OBJECTIVE: SPECIAL PARALLELOGRAMS

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Presentation transcript:

OBJECTIVE: SPECIAL PARALLELOGRAMS 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  , so BAD 1  2 AC bisects  , so 3  4 BCD

Theorem 6.10 The diagonals of a rhombus are perpendicular. THEN IF AC  BD

Rectangle Theorem Theorem 6.11 The diagonals of a rectangle are congruent. THEN IF BD  AC

Parallelogram Theorems If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. IF 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. IF A B C D THEN A B C D

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

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

m2 = 90 Why? Diagonals of a rhombus are . m3 = 12 Why? Triangle sum. AIA are  m4 = 78 Why?

. The diagonals of a rectangle are 8x + 2 = 5x + 11 Diagonal = 3x = 9 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 rectangle are . 8x + 2 = 5x + 11 Diagonal = 3x = 9 8x + 2 x = 3 8(3)+ 2 = 26 Since diagonals are  each is 26.

a. The quadrilateral has congruent diagonals and one angle of 60. 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.