- For the triangle: The angles are 2x, x + 20, and 3x - 35. Sum of angles in a triangle is 180°, so 2x + (x + 20) + (3x - 35) = 180. Simplify: 6x - 15 = 180 → 6x = 195 → x = 32.5. Angles are 65°, 52.5°, and 62.5°. Smallest angle is 52.5°.
- For the quadrilateral: Angles are 3x + 56, 4x, 2x - 24, and 160 - x. Sum of angles in a quadrilateral is 360°, so (3x + 56) + 4x + (2x - 24) + (160 - x) = 360. Simplify: 8x + 192 = 360 → 8x = 168 → x = 21.
- For the intersecting lines: AB, CD, EF are straight lines. Angle at E: 7x - 35 and 3x + 15 are adjacent on line AB, so they sum to 180°: (7x - 35) + (3x + 15) = 180 → 10x - 20 = 180 → 10x = 200 → x = 20. Then angle 7x - 35 = 105°, angle 3x + 15 = 75°. Angle at C: x + 55 = 75°. Since angle 3x + 15 = 75° and angle x + 55 = 75°, they are equal and are alternate interior angles for lines AB and CD with transversal EF, so AB is parallel to CD.
- For the parallelogram: Opposite angles are equal. So 4x - 10 = 2x + 5y and 6x + 10y = 3x + 56? Wait, no — in a parallelogram, opposite angles are equal, and consecutive angles sum to 180°. Actually, from diagram, top right is 2x + 5y, bottom left is 4x - 10 — these are opposite? No, in standard labeling, if it’s labeled clockwise or counterclockwise, opposite angles are equal. Assuming the given angles are opposite: 2x + 5y = 4x - 10 → 5y - 2x = -10. Also, adjacent angles sum to 180°: (2x + 5y) + (6x + 10y) = 180 → 8x + 15y = 180. Solve system: From first equation, 2x = 5y + 10 → x = (5y + 10)/2. Plug into second: 8*(5y+10)/2 + 15y = 180 → 4*(5y+10) + 15y = 180 → 20y + 40 + 15y = 180 → 35y = 140 → y = 4. Then x = (5*4 + 10)/2 = 30/2 = 15. Check: 2x+5y=30+20=50, 4x-10=60-10=50, 6x+10y=90+40=130, and 50+130=180, good. So x=15, y=4.
Parent Tip: Review the logic above to help your child master the concept of angles and algebra worksheet.
