Question

two parallel lines are crossed by a transversal. what is the value of a? a = 43, a = 63, a = 117, a = 137

Explanation:

Step1: Identify angle relationship

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary (sum to \(180^\circ\))? Wait, no, wait. Wait, the angle given is \(63^\circ\), and angle \(a\) and the angle adjacent to \(63^\circ\) (vertical angles or same - side? Wait, no, let's look at the diagram. The two parallel lines are \(g\) and \(f\), cut by transversal \(h\). The angle \(63^\circ\) and angle \(a\) are same - side interior angles? Wait, no, actually, the angle supplementary to \(63^\circ\) (since they are adjacent and form a linear pair) would be \(180 - 63=117\)? Wait, no, wait. Wait, the angle \(a\) and the angle that is vertical to the angle supplementary to \(63^\circ\)? Wait, no, let's re - examine.

Wait, the two parallel lines \(g\) and \(f\), transversal \(h\). The angle marked \(63^\circ\) and angle \(a\): since \(g\parallel f\), and the transversal \(h\), angle \(a\) and the angle adjacent to \(63^\circ\) (the one that is a same - side interior angle? Wait, no, actually, the angle \(a\) and the angle that is supplementary to \(63^\circ\) are equal? Wait, no, let's think again.

Wait, the angle of \(63^\circ\) and angle \(a\): when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Wait, the angle \(63^\circ\) and the angle that is adjacent to \(a\) (on the same side of the transversal) – no, actually, the angle \(a\) and the angle that is vertical to the angle which is supplementary to \(63^\circ\). Wait, no, let's calculate the supplementary angle first. The angle supplementary to \(63^\circ\) is \(180 - 63=117^\circ\)? No, wait, no. Wait, if we look at the diagram, the angle \(63^\circ\) and angle \(a\): since \(g\) and \(f\) are parallel, and the transversal \(h\), angle \(a\) and the angle that is equal to \(180 - 63\)? Wait, no, I think I made a mistake. Wait, the angle \(a\) and the angle of \(63^\circ\): are they same - side interior angles? Wait, no, same - side interior angles sum to \(180^\circ\). Wait, no, the angle \(a\) and the angle that is vertical to the angle adjacent to \(63^\circ\). Wait, let's start over.

The two parallel lines are \(g\) (top) and \(f\) (bottom), cut by transversal \(h\). The angle at the bottom line \(f\) is \(63^\circ\). The angle \(a\) is at the top line \(g\). Since \(g\parallel f\), the angle \(a\) and the angle that is supplementary to \(63^\circ\) are equal? Wait, no, the angle \(63^\circ\) and angle \(a\): if we consider the linear pair, the angle adjacent to \(63^\circ\) is \(180 - 63 = 117^\circ\), and since \(g\parallel f\), the corresponding angle to that \(117^\circ\) angle is \(a\). Wait, no, actually, the angle \(a\) and the angle of \(63^\circ\): are they same - side interior angles? Wait, same - side interior angles are supplementary. Wait, no, the angle \(a\) and the angle that is vertical to the angle which is supplementary to \(63^\circ\). Wait, I think I messed up. Let's use the property of parallel lines and transversals: consecutive interior angles are supplementary. Wait, the angle \(63^\circ\) and angle \(a\): if we look at the diagram, the angle \(a\) and the angle of \(63^\circ\) are same - side interior angles? Wait, no, the angle \(a\) and the angle that is adjacent to \(63^\circ\) (forming a linear pair) – the angle adjacent to \(63^\circ\) is \(180 - 63=117^\circ\), and since \(g\parallel f\), the alternate interior angle or corresponding angle? Wait, no, the angle \(a\) and the angle of \(117^\circ\) (the supplementary angle of \(63^\circ\)): are they equal? Wait, no, let's check the options. The options are 43, 63, 117, 137. Wait, the angle \(a\) and the angle of \(63^\circ\): if they are same - side interior angles, they should sum to \(180^\circ\). Wait, \(a + 63=180\)? Then \(a = 180 - 63=117\). Yes, that makes sense. Because when two parallel lines are cut by a transversal, consecutive interior angles are supplementary. So angle \(a\) and the \(63^\circ\) angle are consecutive interior angles, so they sum to \(180^\circ\).

Step2: Calculate the value of \(a\)

We know that for two parallel lines cut by a transversal, consecutive interior angles are supplementary. So \(a+63 = 180\).
To find \(a\), we solve for \(a\): \(a=180 - 63\)
\(a = 117\)

Answer:

\(a = 117\) (corresponding to the option \(a = 117\))