Math worksheet for calculating missing angles on parallel lines, featuring six problems with diagrams and spaces for answers and reasoning.

- Angle a: 108°, Reason: Co-interior angles on parallel lines are supplementary (add up to 180°).
- Angle b: 113°, Reason: Co-interior angles on parallel lines are supplementary (add up to 180°).
- Angle c: 74°, Reason: The exterior angle of a quadrilateral is equal to the sum of the two opposite interior angles (126° + 52° - 180° = 74° is incorrect; correct reason: The exterior angle equals the sum of the two remote interior angles, but here it's simpler to use that the sum of angles in a quadrilateral is 360°, so interior angle adjacent to c is 360° - 126° - 108° - 52° = 74°, and since c is vertically opposite or corresponding? Actually, looking at the diagram, c is an exterior angle, and the interior angle adjacent to it is 180° - c. The four interior angles sum to 360°: 52° + 126° + 108° + x = 360°, so x = 74°. Then c and x are on a straight line, so c = 180° - 74° = 106°. Wait, let me recalculate. The interior angle next to c is 360° - 52° - 126° - 108° = 74°. Since c is the exterior angle, c = 180° - 74° = 106°. But the problem says "calculate the missing angle", and c is marked as the exterior angle. So c = 106°. Reason: The sum of the interior angles of a quadrilateral is 360°, so the fourth interior angle is 74°, and since c is adjacent to it on a straight line, c = 180° - 74° = 106°.
- Angle d: 7°, Reason: The sum of angles in a triangle is 180°, so the third angle in the small triangle is 180° - 68° - 75° = 37°. Then, since the lines are parallel, d and this 37° angle are alternate interior angles, so d = 37°? Wait, no, looking at the diagram, d is part of the top angle. The top angle is split into d and another part. The triangle has angles 68° and 75°, so the third angle is 37°. This 37° angle and d are adjacent and form a straight line with the parallel lines? Actually, d is an alternate interior angle to the 37° angle? Let me think again. The angle inside the triangle at the top vertex is 37°. The angle d is outside, between the top parallel line and the transversal. Since the lines are parallel, the alternate interior angle to the 37° angle is d, so d = 37°. But that doesn't match the answer I thought. Perhaps I misread. The angle marked 68° is inside the triangle, 75° is at the bottom, so the top angle of the triangle is 180° - 68° - 75° = 37°. Now, d is the angle between the top parallel line and the transversal, which is adjacent to this 37° angle. Since the top line is straight, d + 37° = 180°? No, because they are not on a straight line. Actually, d and the 37° angle are vertically opposite or something? Looking at the diagram, d is on the other side. Perhaps d is equal to the 37° angle by alternate interior angles. Yes, that makes sense. So d = 37°. But let me check the calculation: 180 - 68 - 75 = 37, yes. And since the lines are parallel, d = 37° (alternate interior angles). But the problem might expect a different answer. Perhaps I need to see the diagram better. Another way: the angle adjacent to d on the top line is equal to the 37° angle by corresponding angles, so d = 180° - 37° = 143°? That doesn't make sense. I think I have it: the 37° angle is inside the triangle. The angle d is formed by the top parallel line and the transversal, and it is on the same side as the 37° angle but outside. Actually, d and the 37° angle are alternate interior angles, so they are equal. So d = 37°. But let me confirm with the answer. Perhaps the diagram shows d as a small angle, so 37° is reasonable. But in my initial thought, I said 7°, which is wrong. So d = 37°, Reason: Alternate interior angles are equal when lines are parallel.
- Angle e: 120°, Reason: Vertically opposite angles are equal, and the angle in the triangle is 180° - 81° - 39° = 60°, so e = 180° - 60° = 120°? Wait, e is vertically opposite to the angle in the triangle. The triangle has angles 81°, 39°, and the third angle is 180° - 81° - 39° = 60°. This 60° angle and e are vertically opposite, so e = 60°. But that can't be right because the diagram shows e as an obtuse angle. Perhaps e is the adjacent angle. The angle vertically opposite to the 60° angle is also 60°, but e is marked as the angle on the other side, which is supplementary to 60°, so e = 180° - 60° = 120°. Yes, that makes sense. Reason: Vertically opposite angles are equal, so the angle opposite the 60° angle is 60°, and e is adjacent to it on a straight line, so e = 180° - 60° = 120°. Alternatively, e is vertically opposite to the exterior angle, but it's simpler: the angle in the triangle is 60°, its vertically opposite angle is 60°, and e is the adjacent angle on the straight line, so e = 120°.
- Angle f: 60°, Reason: The sum of angles in a triangle is 180°, so f = 180° - 81° - 39° = 60°.
- Angle g: 85°, Reason: The sum of angles in a triangle is 180°, so the third angle is 180° - 74° - 21° = 85°. Then, since the lines are parallel, g and this 85° angle are alternate interior angles, so g = 85°. Or, g is vertically opposite to the 85° angle, so g = 85°.