1a) Yes, the diagonals bisect each other (marked with congruent segments), which is a sufficient condition for a quadrilateral to be a parallelogram.
1b) Yes, consecutive angles are supplementary (120° + 60° = 180°), which implies opposite sides are parallel, making it a parallelogram.
1c) Yes, one pair of opposite sides is both congruent and parallel (marked with arrows and length 14), which is sufficient to prove it is a parallelogram.
2a) Set the expressions for the bisected diagonal segments equal: 7x + 5 = 3x + 17 → 4x = 12 → x = 3.
2b) In a parallelogram, consecutive angles are supplementary. So (3x + 5) + 70 = 180 → 3x = 105 → x = 35. Also, opposite angles are equal: 2x = x + 3y → 2(35) = 35 + 3y → 70 = 35 + 3y → 3y = 35 → y = 35/3.
2c) Opposite sides must be equal. So x + 2 = 6 → x = 4, and 3x = y - 1 → 3(4) = y - 1 → 12 = y - 1 → y = 13.
3) Since ΔMNP ≅ ΔNOP, corresponding parts are congruent: MN ≅ OP and MP ≅ ON. A quadrilateral with both pairs of opposite sides congruent is a parallelogram. Therefore, MNOP is a parallelogram.
4)
1) Sufficient — if diagonals bisect each other, the quadrilateral is a parallelogram.
2) Insufficient — this only shows that XW and YZ are parallel (consecutive interior angles supplementary), but not enough to prove both pairs of opposite sides are parallel or congruent.
3) Sufficient — if both pairs of opposite sides are congruent, it is a parallelogram.
4) Sufficient — if opposite angles are congruent, it is a parallelogram.
5) Sufficient — if triangles XWO and YZO are congruent, then XO ≅ OZ and WO ≅ OY, so diagonals bisect each other, proving it’s a parallelogram.
6) Insufficient — this proves triangles XWO and ZYO are congruent, which gives XO ≅ OY and WO ≅ OZ, but these are adjacent segments, not necessarily showing diagonals bisect each other or opposite sides are congruent/parallel.
Parent Tip: Review the logic above to help your child master the concept of proofs geometry worksheet.
