Polygons and Quadrilaterals

Chapter 6 JournalChristian Aycinena 9-5

Polygons A polygon is a •closed figure •formed by three or more segments •each segment meet with another segment at a vertex •no two segments with a common endpoint are collinear.

Polygons Not Polygons

Parts of a PolygonEach segment of a polygon is a side of the polygon. The vertex of the polygon is a common endpoint of two sides. A diagonal is a segment that connects any two nonconsecutive vertices.

A

B

E

D

C

FVertex

Diagonal

Side

Sides Polygon Name

34567891012n

-Triangle-Quadrilateral-Pentagon-Hexagon-Heptagon-Octagon-Nonagon-Decagon-Dodecagon-n-gon

Polygon names by the number of its sides.

Polygon ABCDEF is a hexagon.

Examples

Polygon, Heptagon

Polygon, Triangle

Polygon, octagon

Polygon, decagonNot a polygon

Polygon, Quadrilateral

Polygon, 17-gon Not a polygon

Not a polygon

Regular PolygonIn an equilateral polygon all the sides are congruent.In a equiangular polygon all the angles are congruentA regular polygon is one that is both equilateral and equiangular, otherwise it is irregular.

Concave and ConvexA convex polygon has all of the vertices pointing outward. (no internal angles can be more than 180°). A regular polygon is always convex.

When one of the vertices is pointing inward the it is a concave polygon. (If there are any internal angles greater than 180° then it is also concave). *Helpful(Think: concave has a "cave" in it)

ExamplesTell if it is convex or concave.

Convex Concave

Convex Concave

ConcaveConvex

Interior Angles TheoremTo find the sum of the interior angles of a convex polygon, draw all the diagonals from one vertex of that polygon. This creates triangles and the sum of the angle measure of all the triangles created equals the sum of the angle measure of the polygon.

Triangle

Quadrilateral

1

2

Pentagon

1 2 3

Hexagon

1 2 3 4 1 2 3 4

5

Octagon

And more--- n-gon

Polygon # of Sides

# of Triangles

Sum of Int. < Measures

Triangle 3 1 (1)180=180

Quadrilateral 4 2 (2)180=180

Pentagon 5 3 (3)180=180

Hexagon 6 4 (4)180=180

Heptagon 7 5 (5)180=180

N-gon n n-2 (n-2)180=180

In each convex polygon, the number of triangles is two less than the number of sides n. So the sum of the angle measure of all these triangles is (n-2)180.

Theorem 6-1-1-The sum of the interior angle measures of a convex polygon with n sides is (n-2)180. So for example in a quadrilateral—(4-2)=2*180=360 is what all the angles measure in a quadrilateral. To find the measure of each angle you divide the sum of the interior angle by the number of sides of the polygon.

Examples:1. Sum of an Pentagon interior angle measures:(n-2)180(5-2)1803*180540°2. Measure of Nonagon interior angle:(n-2)180(9-2)180 = 1260° A D 1260°/9 = 140° each interior angle of a nonagon.3.Find the measure of each interior angle in the fountain.

(4-2)180 = 360 B m<A+ m<B + m<C+ m<D = 360 C x + 5x + x +5x =36012x = 360 x = 30

m<A = m< C = 30° m<B = m<D = 150°

x°

x°5x°

5x°

Exterior AngleThe sum of the exterior angle with one angle at each vertex is always 360° in a convex polygon. (Hint: Usually you add the exterior angle going clockwise in the convex polygon.)

139

141

80

139 + 141 +80 = 360

36

41

38

49

47

57

52

40

36+38+49+52+41+57+47+40 = 360

110

49

86

54

61

54+61+49+110

ParallelogramsFirst of all a parallelogram is a quadrilateral with two pairs of parallel sides. It follows some properties that they will be shown on the next page. The symbol of a parallelogram is

B

A

C

D

AB || CD, BC|| DAParallelogram ABCD ABCD

Here is the first property, by the definition of a parallelogram you can conclude:-If a quadrilateral is a parallelogram, then it has two pairs of parallel sides.Converse: If a quadrilateral has two pairs of parallel sides, then it is as parallelogram.

A B

CD

ABCD is a parallelogram

E

F G

HEFGH is a parallelogram

W X

YZ WXYZ is a parallelogram

After you could state that AB || DC, AD || BCEF || GH, FG || EHWX || ZY, WZ || XY

A B

CD

E

F G

H

W

Z

X

Y

After you could state thatABCD is a parallelogramEFGH is a parallelogramWXYZ is a parallelogram

TheoremsIf a quadrilateral is a parallelogram, then its opposite sides are congruent ( --- opp. Sides =)If a quadrilateral has opposite sides congruent, then it is a parallelogram.

Examples:A B

CD

E

F G

EFGH is a parallelogram

W X

YZWXYZ is a parallelogram

ABCD is a parallelogram

H

You could state thatAB = DC, AD = BCEF = GH, FG = EHWX = ZY, WZ = XY

1.

