze each option

• For \(a = c\):

\(a\) and \(c\) are not a standard angle - pair (like corresponding, alternate - interior) for parallel line determination.

• For \(c = d\):

\(c\) and \(d\) are not a standard angle - pair (like corresponding, alternate - interior) for parallel line determination.

• For \(b + c=180^{\circ}\):

\(b\) and \(c\) are consecutive interior angles. If consecutive interior angles are supplementary (\(b + c = 180^{\circ}\)), then lines \(m\) and \(n\) are parallel by the consecutive - interior - angles converse theorem.

• For \(a = d\):

\(a\) and \(d\) are alternate exterior angles. If alternate exterior angles are equal (\(a = d\)), then lines \(m\) and \(n\) are parallel by the alternate - exterior - angles converse theorem.

• For \(b + d=180^{\circ}\):

\(b\) and \(d\) are corresponding angles. If \(b + d = 180^{\circ}\), and since \(a + b=180^{\circ}\) (linear pair), then \(a = d\) (subtraction property of equality). So, by the alternate - exterior - angles converse (or we can also use the fact that if we rewrite \(b + d=180^{\circ}\) and know \(a + b = 180^{\circ}\), then \(a=d\) and use the corresponding - angles concept in a more complex way), lines \(m\) and \(n\) are parallel.

Answer:

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