- For the first quadrilateral (top-left): The sum of interior angles is 360°. The angles are 160°, 90°, 2x°, and 3x°. So, 160 + 90 + 2x + 3x = 360. Solving gives 5x = 110, so x = 22.
- For the second quadrilateral (top-middle): The sum of interior angles is 360°. The angles are 90°, 90°, 2x°, and 3x°. So, 90 + 90 + 2x + 3x = 360. Solving gives 5x = 180, so x = 36.
- For the third pentagon (top-right): The sum of interior angles is 540°. The angles are (5x - 50)°, (100 - x)°, 65°, (x/2 - 15)°, and (50 - x)°. So, (5x - 50) + (100 - x) + 65 + (x/2 - 15) + (50 - x) = 540. Simplifying gives 3.5x + 150 = 540, so 3.5x = 390, and x = 780/7 ≈ 111.43.
- For the fourth pentagon (bottom-left): The sum of interior angles is 540°. The angles are (x + 30)°, (7x)°, (31x)°, 150°, and (8x - 25)°. So, (x + 30) + 7x + 31x + 150 + (8x - 25) = 540. Simplifying gives 47x + 155 = 540, so 47x = 385, and x = 385/47 ≈ 8.19.
- For the fifth hexagon (bottom-right): The sum of interior angles is 720°. The angles are (4/3)x°, (0.6x)°, (4x/3)°, (2x/5)°, (2x - 70)°, and (4/5)x°. So, (4/3)x + 0.6x + (4/3)x + (2x/5) + (2x - 70) + (4/5)x = 720. Converting to fractions: (4/3)x + (3/5)x + (4/3)x + (2/5)x + 2x - 70 + (4/5)x = 720. Combining like terms: (8/3)x + (9/5)x + 2x = 790. Finding a common denominator: (40/15)x + (27/15)x + (30/15)x = 790, so (97/15)x = 790, and x = (790 * 15) / 97 = 11850 / 97 ≈ 122.16.
Parent Tip: Review the logic above to help your child master the concept of interior and exterior angles of polygons worksheet with answers.
