Based on the analysis of the worksheet, here are the solutions to each problem:
1.
Find x (Pentagon): The polygon is a pentagon, so the sum of its interior angles is (5-2) * 180° = 540°. The known angles are 90°, 80°, 125°, and 90°. Adding these gives 385°. Therefore, x = 540° - 385° =
155°.
2.
Find x (Quadrilateral Exterior Angles): The sum of the exterior angles of any convex polygon is 360°. The known exterior angles are 62°, 105°, and 73°. Their sum is 240°. Therefore, x = 360° - 240° =
120°.
3.
Find x (Pentagon with Right Angle): This is a pentagon, so the sum of its interior angles is 540°. The known angles are 120°, 70°, and a right angle (90°). The sum of these is 280°. Therefore, x = 540° - 280° =
260°. *(Note: This result suggests the polygon may not be convex, as an interior angle of 260° is reflex.)*
4.
Number of Sides (Regular Polygon): For a regular polygon with interior angle 140°, use the formula: Interior Angle = [(n-2) * 180°] / n. Setting this equal to 140° gives: 140 = (180n - 360) / n. Solving for n: 140n = 180n - 360 → 40n = 360 → n =
9. The polygon has 9 sides.
5.
Find x and y (Pentagon Exterior Angles): The sum of the exterior angles is 360°. The known angles are 62°, 105°, and 73°, which sum to 240°. Therefore, x + y = 360° - 240° =
120°. Without additional information, unique values for x and y cannot be determined.
6.
Exterior Angle (Hexagon and Square): The question asks for the measure of the exterior angle formed by a regular hexagon and a regular quadrilateral (square). The exterior angle of a regular hexagon is always 360°/6 =
60°, regardless of its position relative to the square. The diagram's specific configuration might be misleading, but the exterior angle of the hexagon itself is 60°.
7.
Find x (Square and Triangles): The problem involves a square and two congruent equilateral triangles with corresponding sides parallel. The angle x appears to be an exterior angle. If the triangle is attached to the square such that one side of the triangle is collinear with a side of the square, then the angle x would be the supplement of the triangle's interior angle. Since the interior angle of an equilateral triangle is 60°, the exterior angle x would be 180° - 60° =
120°.
8.
Interior Angle of 20-sided Polygon: Using the formula [(n-2) * 180°] / n for n=20: [(20-2) * 180°] / 20 = [18 * 180°] / 20 = 3240° / 20 =
162°.
9.
Sum of Interior Angles (Convex Heptagon): A heptagon has 7 sides. The sum of the interior angles is (n-2) * 180° = (7-2) * 180° = 5 * 180° =
900°.
Parent Tip: Review the logic above to help your child master the concept of exterior angles of a polygon worksheet.
