1.6-6.1 Polygon notes

A Polygon:

• •

•

•

Examples: Nonexamples:

Named by the letters of the vertices written in order

A polygon will be: Or:

Concave - Convex-

Regular Polygon:

1.6-6.1 Polygon notes

A Diagonal is a segment that connects any two vertices not already

connected by a side of the polygon.

To find the sum of the angles inside a polygon we can use the

following:

1.6-6.1 Polygon notes

Ex. 1: Find the sum of the interior angles of an:

A. 11-gon. B. 3x-gon

C. 15-gon

Ex. 2: Find the number of sides of a regular polygon if the measure

of an interior angle is:

A. 108 B. 170

If a polygon is convex, then the sum of the measures of the

exterior angles, one at each vertex, is __________.

1.6-6.1 Polygon notes

Ex. 1: Find the measures of an exterior angle of a:

A. regular dodecagon

B. polygon with 20 sides

Ex. 2: Find the number of sides of the regular polygon with

an exterior angle of:

A. 15

B. 60

1.6-6.1 Practice

Name:______________________________________Pd:______Date:_________

State if each shape is a polygon. If not, explain why it is not a polygon.

1. 2. 3. 4.

State if each figure is regular or irregular:

9. 10. 11. 12.

For each polygon name it by its number of sides, classify it as concave or convex,

and state if it is regular or irregular:

13. 14. 15. 16.

Find the sum of the measures of the interior angles of each convex polygon

1. 13-gon 2. heptagon

3. 24-gon 4. 4y-gon

1.6-6.1 Practice

Find the number of sides in the regular polygon given the measure of an interior angle

9. 144 10. 120

11. 150 12. 176.4

Find the measure of an exterior angle and an interior angle of the regular polygon

17. quadrilateral 18. pentagon

19. decagon 20. 18-gon

Find the number of sides of a regular polygon given an exterior angle measure

21. 7.5 22. 45

23. 22.5 24. 9

6.2

Parallelograms

Notes

Parallelogram: Properties:

Ex 1: Use parallelogram

ABCD to find x, y, and z

Ex 2: Use parallelogram

ABCD to find x and y

D C

B A

3x 21

2y + 3

45

2y + 5

35

3x - 4

20

D C

B A

Chapter 6 13 Glencoe Geometry

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Study Guide and Intervention

Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

6-2

Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are fourimportant properties of parallelograms.

If PQRS is a parallelogram, then

The opposite sides of a PwQw > SwRw and PwSw >QwRw

parallelogram are congruent.

The opposite angles of a /P > /R and /S > /Q

parallelogram are congruent.

The consecutive angles of a /P and /S are supplementary; /S and /R are supplementary;

parallelogram are supplementary. /R and /Q are supplementary; /Q and /P are supplementary.

If a parallelogram has one right If m/P 5 90, then m/Q 5 90, m/R 5 90, and m/S 5 90.

angle, then it has four right angles.

If ABCD is a parallelogram, find a and b.

AwBw and CwDw are opposite sides, so AwBw > CwDw.

2a 5 34

a 5 17

/A and /C are opposite angles, so /A > /C.

8b 5 112

b 5 14

Find x and y in each parallelogram.

1. 2.

3. 4.

5. 6.

72x

30x 150

2y

608

558 5x8

2y8

6x8 12x8

3y83y

12

6x

8y

88

6x8

4y8

3x8

BA

D C34

2a

8b8

1128

P Q

RS

Less

on

6-2

Example

Exercises

Chapter 6 14 Glencoe Geometry

Study Guide and Intervention (continued)

Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

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Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.

If ABCD is a parallelogram, then:

The diagonals of a parallelogram bisect each other. AP 5 PC and DP 5 PB

Each diagonal separates a parallelogram nACD > nCAB and nADB > nCBD

into two congruent triangles.

Find x and y in parallelogram ABCD.

The diagonals bisect each other, so AE 5 CE and DE 5 BE.

6x 5 24 4y 5 18

x 5 4 y 5 4.5

Find x and y in each parallelogram.

1. 2. 3.

4. 5. 6.

Complete each statement about ~ABCD.

Justify your answer.

7. /BAC >

8. DwEw >

9. nADC >

10. AwDw ||

A B

CDE

x

17

4y

3x82y

12

3x

30810

y

2x8

608 4y8

4x

2y28

3x4y

812

A B

E

CD

6x

18

24

4y

A B

CD

P

Example

Exercises

6.3 Tests for Parallelograms Notes

If one of the following is true…

1.

