1. a) Yes, because the diagonals bisect each other (indicated by congruent segments on each diagonal).
b) No, not enough information; only three angles are given, and while they sum to 240°, the fourth angle is implied to be 120°, but no side or diagonal relationship is provided to confirm parallel sides.
c) Yes, because one pair of opposite sides is both congruent and parallel (indicated by tick marks and arrows).
2. a) x = 6; since diagonals bisect each other: 7x + 5 = 3x + 17 → 4x = 12 → x = 6.
b) x = 20, y = 10; consecutive angles are supplementary: (3x+5) + 70 = 180 → 3x = 105 → x = 35? Wait, correction: Actually, adjacent angles must sum to 180°, so (3x+5) + 2x = 180 → 5x = 175 → x = 35. Then (x+3y) + 70 = 180 → x+3y = 110 → 35 + 3y = 110 → 3y = 75 → y = 25. But also check opposite angles: (3x+5) should equal (x+3y) if it’s a parallelogram? Actually, in a parallelogram, opposite angles are equal, so set 3x+5 = x+3y → 2x - 3y = -5. And from consecutive angles: 3x+5 + 2x = 180 → 5x = 175 → x = 35. Plug into 2(35) - 3y = -5 → 70 - 3y = -5 → 3y = 75 → y = 25. So x=35, y=25.
Correction: The problem says “ensure each quadrilateral is a parallelogram.” For part b), we have two pairs of adjacent angles. Since consecutive angles must be supplementary, we can use:
(3x + 5) + 2x = 180 → 5x + 5 = 180 → 5x = 175 → x = 35.
Also, (x + 3y) + 70 = 180 → x + 3y = 110 → 35 + 3y = 110 → 3y = 75 → y = 25.
So x=35, y=25.
c) x=1, y=7; opposite sides must be equal: x+2 = 6 → x=4? Wait, no: the top side is x+2, bottom is y-1; left is 3x, right is 6. So for parallelogram: opposite sides equal → x+2 = y-1 and 3x = 6. From 3x=6 → x=2. Then x+2 = y-1 → 4 = y-1 → y=5. So x=2, y=5.
3. Given ΔMNP ≅ ΔNOP, then corresponding parts are congruent: MN ≅ NO, MP ≅ OP, NP ≅ NP (reflexive). Also, ∠MNP ≅ ∠NOP, ∠MPN ≅ ∠OPN, ∠PMN ≅ ∠PON. Since ∠MNP ≅ ∠NOP, these are alternate interior angles for lines MN and PO with transversal NP, so MN ∥ PO. Similarly, ∠MPN ≅ ∠OPN implies MP ∥ NO (alternate interior angles with transversal PN). Thus, both pairs of opposite sides are parallel, so MNOP is a parallelogram.
4. 1) Yes — if O is midpoint of both diagonals, then diagonals bisect each other, which proves it’s a parallelogram.
2) No — this only tells us that angles at W are supplementary, which could be true in a trapezoid or other quadrilateral; doesn’t guarantee parallel sides.
3) Yes — if both pairs of opposite sides are congruent, then it’s a parallelogram.
4) Yes — if both pairs of opposite angles are congruent, then it’s a parallelogram.
5) Yes — if ΔXWO ≅ ΔYZO, then XW ≅ YZ and XO ≅ YO, WO ≅ ZO, so diagonals bisect each other → parallelogram.
6) Yes — if ΔXWO ≅ ΔZYO, then XW ≅ ZY, XO ≅ ZO, WO ≅ YO → again, diagonals bisect each other → parallelogram.
Parent Tip: Review the logic above to help your child master the concept of proving quadrilaterals are parallelograms worksheet.
