Final answer:
The sum of exterior angles for any polygon, including a quadrilateral, is always 360°. The sum of the interior angles of a quadrilateral also adds up to 360°, since each pair of an interior and its corresponding exterior angle sums to 180°. Statements (a) and (b) are true, while (c) and (d) are false.
Step-by-step explanation:
Exterior Angles of a Quadrilateral
The question addresses the concept of exterior angles in relation to the interior angles of a quadrilateral and how their sums relate. When considering a quadrilateral, or any polygon, the sum of the exterior angles taken one at each vertex always adds up to 360°. This is a fundamental property of polygons and does not depend on the number of sides.
Now, to connect the sum of exterior angles to the sum of interior angles of a quadrilateral (PQRS): Let's consider an exterior angle at each vertex of the quadrilateral. Since each exterior angle forms a linear pair with its adjacent interior angle, and a linear pair of angles sum up to 180°, the sum of an exterior angle and its corresponding interior angle is 180°. For example, if one interior angle is a, then its corresponding exterior angle will be 180° - a.
Since there are four vertices in a quadrilateral, the sum of its exterior angles is 4*180° = 720°. However, we must subtract the interior angles (4 in total, one at each vertex) to get the sum of the exterior angles alone, which brings us back down to 360°. Correspondingly, the sum of interior angles is the difference between the summed angles of the linear pairs (720°) and the exterior angle sum (360°), which leaves us with 360° for the sum of the interior angles of quadrilateral PQRS.
Therefore, the correct statement is:
- (a) The sum of exterior angles is always 360° - True.
- (b) The sum of interior angles is always 360° - True.
- (c) The sum of exterior angles is twice the sum of interior angles - False.
- (d) The sum of exterior angles is half the sum of interior angles - False.