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 Theorems Theorem 6.9 Each diagonal of a rhombus bisects two angles of the rhombus. AC bisects BAD , so 1 2 ACbisects BCD , so 3 4

Theorem 6.10 The diagonals of a rhombus are perpendicula r. THEN IF AC BD

Rectangle Theorem 6.11 Theorem 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 IF rhombus. THEN

Theorem 6.13 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a B B C rhombus. IF THEN A D A C D

Theorem 6.14 If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. IF B C THEN B A D AC BD A C D

EXAMPLE #1 Find the measures of the numbered angles in the rhombus. m 1 78 , Why? 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 8x 2. The other has length 5x 11. Find the length of each diagonal. The diagonals of a 8x 2 5x rectangle are Diagonal . 113x 8x 9x 2 8(3) 2 3 diagonals are 26 each is Since 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 . Impossibl A e. 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.