1a) Not enough information. The markings show that one diagonal bisects the angles at two opposite vertices, which is not a standard condition for proving a parallelogram.
1b) Yes, it is a parallelogram. The consecutive angles are supplementary (120° + 60° = 180°), which implies that the opposite sides are parallel, satisfying the definition of a parallelogram.
1c) Yes, it is a parallelogram. The diagram shows one pair of opposite sides are both congruent and parallel (indicated by the arrow marks and the length "14"), which is sufficient to prove it is a parallelogram.
2a) x = 3. For the diagonals to bisect each other, set the expressions equal: 7x + 5 = 3x + 17. Solving gives 4x = 12, so x = 3.
2b) x = 25, y = 15. Opposite angles are equal, so 3x + 5 = x + 3y. Also, consecutive angles are supplementary, so 2x + 70 = 180, giving x = 55. Substituting x = 55 into 3x + 5 = x + 3y gives 170 = 55 + 3y, so 3y = 115, y = 115/3 ≈ 38.33. Correction: Consecutive angles must be supplementary. Using 2x + (3x+5) = 180 gives 5x + 5 = 180, so 5x = 175, x = 35. Then using 70 + (x+3y) = 180 gives 70 + 35 + 3y = 180, so 105 + 3y = 180, 3y = 75, y = 25. Final answer: x = 35, y = 25.
2c) x = 2, y = 7. Opposite sides must be equal. So, x + 2 = 6 gives x = 4. And 3x = y - 1. Substituting x = 4 gives 12 = y - 1, so y = 13. Correction: The pairs of opposite sides are (x+2 and 6) and (3x and y-1). Set x+2 = 6, so x = 4. Set 3x = y-1, so 12 = y-1, y = 13. Final answer: x = 4, y = 13.
3) Since ΔMNP ≅ ΔNOP, corresponding parts are congruent. Thus, 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 O is the midpoint of both diagonals XZ and WY, then the diagonals bisect each other, which proves XYZW is a parallelogram.
2) Insufficient. This only tells us that angles ∠XWZ and ∠WZY are supplementary, which may imply WX is parallel to YZ, but does not guarantee the other pair of sides is parallel or any other condition for a parallelogram.
3) Sufficient. If XW is parallel to YZ and WZ is congruent to YX, this satisfies the condition of one pair of opposite sides being both parallel and congruent, which proves it is a parallelogram.
4) Insufficient. These angle congruences do not directly establish parallel sides or congruent opposite sides. They might relate to properties within triangles but not sufficient to prove the whole quadrilateral is a parallelogram.
5) Sufficient. If ΔXWO ≅ ΔYZO, then XO ≅ ZO and WO ≅ YO, meaning the diagonals bisect each other, proving XYZW is a parallelogram.
6) Sufficient. If ΔXWO ≅ ΔZYO, then XO ≅ ZO and WO ≅ YO, again meaning the diagonals bisect each other, proving XYZW is a parallelogram.
Parent Tip: Review the logic above to help your child master the concept of proving a quadrilateral is a parallelogram worksheet.
