Question

which set of equations is enough information to prove that lines a and b are parallel lines cut by transversal f? ( mangle1 = 110^{circ}) and ( mangle2 = 110^{circ}) ( mangle2 = 110^{circ}) and ( mangle3 = 110^{circ}) ( mangle4 = 110^{circ}) and ( mangle3 = 70^{circ}) ( mangle1 = 110^{circ}) and ( mangle3 = 70^{circ})

Explanation:

Step1: Recall the properties of parallel lines

If two parallel lines are cut by a transversal, then consecutive - interior angles are supplementary (\(m\angle2 + m\angle3=180^{\circ}\)), alternate - interior angles are equal (\(m\angle2 = m\angle4\)), and corresponding angles are equal. Also, if \(m\angle1 + m\angle3 = 180^{\circ}\), we can use the converse of the consecutive - interior angles theorem.

Step2: Analyze each option

• Option 1 (\(m\angle1 = 110^{\circ}\) and \(m\angle2 = 110^{\circ}\)):

This only tells us about two angles that are not in a position (like corresponding, alternate - interior, or consecutive - interior) to prove \(a\parallel b\).

• Option 2 (\(m\angle2 = 110^{\circ}\) and \(m\angle3 = 110^{\circ}\)):

This also does not fit the angle - relationship (corresponding, alternate - interior, consecutive - interior) criteria for proving parallel lines.

• Option 3 (\(m\angle4 = 110^{\circ}\) and \(m\angle3 = 70^{\circ}\)):

Since \(m\angle4+m\angle3 = 110^{\circ}+70^{\circ}=180^{\circ}\), and \(\angle4\) and \(\angle3\) are consecutive - interior angles. By the converse of the consecutive - interior angles theorem, if the sum of consecutive - interior angles is \(180^{\circ}\), then the two lines (\(a\) and \(b\)) are parallel.

• Option 4 (\(m\angle1 = 110^{\circ}\) and \(m\angle3 = 70^{\circ}\)):

\(\angle1\) and \(\angle3\) are not in a standard angle - pair (corresponding, alternate - interior, consecutive - interior) relationship to prove \(a\parallel b\).

Answer:

\(m\angle4 = 110^{\circ}\) and \(m\angle3 = 70^{\circ}\)