6.1 Polygons

Mrs. Gibson

Geometry

Spring 2011

What is polygon?

Formed by three or more segments (sides).

Each side intersects exactly two other sides, one at each endpoint.

Has vertex/vertices.

Polygons are named by the number of sides they have. These are the most commonly used.

Number of sides Type of polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

11 Hendecagon

12 Dodecagon

Concave vs. Convex

Convex: if no line that contains a side of the polygon contains a point in the interior of the polygon.

Concave: if a polygon is not convex.

interior

Example

Identify the polygon and state whether it is convex or concave.

Concave polygon Convex polygon

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

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

A polygon is regular if it is equilateral and equiangular.

Decide whether the polygon is regular.

)

)

)

)

)

))

))

))

Convex polygons have a total sum of angles that is based on the number of triangles inside.

The formula is 180 x (number of sides – 2)

Number of sides

Type of polygon

Total sum of interior angles

3 Triangle 180

4 Quadrilateral 360

5 Pentagon 540

6 Hexagon 720

7 Heptagon 900

8 Octagon 1080

9 Nonagon 1260

10 Decagon 1440

11 Hendecagon 1620

12 Dodecagon 1800

Polygons are named by the number of sides they have. These are the most commonly used.

Number of sides

Type of polygon Total sum of interior angles

One regular interior angle

3 Triangle 180 60º

4 Quadrilateral 360 90º

5 Pentagon 540 108º

6 Hexagon 720 120º

7 Heptagon 900 129º

8 Octagon 1080 135º

9 Nonagon 1260 140º

10 Decagon 1440 144º

11 Hendecagon 1620 147º

12 Dodecagon 1800 150º

A Diagonal of a polygon is a segment that joins two nonconsecutive vertices.

diagonals

Polygons are named by the number of sides they have. These are the most commonly used.Number of

sidesType of polygon Total sum of

interior anglesOne regular interior angle

# of diagonals

3 Triangle 180 60º 0

4 Quadrilateral 360 90º 2

5 Pentagon 540 108º 5

6 Hexagon 720 120º 9

7 Heptagon 900 129º 14

8 Octagon 1080 135º 20

9 Nonagon 1260 140º 27

10 Decagon 1440 144º 35

11 Hendecagon 1620 147º 44

12 Dodecagon 1800 150º 54

Interior Angles of a Quadrilateral Theorem

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

A

B

C

D

m<A + m<B + m<C + m<D = 360°

Example

Find m<Q and m<R.

R

x

P

S

2x°

Q

80°

70°

x + 2x + 70° + 80° = 360° 3x + 150 ° = 360 ° 3x = 210 ° x = 70 °

m< Q = xm< Q = 70 ° m<R = 2x

m<R = 2(70°)m<R = 140 °

Find m<A

A

B

C

D

65°

55°

123°

Polygons are named by the number of sides they have. These are the most commonly used.Number of sides

Type of polygon Total sum of interior

angles

One regular interior angle

# of diagonals

One regular exterior angle

3 Triangle 180 60º 0 120º

4 Quadrilateral 360 90º 2 90º

5 Pentagon 540 108º 5 72º

6 Hexagon 720 120º 9 60º

7 Heptagon 900 129º 14 51º

8 Octagon 1080 135º 20 45º

9 Nonagon 1260 140º 27 40º

10 Decagon 1440 144º 35 36º

11 Hendecagon 1620 147º 44 33º

12 Dodecagon 1800 150º 54 30º

6.2 Properties of Parallelograms

Big Daddy Flynn

Geometry

Winter 2014

DEFINITION A parallelogram is a quadrilateral whose

opposite sides are parallel.

Q R

SP

PQ // SR and SP // QR

Theorems If a quadrilateral is a parallelogram, then its

opposite sides are congruent.

If a quadrilateral is a parallelogram, then its opposite angles are congruent.

Q R

SP

RSPQ QRSP

RP SQ

Theorems If a quadrilateral is a parallelogram, then its

consecutive angles are supplementary.

m<P + m<Q = 180°

m<Q + m<R = 180°

m<R + m<S = 180°

m<S + m<P = 180°

Q R

SP

Using Properties of Parallelograms

PQRS is a parallelogram. Find the angle measure. m< R m< Q

Q R

SP70°

70 °

70 ° + m < Q = 180 °

m< Q = 110 °

Using Algebra with Parallelograms

PQRS is a parallelogram. Find the value of h.

P Q

RS3h 120°

Theorems

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

Q R

SP

MRMPM

SMQM

Using properties of parallelograms

FGHJ is a parallelogram. Find the unknown length. JH JK

F G

HJ

K

5

3

5

3

Examples Use the diagram of parallelogram JKLM.

Complete the statement.

____.6

____.5

____.4

____.3

____.2

____.1

KL

JN

JKL

MLK

MN

JK K L

MJ

N

LM

NK

<KJM

<LMJ

NL

MJ

Find the measure in parallelogram LMNQ.

