Lesson 6-1: Parallelogram 1

Lesson 6-1

Parallelograms

Lesson 6-1: Parallelogram 2

Parallelogram

AB CD and BC AD

Definition: A quadrilateral whose opposite sides are parallel.

Symbol: a smaller versionof a parallelogram

Naming: A parallelogram is named using all four vertices. You can start from any one vertex, but you must

continue in a clockwise or counterclockwise direction. For example, the figure above can be either

ABCD or ADCB.

CB

A D

Lesson 6-1: Parallelogram 3

Properties of Parallelogram

AB CD and BC AD

A C and B D

180 180

180 180

m A m B and m A m D

m B m C and m C m D

2. Both pairs of opposite sides are congruent.

3. Both pairs of opposite angles are congruent.

4. Consecutive angles are supplementary.

5. Diagonals bisect each other but are not congruent

AP PC BP PDAC and BDP is the midpoint of .

A B

CDP

1. Both pairs of opposite sides are parallel

Lesson 6-1: Parallelogram 4

Examples1. Draw HKLP.

2. HK = _______ and HP = ________ .

3. m<K = m<______ .

4. m<L + m<______ = 180.5. If m<P = 65, then m<H = ____,m<K = ______ and m<L =____.

6. Draw the diagonals with their point of intersection labeled M.

7. If HM = 5, then ML = ____ .

8. If KM = 7, then KP = ____ .

9. If HL = 15, then ML = ____ .

10. If m<HPK = 36, then m<PKL = _____ .

H K

LP

PL KL

P

P or K

115° 65 115°

M

5 units

14 units7.5 units

36; (Alternate interior angles are congruent.)

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 6-3: Rectangles 7

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.

Lesson 6-3: Rectangles 8

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

Theorem:

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

Lesson 6-3: Rectangles 9

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

6

54

321

E

D C

BA

Lesson 6-4: Rhombus & Square 10

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

≡

≡

Lesson 6-4: Rhombus & Square 11

Properties of a Rhombus

Theorem: The diagonals of a rhombus are perpendicular.

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

Lesson 6-4: Rhombus & Square 12

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 13

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 14

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 15

TrapezoidA quadrilateral with exactly one pair of parallel sides.Definition:

BaseLeg

An Isosceles trapezoid is a trapezoid with congruent legs.

Trapezoid

The parallel sides are called bases and the non-parallel sides are called legs.

Isosceles trapezoid

Lesson 6-5: Trapezoid & Kites 16

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

Lesson 6-5: Trapezoid & Kites 17

The median of a trapezoid is the segment that joins the midpoints of the legs.

The median of a trapezoid is parallel to the bases, and its measure is one-half the sum of the measures of the bases.

Median

1b

2b

1 2

1( )

2median b b

Median of a Trapezoid

Lesson 6-5: Trapezoid & Kites 18

IsoscelesTrapezoid

Quadrilaterals

Rectangle

Parallelogram

Rhombus

Square

Flow Chart

Trapezoid