G.9 Quadrilaterals Part 1

G.9Quadrilaterals

Part 1Parallelograms

Modified by Lisa Palen

Definition• A parallelogram is a quadrilateral whose

opposite sides are parallel.

• Its symbol is a small figure:

CB

A D

AB CD and BC AD

Naming a Parallelogram

• 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, this can be either ABCD or ADCB. CB

A D

Basic Properties• There are four basic properties of all

parallelograms.• These properties have to do with the angles,

the sides and the diagonals.

Opposite Sides

Theorem Opposite sides of a parallelogram are congruent.

• That means that .• So, if AB = 7, then _____ = 7?

CB

A D

ABCD and BC AD

Opposite Angles

• One pair of opposite angles is A and C. The other pair is B and D.

CB

A D

Opposite Angles

Theorem Opposite angles of a parallelogram are congruent.

• Complete: If m A = 75 and m B = 105, then m C = ______ and m D = ______ .

CB

A D

Consecutive Angles

• Each angle is consecutive to two other angles. A is consecutive with B and D.

CB

A D

Consecutive Angles in Parallelograms

Theorem Consecutive angles in a parallelogram are supplementary.

• Therefore, m A + m B = 180 and m A + m D = 180.

• If m<C = 46, then m B = _____?

CB

A D

Consecutive INTERIOR Angles are

Supplementary!

Diagonals• Diagonals are segments that join non-

consecutive vertices.• For example, in this diagram, the only two

diagonals are .AC and BD

CB

A D

Diagonal PropertyWhen the diagonals of a parallelogram intersect, they

meet at the midpoint of each diagonal.• So, P is the midpoint of .• Therefore, they bisect each other;

so and .• But, the diagonals are not congruent!

AC and BD

AP PC BPPD

P

CB

A D

AC BD

Diagonal PropertyTheorem The diagonals of a parallelogram bisect each

other.

P

CB

A D

Parallelogram Summary • By its definition, opposite sides are parallel. Other properties (theorems): • Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• The diagonals bisect each other.

Examples

• 1. Draw HKLP. • 2. Complete: 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 =______ .

Examples (cont’d)

• 6. Draw in the diagonals. They intersect at M.• 7. Complete: 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 = _____ .

Part 2

Tests for Parallelograms

Review: Properties of Parallelograms

• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• The diagonals bisect each other.

How can you tell if a quadrilateral is a parallelogram?

• Defn: A quadrilateral is a parallelogram iff opposite sides are parallel.

• Property If a quadrilateral is a parallelogram, then opposite sides are parallel.

• Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram.

Proving Quadrilaterals as Parallelograms

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram .

Theorem 1:

H G

E FIf one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

Theorem 2:

If EF GH; FG EH, then Quad. EFGH is a parallelogram.

If EF GH and EF || HG, then Quad. EFGH is a parallelogram.

Theorem:

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 3:

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram .

Theorem 4:

H G

EF

M

,If H F and E G

then Quad. EFGH is a parallelogram.

intIf M is themidpo of EG and FH

then Quad. EFGH is a parallelogram. EM = GM and HM = FM

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 values 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

2y = y + 2

y = 2

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

5y + 120 = 180

5y = 60

y = 12

Lesson 6-3: Rectangles 23

Part 3

Rectangles

Lesson 6-3: Rectangles 24

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 quadrilateral with four right angles.

Is a rectangle is a parallelogram?

Thus a rectangle has all the properties of a parallelogram.

Yes, since opposite angles are congruent.

Lesson 6-3: Rectangles 25

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 26

Properties of Rectangles

E

D C

BA

Parallelogram Properties:Opposite sides are parallel.Opposite sides are congruent.Opposite angles are congruent.Consecutive angles are supplementary.Diagonals bisect each other.Plus:All angles are right angles.Diagonals are congruent.

Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles

Lesson 6-3: Rectangles 27

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 28

Part 4

Rhombi and

Squares

Lesson 6-4: Rhombus & Square 29

Rhombus

Definition: A rhombus is a quadrilateral 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.

≡

≡Is a rhombus a parallelogram?

Yes, since opposite sides are congruent.

Lesson 6-4: Rhombus & Square 30

Rhombus

Note: The four small triangles are congruent, by SSS.

≡

≡This means the diagonals form four angles that are congruent, and must measure 90 degrees each.

So the diagonals are perpendicular.

This also means the diagonals bisect each of the four angles of the rhombus

So the diagonals bisect opposite angles.

Lesson 6-4: Rhombus & Square 31

Properties of a RhombusTheorem: The diagonals of a rhombus are perpendicular.

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

Note: The small triangles are RIGHT and CONGRUENT!

Lesson 6-4: Rhombus & Square 32

Properties of a Rhombus

.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.Plus:• All four sides are congruent.• Diagonals are perpendicular.• Diagonals bisect opposite angles.• Also remember: the small triangles are RIGHT and CONGRUENT!

≡

≡

Lesson 6-4: Rhombus & Square 33

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

34

Square

• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.Plus:• Four right angles.• Four congruent sides.• Diagonals are congruent.• Diagonals are perpendicular.• Diagonals bisect opposite angles.

Definition:A square is a quadrilateral 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 35

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 36

Part 5

Trapezoids

and Kites

Lesson 6-5: Trapezoid & Kites 37

Trapezoid

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

Base

Leg Trapezoid

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

Leg

Base

Lesson 6-5: Trapezoid & Kites 38

The median of a trapezoid is the segment that joins the midpoints of the legs. (It is sometimes called a midsegment.)

• Theorem - The median of a trapezoid is parallel to the bases.

• Theorem - The length of the median is one-half the sum of the lengths of the bases.

Median

1b

2b

1 2

1( )

2median b b

Median of a Trapezoid

Lesson 6-5: Trapezoid & Kites 39

Isosceles Trapezoid

A trapezoid with congruent legs.Definition:

Isosceles trapezoid

Lesson 6-5: Trapezoid & Kites 40

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

AC DB

Lesson 6-5: Trapezoid & Kites 41

Kite

A quadrilateral with two distinct pairs of congruent adjacent sides.

Definition:

Theorem: Diagonals of a kite are perpendicular.

Lesson 6-5: Trapezoid & Kites 42

IsoscelesTrapezoid

Quadrilaterals

Rectangle

Parallelogram

Rhombus

Square

Flow Chart

TrapezoidKite