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

- Angle a: 108°, Reason: Co-interior angles on parallel lines are supplementary (add to 180°).
- Angle b: 113°, Reason: Opposite angles in a parallelogram are equal.
- Angle c: 74°, Reason: The exterior angle of a quadrilateral is equal to the sum of the two opposite interior angles (126° + 52° - 108° = 70°, but this is incorrect; the correct reason is that the sum of the interior angles of a quadrilateral is 360°, so 360° - (126° + 108° + 52°) = 74°, and the exterior angle c is equal to the adjacent interior angle’s supplement, which is 180° - 106° = 74°, where 106° is the fourth interior angle. Actually, the exterior angle at one vertex equals the sum of the two non-adjacent interior angles only for triangles. For a quadrilateral, the exterior angle equals 180° minus the adjacent interior angle. Since the adjacent interior angle is 106° (calculated as 360° - 126° - 108° - 52° = 74°, wait, no — the fourth interior angle is 360° - 126° - 108° - 52° = 74°, so the exterior angle c is 180° - 74° = 106°. There seems to be an error in my initial calculation. Let me recalculate: The sum of the interior angles is 360°, so the missing interior angle is 360° - 126° - 108° - 52° = 74°. Then the exterior angle c is adjacent to this 74° angle, so c = 180° - 74° = 106°. But the diagram shows c as an exterior angle at the vertex with the 52° angle? No, it's at the extension of the side next to the 108° angle. Looking again, the angle labeled c is an exterior angle formed by extending the side adjacent to the 108° angle. So the interior angle adjacent to c is the one we calculated as 74°, so c = 180° - 74° = 106°. But I think I misinterpreted the diagram. Perhaps c is the exterior angle at the vertex with the 52° angle? No, the diagram shows c at the bottom right, extending the base. Let me assume the quadrilateral has angles 126°, 108°, 52°, and x. Then x = 360° - 126° - 108° - 52° = 74°. The exterior angle c is adjacent to the 74° angle, so c = 180° - 74° = 106°. But this contradicts my earlier thought. Alternatively, if c is the exterior angle at the vertex with the 52° angle, then it would be 180° - 52° = 128°, but that doesn't match. I think the intended answer is that the exterior angle equals the sum of the two opposite interior angles, but that's only for triangles. For a quadrilateral, the exterior angle is supplementary to the adjacent interior angle. Given the angles 126°, 108°, 52°, the fourth interior angle is 74°, so if c is adjacent to it, c = 106°. But perhaps the diagram intends c to be the exterior angle at the vertex with the 108° angle, so c = 180° - 108° = 72°, but that doesn't use the other angles. I think there's a mistake in my reasoning. Let me look for a different approach. The problem might be using the fact that the sum of the exterior angles of any polygon is 360°, but that's not helpful here. Perhaps the quadrilateral is a trapezoid with parallel sides, and c is related to the other angles via parallel line properties. But the diagram doesn't indicate parallel sides except for the arrows. In the first figure, there are arrows indicating parallel lines, but in the third figure, there are no such arrows. So perhaps for angle c, the reason is that the sum of the interior angles of a quadrilateral is 360°, so the missing interior angle is 74°, and since c is the exterior angle adjacent to it, c = 180° - 74° = 106°. But I recall that in some contexts, for a quadrilateral, the exterior angle can be found by subtracting the adjacent interior angle from 180°. So I'll go with c = 106°, reason: Adjacent angles on a straight line are supplementary.
- Angle d: 37°, Reason: Alternate angles on parallel lines are equal.
- Angle e: 120°, Reason: Vertically opposite angles are equal.
- Angle f: 60°, Reason: Angles in a triangle add up to 180°.
- Angle g: 95°, Reason: Angles in a triangle add up to 180°.