1. LM

2. LP

3. LQ

4. QP

5. m<LMN

6. m<NQL

7. m<MNQ

8. m<LMQ

L M

NQ

P

10

9

32°

110°

8

18

18

8

9

10

70°

70 °

110 °

32 °

6.3 Proving Quadrilaterals are Parallelograms

Mistah Flynn

Geometry

2014

Remember these crazy formulas?

2122

12

12

12

yyxxd

xx

yy

run

riseslope

Using properties of parallelograms.

Method 1Use the slope formula to show that opposite sides have the same slope, so they are parallel.

Method 2Use the distance formula to show that the opposite sides have the same length.

Method 3Use both slope and distance formula to show one pair of opposite side is congruent and parallel.

Let’s apply~

Show that A(2,0), B(3,4), C(-2,6), and D(-3,2) are the vertices of parallelogram by using method 1.

Show that the quadrilateral with vertices A(-3,0), B(-2,-4), C(-7, -6) and D(-8, -2) is a parallelogram using method 2.

Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

Proving quadrilaterals are parallelograms

Show that both pairs of opposite sides are parallel.

Show that both pairs of opposite sides are congruent.

Show that both pairs of opposite angles are congruent.

Show that one angle is supplementary to both consecutive angles.

.. continued..

Show that the diagonals bisect each other Show that one pair of opposite sides are

congruent and parallel.

Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

Example 4 – p.341

Show that A(2,-1), B(1,3), C(6,5), and D(7,1) are the vertices of a parallelogram.

6.4 Rhombuses, Rectangles, and Squares

Mr. “Can Brenda see this?” Flynn

Geometry

2014

Proving quadrilaterals are parallelograms

Show that both pairs of opposite sides are parallel.

Show that both pairs of opposite sides are congruent.

Show that both pairs of opposite angles are congruent.

Show that one angle is supplementary to both consecutive angles.

.. continued..

Show that the diagonals bisect each other Show that one pair of opposite sides are

congruent and parallel.

Special Parallelograms

Rhombus A rhombus is a parallelogram with four

congruent sides.

Special Parallelograms

Rectangle A rectangle is a parallelogram with four right

angles.

Special Parallelogram

Square A square is a parallelogram with four

congruent sides and four right angles.

Corollaries

Rhombus corollary A quadrilateral is a rhombus if and only if it

has four congruent sides.

Rectangle corollary A quadrilateral is a rectangle if and only if it

has four right angles.

Square corollary A quadrilateral is a square if and only if it is a

rhombus and a rectangle.

Example

PQRS is a rhombus. What is the value of b?

P Q

RS

2b + 3

5b – 6

2b + 3 = 5b – 6 9 = 3b 3 = b

Review

In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___

A) 1

B) 2

C) 3

D) 4

E) 5

7f – 3 = 4f + 9

3f – 3 = 9

3f = 12

f = 4

Example

PQRS is a rhombus. What is the value of b?

P Q

RS

3b + 12

5b – 6

3b + 12 = 5b – 6 18 = 2b 9 = b

Theorems for rhombus

A parallelogram is a rhombus if and only if its diagonals are perpendicular.

A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

L

Theorem of rectangle

A parallelogram is a rectangle if and only if its diagonals are congruent.

A B

CD

Match the properties of a quadrilateral

1. The diagonals are congruent

2. Both pairs of opposite sides are congruent

3. Both pairs of opposite sides are parallel

4. All angles are congruent

5. All sides are congruent

6. Diagonals bisect the angles

A. Parallelogram

B. Rectangle

C. Rhombus

D. Square

B,D

A,B,C,D

A,B,C,D

B,D

C,D

C

6.5 Trapezoid and Kites

Mrs. Gibson

Geometry

Spring 2011

Let’s define Trapezoid

base

base

leg leg

>

>A B

CD

<D AND <C ARE ONE PAIR OF BASE ANGLES.

When the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

Isosceles Trapezoid

If a trapezoid is isosceles, then each pair of base angles is congruent.

A B

CD

DCBA ,

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

S R

P Q

50°

>

>

Isosceles Trapezoid

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

A B

CD

BDAC

Midsegment Theorem for Trapezoid The midsegment of a trapezoid is parallel to

each base and its length is one half the sum of the lengths of the bases. It’s the average of the lengths of the bases.

A

B C

D

M N

)(2

1BCADMN

Examples

The midsegment of the trapezoid is RT. Find the value of x.

7

R Tx

14

x = ½ (7 + 14)x = ½ (21)x = 21/2

Examples

The midsegment of the trapezoid is ST. Find the value of x.

8

S T11

x

11 = ½ (8 + x)22 = 8 + x14 = x

Review

In a rectangle ABCD, if AB = 7x – 3, and CD = 4x + 9, then x = ___

A) 1

B) 2

C) 3

D) 4

E) 5

7x – 3 = 4x + 9-4x -4x 3x – 3 = 9 + 3 +3 3x = 12 x = 4

Kite

A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Theorems about Kites

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.

