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LESSON 6.4 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. AC. AC. Rhombus Theorems. Theorem 6.9. Each diagonal of a rhombus. bisects two angles of the rhombus. BAD.

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LESSON 6.4 SPECIAL PARALLELOGRAMS

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  1. LESSON 6.4 SPECIAL PARALLELOGRAMS OBJECTIVE: Use properties of diagonals of rhombuses and rectangles Determine whether a parallelogram is a rhombus or rectangle

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

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

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

  5. 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

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

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

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

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

  10. 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.

  11. 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.

  12. 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.

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

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