Special Quadrilaterals

Keystone Geometry

Special Quadrilaterals (four sides)

A parallelogram has two pairs of opposite sides parallel.

A rectangle has two pairs of opposite sides parallel and four right angles.

A square has two pairs of opposite sides parallel, four right angles, and four equal sides.

A rhombus has two pairs of opposite sides parallel and four equal sides.

A trapezoid has one pair of parallel sides.

Venn Diagram: QuadrilateralsParallelograms in White Circle

Rectangles

Rhombuses

SquaresTrapezoids

Rectangles

Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. Diagonals are congruent NEW

Definition: A rectangle is a parallelogram with four right angles.

A rectangle is a special type of parallelogram.

Thus a rectangle has all the properties of a parallelogram.

Examples1. If AE = 3x +2 and BE = 29, find the value of x.

2. If AC = 21, then BE = _______.

3. If m<1 = 4x and m<4 = 2x, find the value of x.

4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.

m<1=50, m<3=40, m<4=80, m<5=100, m<6=40

10.5 units

x = 9 units

x = 18 units

6

54

321

E

D C

BA

RhombusDefinition: A rhombus is a parallelogram with four congruent sides.

Since a rhombus is a parallelogram the following are true: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other Diagonals bisect the angles and are perpendicular NEW

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Rhombus Examples Given: ABCD is a rhombus. Complete the following.

1. If AB = 9, then AD = ______.

2. If m<1 = 65, the m<2 = _____.

3. m<3 = ______.

4. If m<ADC = 80, the m<DAB = ______.

5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.

9 units

65°

90°

100°

10

Square

Opposite sides are parallel. Four right angles. Four congruent sides. Consecutive angles are supplementary. Diagonals bisect each other. Diagonals are congruent. NEW Diagonals are perpendicular. NEW Each diagonal bisects a pair of opposite angles. NEW

Definition: A square is a parallelogram with four congruent angles and four congruent sides.

Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.

Squares ExamplesGiven: ABCD is a square. Complete the following.

1. If AB = 10, then AD = _______ and DC = _______.

2. If CE = 5, then DE = _____.

3. m<ABC = _____.

4. m<ACD = _____.

5. m<AED = _____.

10 units 10 units

5 units

90°

45°

90°