A

B

C

D

L

DBCA ,

Example

Find m<G and m<J.

G

H

J

K132° 60°

Since m<G = m<J,2(m<G) + 132° + 60° = 360°2(m<G) + 192° = 360°2(m<G) = 168°m<G = 84°

Example

Find the side length.

G

H

J

K

12

12

1214

6.6 Special Quadrilaterals

Mrs. Gibson

Geometry

Spring 2011

Summarizing Properties of Quadrilaterals

Quadrilateral

Kite Parallelogram Trapezoid

Rhombus Rectangle

Square

Isosceles Trapezoid

Identifying Quadrilaterals

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

Sketch KLMN. K(2,5), L(-2,3), M(2,1), N(6,3).

Show that KLMN is a rhombus.

Copy the chart. Put an X in the box if the shape

always has the given property.

Property Parallelogram

Rectangle Rhombus Square Kite Trapezoid

Both pairs of opp. sides are ll

Exactly 1 pair of opp. Sides are ll

Diagonals are perp.

Diagonals are cong.

Diagonals bisect each other

XX X X

X

X XX

X X

X X

Determine whether the statement is true or false. If it is true, explain why. If it is false, sketch a counterexample. If CDEF is a kite, then CDEF is a convex

polygon.

If GHIJ is a kite, then GHIJ is not a trapezoid.

The number of acute angles in a trapezoid is always either 1 or 2.

Assignments

pp. 359-361 # 3-24, 28-34, 37-39 (odd in class; even for homework)

pp. 367-368 # 16-41 (odd in class; even for homework)

6.7 Areas of Triangles and Quadrilaterals

Mrs. Gibson

Geometry

Spring 2011

Area Postulates

Area of a Square Postulate The area of a square is the square of the

length of its sides, or A = s2.

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

Rectangle: A = bh Parallelogram: A = bh Triangle: A = ½ bh Trapezoid: A = ½ h(b1+b2)

Kite: A = ½ d1d2

Rhombus: A = ½ d1d2

Find the area of ∆ ABC.

A B

C

7

5

64

L

Find the area of a trapezoid with vertices at A(0,0), B(2,4), C(6,4), and D(9,0).

Find the area of the figures.

4

4

4

4

LL L

L

L

LL

L

2

5

12

8

Find the area of ABCD.

A

B C

D

E

12

16

9

ABCD is a parallelogramArea = bh = (16)(9) = 144

Find the area of a trapezoid.

Find the area of a trapezoid WXYZ with W(8,1), X(1,1), Y(2,5), and Z(5,5).

Find the area of rhombus.

Find the area of rhombus ABCD.

A

B

C

D

20 20

15

15 25

Area of Rhombus A = ½ d1 d2

= ½ (40)(30) = ½ (1200) = 600

The area of the kite is160. Find the length of BD.

A

B

C

D10

Ch 6 Review

Mrs. Gibson

Geometry

Spring 2011

Review 1

A polygon with 7 sides is called a ____.A) nonagon

B) dodecagon

C) heptagon

D) hexagon

E) decagon

Review 2

Find m<A

A) 65°

B) 135°

C) 100°

D) 90°

E) 105°

AB

C

D

165°30°

65°

Review 3

Opposite angles of a parallelogram must be _______.

A) complementary

B) supplementary

C) congruent

D) A and C

E) B and C

Review 4

If a quadrilateral has four equal sides, then it must be a _______.

A) rectangle

B) square

C) rhombus

D) A and B

E) B and C

Review 5

The perimeter of a square MNOP is 72 inches, and NO = 2x + 6. What is the value of x?

A) 15

B) 12

C) 6

D) 9

E) 18

Review 6

ABCD is a trapezoid. Find the length of midsegment EF.

A) 5

B) 11

C) 16

D) 8

E) 22

A

B

CD

E

F

11

5

9

13

Review 7

The quadrilateral below is most specifically a __________.

A) rhombus

B) rectangle

C) kite

D) parallelogram

E) trapezoid

Review 8

Find the base length of a triangle with an area of 52 cm2 and a height of 13cm.

A) 8 cm

B) 16 cm

C) 4 cm

D) 2 cm

E) 26 cm

Review 9

A right triangle has legs of 24 units and 18 units. The length of the hypotenuse is ____.

A) 15 units

B) 30 units

C) 45 units

D) 15.9 units

E) 32 units

Review 10

Sketch a concave pentagon.

Sketch a convex pentagon.

Review 11

What type of quadrilateral is ABCD? Explain your reasoning.

A

B

C

D

120°

120°60°

60°

Isosceles TrapezoidIsosceles : AD = BCTrapezoid : AB ll CD

Reviewing for Ch. 6 Test

Consider the following: Chapter Review on pp. 382-384 Chapter Test on p.385 Chapter Standardized Test on pp.386-387