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Polygons

5 ways to prove that a quadrilateral is a parallelogram.

1. Show that both pairs of opposite sides are || . [definition]

2. Show that both pairs of opposite sides are .

3. Show that one pair of opposite sides are both and || .4. Show that both pairs of opposite angles are .

5. Show that the diagonals bisect each other .

Examples ……Find the value of x and y that ensures the quadrilateral is a parallelogram.

Example 1:

6x4x+8

y+2

2y

6x = 4x+8

2x = 8

x = 4 units

2y = y+2

y = 2 unit

Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.

120° 5y°

(2x + 8)°2x + 8 = 120

2x = 112

x = 56 units

5y + 120 = 180

5y = 60

y = 12 units

Lesson 6-4: Rhombus & Square 5

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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Lesson 6-4: Rhombus & Square 6

Rhombus Examples .....

Given: ABCD is a rhombus. Complete the following.

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

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

3. m<3 = ______.

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

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

54

3

21E

D C

BA9 units

65°

90°

100°

10

Lesson 6-4: Rhombus & Square 7

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.

Lesson 6-4: Rhombus & Square 8

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

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

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

3. m<ABC = _____.

4. m<ACD = _____.

5. m<AED = _____.

8 7 65

4321

E

D C

BA10 units 10 units

5 units

90°

45°

90°

Lesson 6-5: Trapezoid & Kites 9

Properties of Isosceles Trapezoid

A B and D C

2. The diagonals of an isosceles trapezoid are congruent.

1. Both pairs of base angles of an isosceles trapezoid are congruent.

A B

CD

Base Angles

AC DB

KITE

1. Two sides of adjacent sides congruent

2. Diagonals are perpendicular

Note: opposite sides are not congruent Note: diagonals do not bisect each other