1. In a parallelogram, opposite angles are equal and consecutive angles are supplementary.
- Set opposite angles equal: x + 10 = 5x - 4 → 14 = 4x → x = 3.5
- Substitute x = 3.5:
- ∠P = ∠S = 3.5 + 10 = 13.5°
- ∠Q = ∠R = 5(3.5) - 4 = 17.5 - 4 = 13.5°
- But 13.5° + 13.5° = 27° ≠ 180° — contradiction.
- Correction: Consecutive angles are supplementary.
- (x + 10) + (5x - 4) = 180 → 6x + 6 = 180 → 6x = 174 → x = 29
- Angles:
- ∠P = ∠S = 29 + 10 = 39°
- ∠Q = ∠R = 5(29) - 4 = 145 - 4 = 141°
- Check: 39° + 141° = 180° — correct.
2. In rhombus ABCD, all sides equal, diagonals bisect vertex angles and are perpendicular.
- Diagonal BD bisects ∠ABC and ∠ADC.
- ∠A = 68°, so ∠C = 68° (opposite angles equal).
- Sum of interior angles = 360° → ∠B + ∠D = 224° → ∠B = ∠D = 112°.
- Diagonal BD bisects ∠B → x = 112° / 2 = 56°.
- Diagonal BD also bisects ∠D → y = 112° / 2 = 56°.
- So x = 56°, y = 56°.
3. Isosceles trapezoid PQRS with m∠C = 48°.
- Assume C is the lower-left base angle → ∠C = 48°.
- In isosceles trapezoid, base angles are equal: ∠C = ∠D = 48°.
- Consecutive angles between parallel bases are supplementary.
- So ∠A = ∠B = 180° - 48° = 132°.
- Angles: ∠A = 132°, ∠B = 132°, ∠C = 48°, ∠D = 48°.
4. Rhombus diagonals are 16 and 18.
- Diagonals bisect each other at right angles.
- Each half-diagonal: 8 and 9.
- Side length = hypotenuse of right triangle with legs 8 and 9.
- Side = √(8² + 9²) = √(64 + 81) = √145.
5. Rectangle diagonals are equal.
- Set diagonals equal: 6b - 2 = 4b + 2 → 2b = 4 → b = 2.
- Diagonal lengths: 6(2) - 2 = 12 - 2 = 10; 4(2) + 2 = 8 + 2 = 10.
- Both diagonals are 10 units.
6. In a square, all angles are 90°.
- Set 2x + 44 = 90 → 2x = 46 → x = 23.
7. In parallelogram ABCE (assuming E is connected to form parallelogram), find ∠EBC.
- Given ∠ABE = 37°, ∠BCE = 120°.
- Since ABCE is a parallelogram, AB ∥ CE, AE ∥ BC.
- ∠EBC is part of ∠ABC.
- Consider triangle BEC or use properties.
- Note: ∠BCE = 120° is an interior angle at C.
- In parallelogram, consecutive angles are supplementary.
- ∠ABC + ∠BCE = 180° → ∠ABC = 180° - 120° = 60°.
- ∠ABC = ∠ABE + ∠EBC → 60° = 37° + ∠EBC → ∠EBC = 23°.
Parent Tip: Review the logic above to help your child master the concept of quadrilateral practice worksheet.
