Polygons6-1 to 6-5

Describe Polygons.Recognize and apply properties of sides and angles ofquadrilaterals.

A polygon is an enclosed plane figure that is made up of segments.

Polygons• 3 sided Triangle• 4 sided Quadrilateral• 5 sided Pentagon• 6 sided Hexagon• 7 sided Heptagon• 8 sided Octagon• 9 sided Nonagon• 10 sided Decagon• 11 sided hendecagon• 12 sided Dodecagon

FYI• Names of Polygons • 13 triskaidecagon 14 tetrakaidecagon, tetradecagon 15

pentakaidecagon, pentadecagon 16 hexakaidecagon, hexadecagon 17 heptakaidecagon 18 octakaidecagon 19 enneakaidecagon

• 20 icosagon 21 icosikaihenagon, icosihenagon 22 icosikaidigon 23 icosikaitrigon 24 icosikaitetragon 25 icosikaipentagon 26 icosikaihexagon 27 icosikaiheptagon 28 icosikaioctagon 29 icosikaienneagon

• 30 triacontagon 31 triacontakaihenagon 32 triacontakaidigon 33 triacontakaitrigon 34 triacontakaitetragon 35 triacontakaipentagon 36 triacontakaihexagon 37 triacontakaiheptagon 38 triacontakaioctagon 39 triacontakaienneagon

• 40 tetracontagon 41 tetracontakaihenagon 42 tetracontakaidigon 43 tetracontakaitrigon 44 tetracontakaitetragon 45 tetracontakaipentagon 46 tetracontakaihexagon 47 tetracontakaiheptagon 48 tetracontakaioctagon 49 tetracontakaienneagon

• 50 pentacontagon ... 60 hexacontagon ... 70 heptacontagon ... 80 octacontagon ... 90 enneacontagon ...

Identify the polygon.

PolygonsQuadrilateral

Kite Trapezoid Parallelogram

RectangleRhombus

Square

Isoscelestrapezoid

Formulas• The sum of the interiors angle of a

convex polygon is (n-2)180.• The measure of each interior angle of a

regular n-gon is (n-2)180/n• The sum of the measures of the

exterior angles of a convex polygon, one angle at each vertgex is 360.

• The measure of each exterior angle of a regular n-gon is 360/n.

Parallelogram• A parallelogram is a four-sided

figure with both pairs of opposite sides parallel.

Quadrilaterals• Quadrilaterals are four-sided

polygons.• <A + <B + <C + <D = 360°

A B

D C

Properties of a Parallelogram

1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are

congruent.3. Both pairs of opposite angles are

congruent.4. The diagonals bisect each other.5. Consecutive angles are supplementary.

Diagonal

• The diagonals of a polygon are the segments that connect any two nonconsecutive vertices.

• 1. AB // DC, AD // BC• 2. AB =DC, AD = BC• 3. <A = <C and <B = <D• 4. AM = MC and MD = MB• 5. <A + <B = 180 and <B + <C = 180• <C + <D = 180 and <D + <A = 180

A B

CD

WXYZ is a parallelogram, m<ZWX = b, and m<WXY = d. Find the values of a, b, c, and d.

W X

YZ

15

18°31

°

a

2c

22

• Ch =• GF //• <DCG =• DC =• <DCG is supplementary to __• ∆HGC =

G

D

C

F

H

In parallelogram ABCD, AB = 2x +5, m<BAC = 2y, m<B = 120, m<CAD = 21, and CD= 21. Find the values of x and y.

Quadrilateral WXYZ is a parallelogram with m<W = 47. Find the measure of angles X, Y, and Z.

Assignment• Class work on page 407• problems 9-20• Homework page 409, problems

31-36

6-3 Tests for Parallelogram

• A Quadrilateral is a parallelogram if any of the following is true.

• Both pairs of opposite sides are parallel.• Both pairs of opposite sides are congruent.• Both pairs of opposite angles are congruent.• Diagonals bisect each other.• A pair of opposite sides is both parallel and

congruent.

PolygonsQuadrilateral

Kite Trapezoid Parallelogram

RectangleRhombus

Square

Isoscelestrapezoid

Rectangle• A rectangle is a quadrilateral with

four right angles.

