6.4:Special Parallelograms

Objectives:•To use the properties of diagonals of rhombuses and rectangles•Determine whether a parallelogram is a rhombus or a rectangle

Special Parallelograms

Review of the Properties of a parallelogram:• 2 pairs of opposite parallel sides• 2 pairs of opposite congruent sides• Opposite angles are congruent• Consecutive angles are supplementary• Diagonals bisect each other

Special Parallelograms: Rhombus, Rectangle, Square These figures will inherit ALL the properties above,

AND they will each add their own individual properties

Theorems For a Rhombus:

• Each diagonal of a rhombus bisects 2 angles of the rhombus

• The diagonals of a rhombus are perpendicular bisects &

bisects &

A B

CD DCB BADACBD ADC CBA

BDAC

Examples: Find the measures of the numbered angles in the rhombus.

1

2

3 4

12°

The figure below is a rhombus. Find TQ, TP, and SQ.

Finding Angle Measures• What are the measures of the numbered angles in

rhombus ABCD?

1 90m

2 58m

3 58m

1 3 4 180m m m 90 58 4 180m 148 4 180m

4 32m

Theorem for Rectangle

The diagonals of a rectangle are congruent. (remember, they also are bisected, so all 4 segments created by the intersection are congruent)

ANDBDAC

DEBECEAE

E

EXAMPLE

BD=5y-7 and AC = y + 5. Find the value of y and the length of BE.

E

THEOREMSIf one diagonal of a bisects 2 angles of the

then the is a RHOMBUS.

If the diagonals of a are perpendicular, then the is a RHOMBUS.

If the diagonals of a are congruent, then theis a RECTANGLE.

SQUARE

Remember, a square is a RECTANGLE and a RHOMBUS, so it inherits ALL the properties of a rectangle, rhombus and parallelogram.

Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain your answer.

1. Each diagonal is 15 cm long, and one angle of the quadrilateral has a measure of 45°.

2. The diagonals are congruent, perpendicular, and they bisect each other.