Parallelograms and Special Quadrilaterals: Geometry Math Test – PDF Download

Question 1: Properties of a parallelogram (2 points)

In parallelogram \(PQRS\), \(PQ=16\text{ cm}\), \(QR=9\text{ cm}\) and \(m\angle P=58^{\circ}\).

1. Find \(RS\) and \(SP\). (1 point)
2. Find \(m\angle Q\), \(m\angle R\) and \(m\angle S\). (1 point)

Question 2: Angles with a variable (2 points)

In parallelogram \(WXYZ\), \(m\angle W=(5x-15)^{\circ}\) and \(m\angle X=(3x+35)^{\circ}\). Find \(x\) (1 point), then find \(m\angle Y\) and \(m\angle Z\) (1 point).

Question 3: Diagonals with two variables (2 points)

The diagonals of parallelogram \(JKLM\) meet at \(N\). Use the labels in the figure to find \(JL\) and \(KM\). (1 point each)

Question 4: Parallelogram or not? (3 points)

Decide whether each quadrilateral must be a parallelogram. Justify each answer. (1 point each)

1. In quadrilateral \(ABCD\), \(\overline{AB}\) is parallel to \(\overline{DC}\) and \(AB=DC\).
2. In quadrilateral \(EFGH\), \(m\angle E=m\angle G\) and \(m\angle F=m\angle H\).
3. In quadrilateral \(PQRS\), \(PQ=PS\), \(RQ=RS\), and the diagonals are perpendicular.

Question 5: Diagonals of a rectangle (2 points)

Rectangle \(RSTU\) has diagonals \(RT=6x+4\) and \(SU=4x+18\). Find \(x\) (1 point) and \(RT\) (1 point).

Question 6: Angles of a rhombus (2 points)

In rhombus \(ABCD\), \(m\angle ABC=76^{\circ}\). Find \(m\angle BAD\) (1 point) and \(m\angle BAC\) (1 point).

Question 7: Isosceles trapezoid (2 points)

In isosceles trapezoid \(ABCD\) with bases \(\overline{AB}\) and \(\overline{DC}\), \(m\angle A=64^{\circ}\).

1. Find \(m\angle B\), \(m\angle C\) and \(m\angle D\). (1 point)
2. The diagonals are \(AC=4x+1\) and \(BD=6x-11\). Find \(AC\). (1 point)

Question 8: Kite (2 points)

Kite \(KLMN\) has \(KL=KN\) and \(ML=MN\). Also \(m\angle K=88^{\circ}\) and \(m\angle M=48^{\circ}\). Find \(m\angle L\) (1 point). The diagonals meet at \(P\). Find \(m\angle KPL\) (1 point).

Question 9: Two-column proof (3 points)

Given: In quadrilateral \(ABCD\), \(\overline{AB}\) is parallel to \(\overline{DC}\) and \(AB=DC\). Prove: \(ABCD\) is a parallelogram. Write a two-column proof. You may draw diagonal \(\overline{AC}\).

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