Chapter 8: Exploring Quadrilaterals

4 sided polygons

8.1 Polygons8.1 Polygons

Def of Polygon: Closed figure formed by a finite number of coplanar segments such that:

-The sides that have a common endpoint are noncollinear.

-Each side intersects exactly 2 other sides at their endpoints

No line containing a side of the polygon contains a point in the interior of the polygon.NOT CONVEX

34567891011121314

# of sides Name Regular Polygon

A Polygon that isBoth equilateralAnd equiangular

(Remember: If a triangleIs equilateral, then it is Equiangular. This only Works for triangles.)

• Triangle• Quadrilateral• Pentagon• Hexagon• Heptagon• Octagon• Nonagon• Decagon• Undecagon• Dodecagon• 13-gon• 14-gon• The pattern continues

-The sum of the measures of the angles in a triangle is?

-Look at the quadrilateral, how many triangles are formed?-So, what is the sum of the measures of the angles in a quadrilateral?

-Look at the pentagon, how many triangles are formed?-So, what is the sum of the measures of the angles in a pentagon?

-Look at the hexagon, how many triangles are formed?-So what is the sum of the measures of the angles in a hexagon?

Do you see a pattern? Howdoes the numberof triangles relate to thenumber of sides?Can you write aformula?

1. Using a template to draw each of these figures on a piece of paper.2. Extend out the sides for each figure (like the triangle.)3. Measure each of the exterior angles that are formed.4. What can you conclude about the sum of the exterior angles of a convex polygon, one at each vertex?

Finding angle measures

# of sides Sum of One interior < sum of One exterior < interior <‘s (if regular) exterior <‘s (if regular) n (n – 2)180 (n – 2)180 360 360

n n

3 (3-2)180 = 180 (3-2)180= 60 360 360=120

3 3 12 (12-2)180 =1800 (12-2)180 =150 360 360=30

12 12

SAMPLES OF APPLYING THE FORMULAS

Examples1. Find the sum of the measures of the

interior angles of a convex 26-gon.

2. Name the regular polygon with an exterior angle measuring 45.

3. Find the measure of an interior angle and an exterior angle for a regular 16-gon.

4. If the measure of an interior angle of a regular polygon is 144, classify the polygon according to the number of sides.

Try p. 518: 7 - 18

7. AB, BC, CD, DE, EF, FA 13. 135 and 45

8. Convex 14. 180(a-2) , 360

a9. 1440

10.3420 15. 6

11.12 16. 12

12.45 17. 34, 102,67,199,138 18. 72

Section 8.2Parallelograms

Definition of Parallelogram:A quadrilateral with both pairs of opposite sides parallel.

II. PROPERTIES OF PARALLELOGRAMS

Parallelogram

Def: Both pairof oppositesides are //

Both pair ofopposite sidesare

Both pair of opposite <‘sare Consecutive

angles aresupplementary

Diagonalsbisect eachother

Examples1. WXYZ is a parallelogram. Find the indicated information.

W X

YZ

V

3a + 5

7a -7

16b -3 8b + 1317

c

M<ZWX = 100

A. Find the value of a. B. Find the value of b.C. Find the value of c. D. Find m<WZYE. Find m<XYZ F. Find m<WXZ

3 217 80100 80

2. Use the definition of a parallelogram to determine if RSTV is a parallelogram.

R S

TV

(1,1) (3,6)

(8, 8)(6, 3)

To use the definition, wemust see if opposite sides are parallel.

Since RS // VT (have the same slope) and RV // ST (have the sameslope), RSTV is a parallelogram.

Slope RS = (6-1)/(3-1) = 5/2

Slope VT = (8-3)/(8-6) = 5/2

Slope RV = (3-1)/(6-1) = 2/5

Slope ST = (8-6)/(8-3) = 2/5

The Probability of an eventis the ratio of the number of favorable outcomes to the total number of possible

outcomes.