Properties of a Rectangle

1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent.3. Both pairs of opposite angles are congruent.4. The diagonals bisect each other.5. Consecutive angles are supplementary6. All angles are congruent7. The diagonals are congruent

1. Explain why a rectangle is a special type of parallelogram.

• All rectangles are parallelograms, but not all parallelograms are rectangles.

Ex. 2 A rectangular park has two walking paths as shown. If PS = 180 meters and PR

= 200 meters, find QT.

• 1A If TS = 120m, find PR• If m<PRS =64, find m<SQR

P Q

RS

Ex. 3 Quadrilateral MNOP is a rectangle.

Find the value of x.

• MO = 2x – 8; NP = 23• MO = 4x – 13; PC = x + 7

M N

OP

Ex. 4 Use rectangle KLMN and the given information to solve each

problem.

• M<1 = 70. Find m<2, M<5, M<6K L

MN

C

12

3456

78

9 10

Ex. 5 Quadrilateral JKLM is a rectangle. If m<KJL = 2x +4 and m<JLK = 7x + 5, find

x.

P

J K

LM

6-4 Rhombus• A rhombus is a quadrilateral with

four congruent sides.

Assignments6-4 Rectangles

• Class work on page 426, problems 10-19

• Homework – problems 26-31

Properties of a Rhombus

1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent.3. Both pairs of opposite angles are congruent.4. The diagonals bisect each other.5. Consecutive angles are supplementary6. All sides are congruent7. The diagonals are perpendicular8. The diagonals bisect the opposite angles

RhombusA B

CD

Use rhombus BCDE and the given information to find each missing value.

• If m<1 = 2x + 20 and m<2 = 5x – 4,• find the value of x.• If BD = 15, find BF.• If m<3 = y2 + 26, find y. B

C

D

E

F

12

3

Square• A square is a quadrilateral with

four right angles and four congruent sides.

Properties of a Square1. Both pairs of opposite sides are parallel.2. Both pairs of opposite sides are congruent.3. Both pairs of opposite angles are congruent.4. The diagonals bisect each other.5. Consecutive angles are supplementary6. All angles are congruent.7. The diagonals are congruent.8. All sides are congruent9. The diagonals are perpendicular.10. The diagonals bisect the opposite angles.

Assignment 6-5

• Page 435• Class work – problems 7-12• Homework – 23-33

PolygonsQuadrilateral

Kite Trapezoid Parallelogram

RectangleRhombus

Square

Isoscelestrapezoid

6-6Trapezoids and Kites

• Properties of a trapezoid• A trapezoid is a quadrilateral with

exactly one pair of parallel sides.• The angles along the legs are

supplementary.

leg leg

base

base

Trapezoid• AB // DC• M<A + m<D = 180• M<B + m<C = 180

A B

CD

Isosceles Trapezoid Properties• The legs are congruent• Both pairs of base angles are

congruent• The diagonals are congruent• Angles along the legs are

supplementary.

Isosceles Trapezoid

AD = BCm<A = m<B, m<D = m<C

AC = BDm<A + m<D = 180m<B + m<C = 180

A B

CD

PQRS is an isosceles trapezoid. Find m<P, m<Q, and m<R.

50°

P Q

RS

Midsegment of a Trapezoid

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

XY = ½(AB + DC)A B

CD

X Y

Find the length of the midsegment

• When the bases are• 7 and 11 • 3 and 7• 12 and 7 • 14 and 16

x

Find x

x

7

4

Find x

x

17

15

Find x

• AB = ½(EZ + IO)

E Z

I O

A B

4x - 10

13

3x + 8

Find x

• AB = ½(EZ + IO)

E Z

I O

A B

3x-1

10

7x+1

Kite

• Two pairs of consecutive congruent sides.

• Diagonals are perpendicular.• Exactly one pair of opposite angles

are congruent.

• Find angles j and k.

• Find each side length.

• Homework • Page 445• Problems 16-27

6-6 Assignments• Class work on page 444• problems 1-11, 16-27