Chapter 6Quadrilaterals

Section 6.1

Polygons

Polygon• A polygon is formed by three or more

segments called sides– No two sides with a common endpoint are

collinear.– Each side intersects exactly two other sides,

one at each endpoint.– Each endpoint of a side is a vertex of the

polygon.– Polygons are named by listing the vertices

consecutively.

Identifying polygons

• State whether the figure is a polygon. If not, explain why.

Polygons are classified by the number of sides they have

NUMBER OF SIDES

TYPE OF POLYGON

3

4

5

6

7

NUMBER OF SIDES

TYPE OF POLYGON

8

9

10

12

N-gon

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

dodecagon

N-gon

Two Types of Polygons:

1. Convex: If a line was extended from the sides of a polygon, it will NOT go through the interior of the polygon.

Example:

2. Concave: If a line was extended from the sides of a polygon, it WILL go through the interior of the polygon.

Example:

I. Use the number of sides to tell what kind of polygon the shape is. Then state whether the polygon is convex or concave. 1. 2. 3.

__________________ __________________ __________________

__________________ __________________ __________________

Regular Polygon• A polygon is regular if it is equilateral

and equiangular

• A polygon is equilateral if all of its sides are congruent

• A polygon is equiangular if all of its interior angles are congruent

Diagonal• A segment that joins two

nonconsecutive vertices.

II. Use the diagram at the right to answer the following.

4. Name the polygon by the number of sides it has. __________________

5. Polygon ABCDEFG is one name for the polygon. State two other names.

Polygon _______________ Polygon _______________

6. Name all the diagonals that have vertex E as an endpoint.

_____________________________________________

7. Name the nonconsecutive angles to A . _______________________________

Interior Angles of a Quadrilateral Theorem

• The sum of the measures of the interior angles of a quadrilateral is 360°

1

4

2

3

3604321 mmmm

Section 6.2Properties of

Parallelograms

Parallelogram• A quadrilateral with both pairs of

opposite sides parallel

Theorem 6.2

• Opposite sides of a parallelogram are congruent.

Theorem 6.3• Opposite angles of a

parallelogram are congruent

Theorem 6.4• Consecutive angles of a

parallelogram are supplementary.

1

2 3

4

18021 mm

Theorem 6.5• Diagonals of a parallelogram

bisect each other.

Examples of P roblems: I. Decide whether the figure is a parallelogram. If it is not, explain why not.

1. 2. 3. ________________________ ________________________ ________________________

________________________ ________________________ ________________________

K

H

J

G

L

6 8

U

W

V

X

Z

II. Use the diagram of the parallelogram to complete the statement. R efer to parallelogram GHJK to complete each statement.

4. ________JH 5. ________LH 6. ________KH

R efer to parallelogram UVWX to complete each statement.

7. If 15XU and 28UW , find WZ . ___________ 8. If 120VWXm , find WXUm . ___________ 9. If 55UVWm and 87xVWXm , find x. ___________

III. In ABCD , 105Cm . Find the measure of each angle.

10. ________Am 11. ________Dm

IV. Find the value of each variable in the parallelogram.

12. 13.

Section 6.3

Proving Quadrilaterals are Parallelograms

Theorem 6.6To prove a quadrilateral is a

parallelogram:

• Both pairs of opposite sides are congruent

Theorem 6.7 To prove a quadrilateral is a

parallelogram:

• Both pairs of opposite angles are congruent.

Theorem 6.8 To prove a quadrilateral is a

parallelogram:

• An angle is supplementary to both of its consecutive angles.

18021 mm

1

2 3

4

18032 mm

Theorem 6.9 To prove a quadrilateral is a

parallelogram:

• Diagonals bisect each other.

Theorem 6.10 To prove a quadrilateral is

a parallelogram:

• One pair of opposite sides are congruent and parallel.

>

>

Are you given enough information to determine whether the quadrilateral is a parallelogram? Explain. 1. 2. 3. 4.

5. 6. 7. 8.

What value of x will make the polygon a parallelogram? 1. 2. 3.

Section 6.4Types of

parallelograms

Rhombus• Parallelogram with four congruent

sides.

Properties of a rhombus

• Diagonals of a rhombus are perpendicular.

Properties of a rhombus

• Each Diagonal of a rhombus bisects a pair of opposite angles.

Rectangle• Parallelogram with four right angles.

Properties of a rectangle

• Diagonals of a rectangle are congruent.

Square• Parallelogram with four congruent

sides and four congruent angles.

• Both a rhombus and rectangle.

Properties of a square

• Diagonals of a square are perpendicular.

Properties of a square

• Each diagonal of a square bisects a pair of opposite angles.

45°45°

45°45° 45°

45°

45°45°

Properties of a square

• Diagonals of a square are congruent.

3-Way Tie

Rectangle

Rhombus Square

Section 6.5Trapezoids and Kites

Trapezoid• Quadrilateral with exactly one pair of

parallel sides.

• Parallel sides are the bases.

• Two pairs of base angles.

• Nonparallel sides are the legs.

>

>Base

Base

Leg Leg

Isosceles Trapezoid

• Legs of a trapezoid are congruent.

Theorem 6.14• Base angles of an isosceles trapezoid are

congruent.

>

>

A B

CD

DCBA ,

Theorem 6.15• If a trapezoid has one pair of congruent

base angles, then it is an isosceles trapezoid.

>

>

A B

CD

ABCD is an isosceles trapezoid

Theorem 6.16

• Diagonals of an isosceles trapezoid are congruent.

>

A B

CD

>

ABCD is isosceles if and only if BDAC

Examples on Board

Midsegment of a trapezoid

• Segment that connects the midpoints of its legs.

Midsegment

Midsegment Theoremfor trapezoids

• Midsegment is parallel to each base and its length is one half the sum of the lengths of the bases.

A

B C

D

MN

MN= (AD+BC)2

1

Examples on Board

Kite

• Quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorem 6.18

• Diagonals of a kite are perpendicular.

A

B

C

D

BDAC

Theorem 6.19• In a kite, exactly one pair of opposite angles are

congruent.

A

B

C

D

DBCA ,

Examples on Board

Pythagorean Theorem

a

b

c

222 cba

Section 6.6Special Quadrilaterals

Properties of QuadrilateralsProperty Rectangle Rhombus Square Trapezoid Kite

Both pairs of opposite sides are congruent

Diagonals are congruent

Diagonals are perpendicular

Diagonals bisect one another

Consecutive angles are supplementary

Both pairs of opposite angles are congruent

X

X

X

X X

X

X X X

X X X X

X

X X X

X X X X

Properties of Quadrilaterals

• Quadrilateral ABCD has at least one pair of opposite sides congruent. What kinds of quadrilaterals meet this condition?

PARALLELOGRAM RHOMBUSRECTANGLE SQUARE ISOSCELES

TRAPEZOID

Section 6.7

Areas of Triangles and Quadrilaterals

Area Congruence Postulate

• If two polygons are congruent, then they have the same area.

Area Addition Postulate

• The area of a region is the sum of the areas of its non-overlapping parts.

Area Formulas

2sA

PARALLELOGRAM RECTANGLE SQUARE

A=bh A=lw

TRIANGLE

bhA2

1

Area Formulas

1d

2d

212

1ddA

RHOMBUS KITE

1d

2d

212

1ddA

Area Formulas

TRAPEZOID

1b

2b

h

)(2

121 bbhA