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2/9/15Unit 8 Polygons and Quadrilaterals

Special Parallelograms

Rectangles, Rhombi and

Squares

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Rectangles

Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.

Definition: A rectangle is a parallelogram with four right angles.

A rectangle is a special type of parallelogram. Thus a rectangle has all the properties of a parallelogram.

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Properties of Rectangles

Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.

If a parallelogram is a rectangle, then its diagonals are congruent.

E

D C

BA

Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.

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Examples…….

1. If AE = 3x +2 and BE = 29, find the value of x.

2. If AC = 21, then BE = _______.

3. If m<1 = 4x and m<4 = 2x, find the value of x.

4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.

m<1=50, m<3=40, m<4=80, m<5=100, m<6=40

10.5 units

x = 9 units

x = 18 units

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54

321

E

D C

BA

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Rhombus

Definition: A rhombus is a parallelogram with four congruent sides.

Since a rhombus is a parallelogram the following are true: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other

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≡

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Properties of a Rhombus

Theorem: The diagonals of a rhombus are perpendicular.

Theorem: Each diagonal of a rhombus bisects a pair of opposite angles.

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Rhombus Examples .....

Given: ABCD is a rhombus. Complete the following.

5. If AB = 9, then AD = ______.

6. If m<1 = 65, the m<2 = _____.

7. m<3 = ______.

8. If m<ADC = 80, the m<DAB = ______.

9. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.

54

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21E

D C

BA9 units

65°

90°

100°

10

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Square

Opposite sides are parallel. Four right angles. Four congruent sides. Consecutive angles are supplementary. Diagonals are congruent. Diagonals bisect each other. Diagonals are perpendicular. Each diagonal bisects a pair of opposite angles.

Definition: A square is a parallelogram with four congruent angles and four congruent sides.

Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.

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Squares – Examples…...Given: ABCD is a square. Complete the following.

10. If AB = 10, then AD = _____ and DC = _____.

11. If CE = 5, then DE = _____.

12. m<ABC = _____.

13. m<ACD = _____.

14. m<AED = _____.

8 7 65

4321

E

D C

BA10 units 10 units

5 units

90°

45°

90°