LESSON 6 4 SPECIAL PARALLELOGRAMS OBJECTIVE Use properties

LESSON 6. 4 SPECIAL PARALLELOGRAMS OBJECTIVE: Use properties of diagonals of rhombuses and rectangles Determine whether a parallelogram is a rhombus or rectangle

Rhombus Theorem 6. 9 Each diagonal of a rhombus bisects two angles of the rhombus. AC ___ bisects � BAD, so � 1 �� 2 AC bisects � BCD, so � 3 �� 4

Theorem 6. 10 The diagonals of a rhombus are perpendicular. IF THEN AC �BD

Rectangle Theorem 6. 11 The diagonals of a rectangle are congruent. THEN IF BD �� AC

Parallelogram Theorems Theorem 6. 12 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. IF THEN

Theorem 6. 13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. B C IF B THEN A D C

Theorem 6. 14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. IF B C C THEN B A D AC �BD A D

EXAMPLE #1 Find the measures of the numbered angles in the rhombus. 78�Why? m� 1 = ____, diagonals of a rhombus bisect the � ’s.

m� 2 = 90� Why? Diagonals of a rhombus are �. m� 3 = 12� Why? Triangle sum. m� 4 = 78� Why? AIA are �

EXAMPLE #2 One diagonal of a rectangle has length 8 x + 2. The other has length 5 x + 11. Find the length of each diagonal. The diagonals of a rectangle are �. 8 x + 2 = 5 x + 11 Diagonal = 3 x = 9 8 x + 2 x=3 8(3)+ 2 = 26 Since diagonals are �each is 26.

EXAMPLE #3 Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain. a. The quadrilateral has congruent diagonals and one angle of 60�. Impossible. A parallelogram with congruent diagonals is a rectangle with four right angles.

The quadrilateral has perpendicular diagonals and four right angles. The figure can be a parallelogram. � diagonals makes it a rhombus, 4 right angles makes it a rectangle. If a parallelogram is both a rhombus and a rectangle then it is a square.

Assignment: pg 315 #1 -15 odd, 16 -21, 48 -49, 57 -58 Must include equations and explanations whenever required.