A B

CD

E

F G

H

W

Z

X

Y

You could state that ABCD is a parallelogramEFGH is a parallelogramWXYZ is a parallelogram

2.

If a quadrilateral is a parallelogram, then its opposite angles are congruent ( --- opp. <´s =)If a quadrilateral has opposite congruent angles, then it is a parallelogram.

A B

CD

E

F G

EFGH is a parallelogram

W X

YZWXYZ is a parallelogram

ABCD is a parallelogram

H

You could state that<A = <C, <B = <D<E = <G, <F = <H<W = <Y, <X = <Z

A B

CD

E

F G

H

W

Z

X

Y

You could state that ABCD is a parallelogramEFGH is a parallelogramWXYZ is a parallelogram

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. ( --- cons. <´s supp.)If a quadrilateral has supplementary consecutive angles , then it is a parallelogram.

A B

CD

E

F G

EFGH is a parallelogram

W X

YZWXYZ is a parallelogram

ABCD is a parallelogram

H

You could state thatm<A + m< B =180m<B + m<C =180m<C + m<D = 180m<D + m<A =180

m<E + m<F= 180 m<F + m<G= 180m<G + m<H= 180m<H + m<E = 180

m<W + m<X=180m<X + m<Y= 180m<Y + m<Z = 180m<Z + m<W= 180

If a quadrilateral is a parallelogram, then its diagonals bisect each other. ( --- diags. Bisect each other)If a quadrilateral has diagonals that bisect each other, then it is a parallelogram.

A B

CD

W X

YZ

K

L

You could state thatAK = KCDK = KB

You could state thatWL = LYXL = ZL

You could state thatEÑ = ÑGHÑ = ÑF

E F

GH

Ñ

A B

CD

K

You could state that ABCD is a parallelogram

E F

GH

Ñ

You could state that EFGH is a parallelogramW X

YZ

L

You could state that WXYZ is a parallelogram

Proving Quadrilaterals areParallelograms

Quadrilaterals have two pairs of opposite sides are parallelOpposite sides are congruentOpposite angles are congruentConsecutive angles are supplementaryDiagonals bisect each otherOne set of congruent and parallel sides

Examples:

2 pairs of parallel sidesOpposite sides congruent Opposite angles congruent

150

30

Consecutive angles are supplementary

Diagonals Bisect each other One set of congruent and parallel sides

Example:

Rhombus, Rectangle, SquareAll of are parallelograms and contain all the properties a parallelogram contains; also they contain their own properties.

Rectangle: is a quadrilateral with 4 right anglesRhombus: is a quadrilateral with four congruent sidesSquare: is a combination of a rectangle and rhombus(4 rt. <´s and 4 = sides)

TheoremsRectangle Properties If a quadrilateral is a rectangle, then it is a parallelogram. ( rect.---- )If a parallelogram is a rectangle, then its diagonals are congruent. (rect. --- diags. =)

A B

CD

E F

GH

W X

YZ

After you could state thatABCD is a parallelogram

EFGH is a parallelogram

WXYZ is a parallelogram

A B

CD

AC = BD

E F

GH

EG = FH

W X

YZ

WY = XZ

Rhombus PropertiesIf a quadrilateral is a rhombus then it is a parallelogram. (rhombus-- )If a parallelogram is a rhombus , then its diagonals are perpendicular. (rhombus—diags. Perpendicular)If a parallelogram, is a rhombus, then each diagonal bisects a pair of opposite angles.

A B

CD

EF

GH

W

X

Y

Z

After you could state thatABCD is a parallelogramEFGH is a parallelogramWXYZ is a parallelogram

A B

CD

EF

GH

AC is perpendicular to BD EG is perpendicular to HF YW perpendicular to XZ

W

X

Y

Z

A B

CD

1 23 4

5 6

7 8

W

X

Y

Z

1 23 4

5 67 8

EF

GH

5 6

1 2

3 4

7 8

<1 = <2<3= <4<5 = <6<7 = <8

<1 = <2<3= <4<5 = <6<7 = <8

<1 = <2<3= <4<5 = <6<7 = <8

Square Properties: Follows all the properties of a parallelogram, rhombus and rectangle.

TrapezoidQuadrilateral with exactly one pair of parallel sides. The parallel sides are called the base and the nonparallel are the legs. Base angles are two consecutive angles whose common side is a base.

Base

Base

Leg

Leg

Base Angles

Isosceles Trapezoid: one pair of congruent legs.

Isosceles Trapezoids If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent.

If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.

A trapezoid is isosceles if and only if its diagonals are congruent.

KiteQuadrilaterals with two pairs of congruent consecutive sides.Properties: If a quadrilateral is a kite, then its diagonals are perpendicular.

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

KITES