2.

3.

4.

5.

…Then the quadrilateral is a parallelogram.

Find x and y so that the quadrilateral is a parallelogram

6y-42

4y

2x+36 6x-12

(56) ̊

(4y+4) ̊ (7x) ̊

(5y-26) ̊

6.3 Tests for Parallelograms Notes

Are the following figures parallelograms? Justify your

answer by telling why or why not.

1. 2.

3. 4.

5. 6.

82

o

82o 98

o

98o

12

12

7

7 8

8

10 10

6.3 Homework

6.4

Rectangle

Notes

Definition: Properties:

ABCD is a rectangle.

Find x and AC if

AC = 6x + 14 and BD = 9x + 5

Use the rectangle to find

x and y given:

284

443

542

2091

+=∠

+=∠

+=∠

+=∠

ym

ym

xm

xm

C

B

E

D

A 1 2

3

4

Chapter 6 29 Glencoe Geometry

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Skills Practice

Rectangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-4

ALGEBRA ABCD is a rectangle.

1. If AC 5 2x 1 13 and DB 5 4x 2 1, find x.

2. If AC 5 x 1 3 and DB 5 3x 2 19, find AC.

3. If AE 5 3x 1 3 and EC 5 5x 2 15, find AC.

4. If DE 5 6x 2 7 and AE 5 4x 1 9, find DB.

5. If m/DAC 5 2x 1 4 and m/BAC 5 3x 1 1, find x.

6. If m/BDC 5 7x 1 1 and m/ADB 5 9x 2 7, find m/BDC.

7. If m/ABD 5 x22 7 and m/CDB 5 4x 1 5, find x.

8. If m/BAC 5 x21 3 and m/CAD 5 x 1 15, find m/BAC.

PRST is a rectangle. Find each measure if m/1 5 50.

9. m/2 10. m/3

11. m/4 12. m/5

13. m/6 14. m/7

15. m/8 16. m/9

COORDINATE GEOMETRY Determine whether TUXY is a rectangle given

each set of vertices. Justify your answer.

17. T(23, 22), U(24, 2), X(2, 4), Y(3, 0)

18. T(26, 3), U(0, 6), X(2, 2), Y(24, 21)

19. T(4, 1), U(3, 21), X(23, 2), Y(22, 4)

T

12 3 4

567 8

9

S

RP

D C

BAE

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6-4

6.5

Rhombus

(Rhombi)

Notes

Definition: Properties:

Use Rhombus ABCD to

find x if: 84

126

−=∠

−=∠

xDECm

xEDCm

Use Rhombus ABCD to

find x and AC if:

AC = 5x – 4 and AE = 2x + 6

E

D C

B A

E

D C

B A

6.5

Square

Notes

Definition: Properties:

Use Square ABCD to

find x if: 234

56

+=∠

−=∠

xBCEm

xBAEm

Use Square ABCD to find

x and y if:

39

34

322

=

+=

+=∠

CD

yAB

xCEDm

E

D C

B A

E

D C

B A

6.5 Homework

6.6 Trapezoid Notes

Trapezoid –

If the legs are congruent then the trapezoid is an ______________________

Median –

Ex 1: If AB = 17 and DC = 29 Ex 2: If DC = 22 and XY = 15

Find XY Find AB

6.6 Trapezoid Notes

Ex 3: WXYZ is an Isosceles Trapezoid

Find the median, X∠ , and Y∠ .

Ex 4: For trapezoid ABCD, G and H are midpoints of the legs.

Find AB, D∠ , and B∠

Ex 5: JKLM is an Isosceles Trapezoid with median QR

Find LM, J∠ , and K∠

W X

Y Z

30

14

D C

H G

B A

54

36

J K

L M

Q R 29

24

54 ̊

6.6 Homework

������� is the median of trapezoid HJKL. Find each indicated value

1. Find MN if HJ = 32 and LK = 60

2. Find LK if HJ = 18 and MN = 28

3. Find MN if HJ + LK = 42

4. Find m∠LMN if m∠LHJ = 116

5. Find m∠JKL if HJKL is isosceles and m∠HLK = 62

6. Find HJ if MN = 5x + 6, HJ = 3x + 6, and LK = 8x