Question

12 fill in the blank 1 point
lines a and b are parallel lines cut by a transversal.
if the ( m angle 1 ) is ( 57^{circ} ), find the measure of ( angle 6 ).
( m angle 6= ) type your answer_ degrees

Explanation:

Step1: Find the measure of ∠3

Since ∠1 and ∠3 are vertical angles, they are equal. So, \(m\angle3 = m\angle1=57^{\circ}\).

Step2: Use the property of consecutive - interior angles

Lines A and B are parallel. ∠3 and ∠6 are consecutive - interior angles. But wait, another approach: ∠3 and ∠5 are alternate - interior angles (\(m\angle3 = m\angle5 = 57^{\circ}\) as alternate - interior angles for parallel lines A and B cut by a transversal). And ∠5 and ∠6 are supplementary (\(\angle5+\angle6 = 180^{\circ}\))? No, wrong. Wait, ∠3 and ∠5 are alternate - interior angles. Then ∠5 and ∠6 are adjacent angles forming a linear pair? No. Wait, correct approach: ∠1 and ∠3 are vertical angles (\(m\angle3=57^{\circ}\)). ∠3 and ∠5 are alternate - interior angles (so \(m\angle5 = 57^{\circ}\)). ∠5 and ∠6 are supplementary (linear pair: \(m\angle5 + m\angle6=180^{\circ}\))? No, wrong. Wait, correct: ∠1 and ∠4 are supplementary (\(m\angle1 + m\angle4=180^{\circ}\)), \(m\angle4 = 123^{\circ}\). ∠4 and ∠6 are alternate - exterior angles (for parallel lines A and B cut by a transversal). So \(m\angle6=m\angle4\).

Another way:
Since \(m\angle1 = 57^{\circ}\), then \(m\angle3=57^{\circ}\) (vertical angles).
Since lines A and B are parallel, \(m\angle3+m\angle6 = 180^{\circ}\) (consecutive - interior angles). So \(m\angle6=180 - 57=123^{\circ}\).

Answer:

\(123\)