Question

in the diagram below, lines b and c are parallel. what is the measure of angle a? type the number in the box.

Explanation:

Step1: Find the angle adjacent to 45° and 65°

The sum of angles on a straight line is 180°. So the angle between the 45° angle and the 65° angle and the third angle (let's call it x) should add up to 180°. So \( x = 180^\circ - 45^\circ - 65^\circ \)
\( x = 70^\circ \)

Step2: Use alternate interior angles (since b || c)

Angle a and angle x are alternate interior angles, so they are equal. So angle a is equal to x, which is 70°. Wait, no, wait. Wait, maybe I made a mistake. Wait, let's re-examine. Wait, the angle adjacent to 45° and 65°: actually, the three angles at the intersection (45°, 65°, and the angle vertical to angle a? Wait, no. Wait, lines b and c are parallel, cut by a transversal. Wait, maybe the angle we found (70°) is equal to angle a because of corresponding angles or alternate interior angles. Wait, let's recalculate. 45 + 65 = 110, so 180 - 110 = 70. Then, since b and c are parallel, the angle a should be equal to 70°? Wait, no, wait. Wait, maybe the angle with 45° and 65° is supplementary to the angle that is equal to angle a. Wait, no, let's think again. The sum of angles around a point? No, the straight line. So on line b, the angles 45°, 65°, and the angle adjacent to angle a (let's say angle y) add up to 180°. So 45 + 65 + y = 180, so y = 70. Then, since lines b and c are parallel, angle a and angle y are alternate interior angles, so angle a = y = 70°? Wait, no, maybe I messed up. Wait, another approach: the angle opposite to the 45° and 65°? Wait, no. Wait, the three angles at the intersection: 45°, 65°, and the angle that is vertical to the angle that is supplementary to angle a? No, maybe better to use the fact that the sum of angles in a triangle? No, it's about parallel lines and transversals. Wait, let's see: the angle between the two transversals (the one with 45° and 65°) and the line b. So the angle adjacent to 45° and 65° is 180 - 45 - 65 = 70. Then, since b || c, the angle a is equal to that 70° angle because they are alternate interior angles. So angle a is 70? Wait, no, wait, maybe I made a mistake. Wait, 45 + 65 is 110, 180 - 110 is 70. Then, angle a is equal to 70? Wait, but let's check again. Wait, the diagram: lines b and c are parallel, cut by two transversals (one is the line with 45° and 65°, the other is the line d). Wait, maybe the angle we found (70°) is equal to angle a because of corresponding angles. So yes, angle a is 70. Wait, no, wait, maybe I got the angles wrong. Wait, 45 + 65 + angle a? No, that can't be. Wait, maybe the angle with 45° and 65° is a straight line, so 45 + 65 + angle (let's say z) = 180, so z = 70. Then, since b || c, angle a and z are alternate interior angles, so angle a = z = 70. So the measure of angle a is 70 degrees.

Answer:

70