- For the top-left quadrilateral: The interior angles are 90°, 100°, 110°, and x. The sum of interior angles in a quadrilateral is 360°. Therefore, 90 + 100 + 110 + x = 360, which simplifies to 300 + x = 360. Solving for x gives x = 60°.
- For the top-right pentagon: The exterior angles are given as x, 200°, 330°, 210°, and 260°. The sum of exterior angles for any polygon is always 360°. Therefore, x + 200 + 330 + 210 + 260 = 360. This simplifies to x + 1000 = 360, which gives x = -640°. However, an exterior angle cannot be negative, indicating a potential error in the problem setup or interpretation. Assuming the values represent reflex angles (greater than 180°), the actual exterior angles would be their supplements (360° minus the given value). Thus, the actual exterior angles are (360-x), 160°, 30°, 150°, and 100°. Summing these gives (360-x) + 160 + 30 + 150 + 100 = 360, which simplifies to 800 - x = 360, so x = 440°. This is also impossible for an exterior angle. Given the inconsistency, the problem may contain an error.
- For the bottom-left quadrilateral: The interior angles are 100°, 90°, 60°, and x. The sum of interior angles in a quadrilateral is 360°. Therefore, 100 + 90 + 60 + x = 360, which simplifies to 250 + x = 360. Solving for x gives x = 110°.
- For the bottom-right quadrilateral: The exterior angles are 50°, 300°, 275°, and x. The sum of exterior angles for any polygon is always 360°. Therefore, 50 + 300 + 275 + x = 360. This simplifies to 625 + x = 360, which gives x = -265°. Again, this is impossible for an exterior angle. Assuming the values represent reflex angles, the actual exterior angles would be 50°, 60°, 85°, and (360-x). Summing these gives 50 + 60 + 85 + (360-x) = 360, which simplifies to 555 - x = 360, so x = 195°. This is still not a valid exterior angle (should be less than 180°). Given the inconsistency, the problem may contain an error.
Parent Tip: Review the logic above to help your child master the concept of interior and exterior angles of polygons worksheet with answers.