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Find the missing angle x in each polygon.

Four geometric figures with labeled angles and missing angle x, showing exterior angles and interior angles for solving geometric problems.

Four geometric figures with labeled angles and missing angle x, showing exterior angles and interior angles for solving geometric problems.

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Show Answer Key & Explanations Step-by-step solution for: Exterior Angles of Convex Polygons - Opinions Nobody Asked For
- 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.
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