Question

lines m and n are parallel. find the measure of \\( \angle a \\).

Explanation:

Step1: Use the property of parallel lines and transversal

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. But here, we can also use the property that the sum of angles on a straight line formed by the transversal is \(180^{\circ}\). Also, note that the angle adjacent to \(55^{\circ}\) (on the same side of the transversal) and \(\angle a\) are equal because of the alternate - interior angles property (since lines \(M\) and \(N\) are parallel).
Another way: The sum of angles on a straight line is \(180^{\circ}\). Let's assume the angle adjacent to \(55^{\circ}\) (on the transversal - line intersection with line \(N\)) is \(x\). So \(x + 55^{\circ}=180^{\circ}\), then \(x = 180^{\circ}- 55^{\circ}=125^{\circ}\). But actually, \(\angle a\) and \(55^{\circ}\) are related by the property of parallel lines. The correct property is that \(\angle a\) and \(55^{\circ}\) are supplementary (consecutive interior angles).

Step2: Calculate \(\angle a\)

Since \(\angle a+55^{\circ}=180^{\circ}\) (consecutive interior angles for parallel lines \(M\) and \(N\) cut by a transversal), then \(\angle a=180^{\circ}-55^{\circ}\)

$$ \angle a = 125^{\circ} $$

Answer:

\(125\)