Geometry Parallelograms and Quadrilaterals Worksheet Solutions (PDF)

Opposite sides are congruent.

\(CD = AB = 14\) and \(AD = BC = 9\).

CD = 14, AD = 9

Opposite angles are congruent: \(m\angle Y = 123^{\circ}\).

Consecutive angles are supplementary: \(m\angle X = 180^{\circ} – 123^{\circ} = 57^{\circ}\).

m∠Y = 123°, m∠X = 57°

Consecutive angles are supplementary.

\((4x + 10) + (2x + 20) = 180\)

\(6x + 30 = 180\)

\(x = 25\)

\(m\angle A = 110^{\circ}\) and \(m\angle B = 70^{\circ}\).

x = 25; 110° and 70°

The diagonals bisect each other.

\(AC = 2 \cdot 7 = 14\) and \(BD = 2 \cdot 5.5 = 11\).

AC = 14, BD = 11

\(3x + 1 = 5x – 9\)

\(10 = 2x\)

\(x = 5\)

\(AE = 16\), so \(AC = 32\).

x = 5; AC = 32

\(4x – 3 = 2x + 9\)

\(2x = 12\)

\(x = 6\)

\(AB = 4(6) – 3 = 21\)

x = 6; AB = 21

\(2(8 + 13) = 42\)

42 inches

The diagonals bisect each other: the midpoint of BD is \((3, 2.5)\), and the midpoint of AC must be the same.

\(\frac{0 + x}{2} = 3\) gives \(x = 6\), and \(\frac{0 + y}{2} = 2.5\) gives \(y = 5\).

C(6, 5)

The diagonals of a rectangle are congruent.

\(2x + 6 = 4x – 10\)

\(16 = 2x\)

\(x = 8\)

\(AC = 2(8) + 6 = 22\)

x = 8; AC = 22

\(\sqrt{9^{2} + 12^{2}} = \sqrt{225} = 15\)

15 units

The diagonals are perpendicular and bisect each other, so each side is the hypotenuse of a right triangle with legs 8 and 15.

\(\sqrt{8^{2} + 15^{2}} = 17\), and the perimeter is \(4 \cdot 17 = 68\).

Side 17; perimeter 68

Consecutive angles are supplementary: \(m\angle B = 116^{\circ}\).

The diagonal bisects angle A: \(64^{\circ} \div 2 = 32^{\circ}\).

m∠B = 116°; 32°

\(10\sqrt{2}\) by the Pythagorean theorem: \(\sqrt{10^{2} + 10^{2}}\).

The diagonal bisects the right angle, giving \(45^{\circ}\).

10√2; 45°

The angles of a quadrilateral add to 360°.

\(360^{\circ} – 70^{\circ} – 95^{\circ} – 95^{\circ} = 100^{\circ}\)

100°

Base angles are congruent: \(m\angle C = 72^{\circ}\).

Each angle at the top base is supplementary to a bottom angle: \(180^{\circ} – 72^{\circ} = 108^{\circ}\).

m∠C = 72°; m∠A = m∠B = 108°

\(\frac{10 + 24}{2} = 17\)

17

Each side: \(\sqrt{4^{2} + 3^{2}} = 5\), so all four sides are congruent and ABCD is a rhombus.

The diagonals are AC = 8 and BD = 6. They are not congruent, so the angles are not right angles and it is not a square.

Rhombus: all sides 5; diagonals 8 and 6 are unequal

\(m\angle C = 72^{\circ}\), and \(m\angle B = m\angle D = 180^{\circ} – 72^{\circ} = 108^{\circ}\).

\(2(14 + 9) = 46\) inches.

72°, 108°, 108°; perimeter 46 in

The diagonals of a rectangle are congruent and given by the Pythagorean theorem.

\(\sqrt{24^{2} + 18^{2}} = \sqrt{900} = 30\)

30 inches

The diagonals split the kite into four right triangles. Area \(= \frac{1}{2} d_{1} d_{2}\).

\(\frac{1}{2} \cdot 60 \cdot 40 = 1200\)

1,200 square centimeters