Favorable Outcomes Total Outcomes

3. Two sides of ABCD are chosen at random. What is the

probability that the two sides are not congruent?

A B

CD

AB and BC not congruentAB and DC are congruentAB and AD not congruentBC and DC not congruentBC and AD are congruentDC and AD not congruent

Not congruent/Total 4 /6 So the probability is 2/3.

4. Find the values of w, x, y, and z for the parallelogram.

110

x

y

zw

Answers:w = 110 (opposite angles are congruent) x = 70 (linear pair with w) y = 70 (consecutive angles are supplementary) z = 70 (opposite angles are congruent and consecutive angles are supplementary)

5. Find the indicated values for the parallelogram

100

20

ab c

de

fgh

i

Answersa = 100 b = 80 c = 80 d = 60e = 30 f = 70 g = 20 h = 60i = 30

70

6. Find all possible values for the 4th vertex of the parallelogram 3 of the vertices are (0,0), (4,4), and (8,0)

(0,0)(8,0)

(4,4)(-4,4) (12,4)

(4, -4)

Answers to 6 – 14 __6. HF __7. DC

8. <DFG __9. GF

10. <CDF and <CGF

11. HDF

12. AB = CD so 2x + 5 = 21 and 2x = 16, so x = 8 Since <B = 120, m<BAC + m<CAD = 60 2y + 21 = 60, then 2y = 39 and y = 19.5

13. m<Y = 47 m<X = 133 m<Z = 133

14. SLOPES PT and QR = 5 QP and TR = -1 therefore opposite sides are // and it is a parallelogram

8-3 TESTS FOR 8-3 TESTS FOR PARALLELOGRAMSPARALLELOGRAMS

I.I. HOW DO YOU KNOW IF A HOW DO YOU KNOW IF A QUADRILATERAL IS A QUADRILATERAL IS A PARALLELOGRAM?PARALLELOGRAM?

PERFORM A

TEST

Opposite sides are parallel

Opposite

Sides are

congruent

Opposite angles are

congruent

Diagonals

Bisect

Each

other

The same

pair of

opposite

sides(both pair)(both pair) (both pair) and //

II. EXAMPLESII. EXAMPLES

1. IS CUTE A PARALLELOGRAM? 1. IS CUTE A PARALLELOGRAM? WHY?WHY?

C

U

T

E

62

118

118 62

YES, BECAUSE OPPOSITE ANGLES ARE CONGRUENT AND CONSECUTIVE ANGLES ARE SUPPLEMENTARY

2. Find x and y so that the 2. Find x and y so that the quadrilateral is a parallelogram.quadrilateral is a parallelogram.

3x + 17

4 4x-8

2y4x – 8 = 4

4 x = 12

X = 3

So 3( 3) + 17 = 26

26 = 2 y

13 = y

A(5,6) B(9,0) C(8, - 5) D (3, -2)A(5,6) B(9,0) C(8, - 5) D (3, -2) AB= AB=

CD=CD=

52)60()59( 22

34)52()83( 22

A

B

C

D

NO- OPPOSITE SIDES ARE NOT CONGRUENT.

3. IS ABCD A PARALLELOGRAM?

4. Determine if PZRD is a parallelogram.

P(-1,9) Z(3,8)

R(6,2)D(2,3)

a. SLOPESslope PZ and DR = -1/4 slope PD and ZR = -2It is a parallelogram because both

pair of opposite sides are //

b. DISTANCES distance PZ and DR = 17distance PD and ZR = 35It is a parallelogram because both

pair of opposite sides are congruent.

c. MIDPOINTS DZ (2.5,5.5) PR (2.5,5.5) It is a parallelogram because thediagonals bisect each other.

d. Slope PZ and DR = -1/4 and Distance PZ and DR = 17. Since the same pair of opposite sides are parallel and congruent it is a parallelogram.

e. Slope PD and ZR = -2 and Distance PD and ZR = 35. Since the same pair of opposite sides are parallel and congruent it is a parallelogram.

7. Since the triangles will be congruent by SAS, the other pair of opposite sides will be congruent and it is a parallelogram because both pair of opposite sides are congruent. 8. No, the same pair are not congruent and parallel.9. 6x = 4x + 8 so 2x = 8 and x = 4 y² = y so y² - y = 0 and y(y – 1) = 0 so either y = 0 or y – 1 = 0 which means y = 0 or y = 1. Distances are positive, so y = 1.10. 2x + 8 = 120 so 2x = 112 and x = 56. 5y = 60 and y = 12.11. False: It could have congruent diagonals and be another

type of quadrilateral (trapezoid).12. No, not a parallelogram. One method of showing this is to

show the diagonals do not bisect each other. midpoint of GJ = (2, 2.5) midpoint of HK = (1.5, -1.5) The diagonals do not bisect each other.

8-4 RectanglesProperties of Rectangles

–How do you know if a quadrilateral is a rectangle?

RECTANGLE

Definition:

Quadrilateral with 4 right

angles

A Parallelogram with congruent diagonals

PROPERTIES

Examples

1. Find x and y.

2x+ y

36

9 x -y SOLUTION

X – Y = 92X + Y = 36

Add the two equations

3X + 0Y = 45

3X = 45 SO X = 15

SINCE X = 15 AND X – Y = 9, THEN15 – Y = 9 OR –Y = -6 OR Y = 6 ORSINCE X = 15 AND 2X + Y = 36, THEN2(15) + Y = 36 OR 30 + Y = 36 Y = 6

• 2.    Find x.• AC = x• DB = 6x - 8

A B

D C

E

2

SOLUTIONSince the diagonals of a rectangle are congruent, x² = 6x – 8

x² = 6x – 8 set equal to 0 and factorx² - 6x + 8 = 0 SO (x – 2)(x – 4) = 0 THEN x – 2 = 0 or x – 4 = 0 SO x = 2 or x = 4

3.Is ABCD a rectangle? Prove it. A (10,4) B (10,8) C (-4,8) D (-4,4)

A(10,4)

B(10, 8)C(-4,8)

D(-4, 4)

SOLUTION:

Slope of AB = (8-4)/(10-10) = 4/0 = undefinedSlope of CD = (8-4)/(-4- -4) = 4/0 = undefinedSlope of BC = (8-8)/(10 - -4) = 0/14 = 0Slope of AD = (4-4)/(10- -4) = 0/14 = 0

Since a slope of 0 and an undefined slope make theconsecutive sides, ABCD is arectangle because it has 4 right angles ( form 4Right angles)

4. Find all of the numbered angles

60º 1

2

3

4

5

6

78

9

1011

SOLUTION: angles 1, 4, 5, and 10 = 30º angles 3, 9, and 11 = 60º angles 2 and 8 = 60º angles 6 and 7 = 120º

5. X = 15.56. X = 5 or –27. X = 13.58. False: This is a property of a parallelogram. The

parallelogram might not be a rectangle.9. Slope of AB and CD = 1 Slope of AD and BC = -1 Since consecutive sides are , the quadrilateral has 4 right

angles and it is a rectangle. 10. m<2 = 20 m<5 = 70 m<6 = 2011. m<6 = 26 m<7 = 26 m<8 = 6412. m<2 = 54 m<3 = 54

8-5 Squares and Rhombi

I. Rhombus

Def:

quadrilateral

with 4 =

sides

Parallelogram with diagonals

Parallelogramwith diagonals.that bisect a pair of opposite angles

II. Square

SquareRectangle

rhombus

+=

III. Examples rhombus DLPM

1. DM = 26 a. OM= _______ b. MD is congruent to PL. True or false? c. <DLO is congruent to <MLO. True or false?

D

P

L

M

O

13

False, this is not a rectangle

True, diagonalsbisect the angles.

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

a. If m<1 = 2x + 20 and m<2 = 5x – 4, find the value of x, m<1 and m<2.

b. If BD = 15, find BF

c. If m<3 = y² + 26, find y.

B C

E D

F

123

ANSWERS:a. 2x + 20 = 5x – 4 so 20 = 3x – 4 then 24 = 3x and 8 = x m<1 = 2(8) + 20 = 36 and m<2 = 5(8) – 4 = 36b. Since it is also a parallelogram, BF = 7.5 (diag. bisect each

other.)c. y² + 26 = 90 so y² = 64 then y = 8 or -8

3. What type of quadrilateral is ABCD?A (-4, 3) B (-2,3) C(-2, 1) D (-4,1)Justify.

A(-4,3) B(-2,3)

C(-2,1)D(-4,1)

AB = 2BC = 2CD = 2AD = 2

Slope AB = 0Slope BC = undefinedSlope CD = 0Slope AD = undefined

Therefore ABCD is a parallelogram, rhombus, rectangle, and square because all sides are and it has 4 right <‘s.

Try p. 316 – 317:4, 10-14, 17, 18

Answers: 4. Yes yes yes yes 17. rectangle no yes no yes square no no yes yes no no yes yes 18. rhombus

square10. m<RSW = 33.511. m<SVT = 22.512. X = 4113. X = 1214. PA = 5 AR = 5 RK = 5 PK = 5 Slopes PA = -1/2 AR = 2 RK = -1/2 PK = 2 It is a parallelogram, rectangle, rhombus, and square

because all sides are congruent and it forms 4 right angles.

8-6 Trapezoids8-6 Trapezoids

I. PropertiesI. Properties

TrapezoidDefinition: Aquadrilateral withexactly one pair of parallel sides.

base

base

leg leg

and

are one pair of base angles

andare another pairof base angles

Isosceles trapezoidIsosceles trapezoid

Congruent legs

Both pair of base <‘s are congruent

Diagonals are congruent

Medians of trapezoidsMedians of trapezoids

MEDIAN The length of a median of a trapezoid is the average of the base lengths.

BASE

BASE

BASE + BASE 2

= MEDIAN

ExamplesExamples1. Verify that MATH is an isosceles 1. Verify that MATH is an isosceles

trapezoid. M(0,0) A(0,3) T(4,4) H (4,-trapezoid. M(0,0) A(0,3) T(4,4) H (4,-1)1)

M(0,0) H(4, -1)

T(4,4)A(0,3) Slope of AM = undefined Slope of AT = 1/4Slope of TH = undefined Slope of MH = -1/4So MATH is a trapezoid(one pair of parallel sides)

AT = 17 MH = 17So MATH is isosceles(legs are congruent)

2. Find x.2. Find x.

3x - 1

29.5

7x + 10

(3x – 1) + (7x + 10) = 29.5 2

10x + 9 = 59

10x = 50

x = 5

3.Find the value of x.

5

12

3x - 5

5 + 3x – 5 = 2(12)

3x = 24

X = 8

4. Decide whether each statement is 4. Decide whether each statement is sometimes, always, or never true.sometimes, always, or never true.a. A trapezoid is a parallelogram.a. A trapezoid is a parallelogram.

b. The length of the median of a trapezoid b. The length of the median of a trapezoid is one-half the sum of the lengths of the is one-half the sum of the lengths of the bases.bases.

c. The bases of any trapezoid are parallel.c. The bases of any trapezoid are parallel.

d. The legs of a trapezoid are congruent. d. The legs of a trapezoid are congruent.

never

always

always

sometimes

Try p. 324: 5 - 135. True: Diagonals of an isosceles trapezoid are congruent.

6. True: The legs of an isosceles trapezoid are congruent.

7. False: If the diagonals bisected each other it would be a

8. x² = 16 so x = 4 or –4

9. 17 the legs are congruent and the median cuts the legs in half

10. 180 – 62 = 118

11. R(5, 3) S(2.5, 8)

12. RS = approx 5.59 (use distance formula for the points in #11)

13. NO, MNPQ is not a trapezoid.