chapter 8 notes 2013-2014x

1

Geometry

Chapter 8 – Quadrilaterals ***In order to get full credit for your assignments they must me done on time and you must SHOW ALL WORK. ***

1.____ (8-1) Angles of Polygons – Day 1- Pages 407-408 13-16, 20-22, 27-32, 35-43 odd

2. ____ (8-2) Parallelograms – Day 1- Pages 415 16-31, 37-39

3. ____ (8-3) Test for Parallelograms – Day 1- Pages 421-422 13-23 odd, 25 -31 odd

4. ____ (8-4) Rectangles – Day 1- Pages 428-429 10, 11, 13, 16-26, 30-32, 36

5. ____ (8-5) Rhombi and Squares – Day 1 – Pages 434-435 12-19, 20, 22, 26 - 31

6.____ (8-6) Trapezoids – Day 1– Pages 10, 13-19, 22-25

7. _____ Chapter 8 Review

Name: ____________________________

2

You need to graph the quadrilaterals and find the following for each: the length of all the sides, length of the diagonals, slope of all the sides, and the slope of the diagonals. 1. J(0,6), K( -4,6), L( -4,0), M(0,0)

JK=

KL=

LM=

JM=

JL= KM= mJK= mKL= mLM= mJM=

mJL= mKM=

3

Compare the results you got on the previous page and check all the boxes that apply.

Type of Quadrilateral

Yes No

Parallelogram Rectangle

Square Rhombus Trapezoid

Properties of the sides Yes No Both pairs of opposite sides are ||

Exactly 1 pair of opposite sides are || Both pairs of opposite sides are ≅

Exactly 1 pair of opposite sides are ≅

Exactly 1 pair of consecutive sides are ≅

All sides are ≅

Properties of the diagonals Yes No Diagonals are ⊥

Diagonals are ≅ Diagonals bisect each other

Diagonals bisect pair of opposite ∠s

Properties of the angles Yes No Both pairs of opposite ∠s are ≅

Exactly 1 pair of opposite ∠s are ≅

All ∠s are ≅

All ∠s are supplementary to 2 consecutive ∠s

Notes:

4

2. T(-3, -1), R( -1, 3), Q( 2, 3), V(4, -1)

TR=

RQ=

QV=

TV=

QT= RV= mTR= mRQ= mQV= mTV=

mQT= mRV=

5



Yes No







All sides are ≅






All ∠s are ≅


Notes:

6

3. A(-7, -2), B( 0, 4), C( 9, 2), D(2, -4)

AB=

BC=

CD=

AD=

AC= BD= mAB= mBC= mCD= mAD=

mAC= mBD=

7



Yes No







All sides are ≅






All ∠s are ≅


Notes:

8

4. E(-2, 4), F( 3, 5), G( 2, -3), H(-3, -4)

EF=

FG=

GH=

EH=

EG= FH= mEF= mFG= mGH= mEH=

mEG= mFH=

9



Yes No







All sides are ≅






All ∠s are ≅


Notes:

10

5. U(1, 2), X( -2, -1), Y( 1, -4), Z(4, -1)

UX=

XY=

YZ=

UZ=

UY= XZ= mUX= mXY= mYZ= mUZ=

mUY= mXZ=

11



Yes No







All sides are ≅






All ∠s are ≅


Notes:

12

6. J(-4, -1), K( 1, -1), L( 4, 3), M(-1, 3)

JK=

KL=

LM=

JM=

JL= KM= mJK= mKL= mLM= mJM=

mJL= mKM=

13



Yes No







All sides are ≅






All ∠s are ≅


Notes:

14

7. U(-1, -1), X( -8, -1), Y( -5, 5), Z(2, 5)

UX=

XY=

YZ=

UZ=

UY= XZ= mUX= mXY= mYZ= mUZ=

mUY= mXZ=

15



Yes No







All sides are ≅






All ∠s are ≅


Notes:

16

8. A(-3, 4), B( 2, 5), C( 3, 3), D(-1, 0)

AB=

BC=

CD=

AD=

AC= BD= mAB= mBC= mCD= mAD=

mAC= mBD=

17



Yes No







All sides are ≅






All ∠s are ≅


Notes:

18

9. E(-2, -1), F( -1, 3), G( 3, 2), H(2, -2)

EF=

FG=

GH=

EH=

EG= FH= mEF= mFG= mGH= mEH=

mEG= mFH=

19



Yes No







All sides are ≅






All ∠s are ≅


Notes:

20

10. T(-2, 3), R( 1, 4), Q( 3, -2), V(0, -3)

TR=

RQ=

QV=

TV=

QT= RV= mTR= mRQ= mQV= mTV=

mQT= mRV=

21



Yes No







All sides are ≅






All ∠s are ≅


Notes:

22

A little background…

Polygon is the generic term for a closed figure with any number of sides. Depending on the

number, the first part of the word “Poly” is replaced by a prefix. The prefix used is from

Greek. The Greek term for 5 is Penta, so a 5-sided figure is called a Pentagon. We can draw

figures with as many sides as we want, but most of us don’t remember all that Greek, so when

the number is over 12, or if we are talking about a general polygon, many mathematicians call

the figure an “n-gon.” So a figure with 46 sides would be called a “46-gon.”

Vocabulary Polygon - A closed plane (two-dimensional) figure made up of several line segments that

are joined together. The sides do not cross each other. Exactly two sides meet at

every vertex. Types of Polygons Regular - all angles are equal and all sides are the same length. Regular polygons are both

equiangular and equilateral.

Irregular – Any polygon with any angles NOT equal and any sides NOT the same length.

Equiangular - all angles are equal.

Equilateral - all sides are the same length.

Convex - a straight line drawn through a convex polygon crosses at most two

sides. Every interior angle is less than 180°.

Concave - you can draw at least one straight line through a concave polygon that

crosses more than two sides. At least one interior angle is more than

180°.

Polygon Parts

Side - one of the line segments that make up the

polygon.

Vertex - point where two sides meet. Two or more of

these points are called vertices.

Diagonal - a line connecting two vertices that isn't a

side.

Interior Angle - Angle formed by two adjacent sides

inside the polygon.

Exterior Angle - Angle formed by one side of a

triangle and the extension of another side.

23

a.) Compare the number of triangle to the number of sides. Do you see a pattern?

b.) How can you use the number of triangles formed by the diagonals to figure out the sum of all the interior angles of a polygon?

c.) Write an expression for the sum of the interior angles of an n-gon, using n and the patterns you found from the table.

Number of sides of the

polygon

Name of the

polygon

Number of

interior angles

Number of

diagonals possible from one

vertex point

Number of

triangles formed

from one vertex point

Sum of the

measures of

interior angles

One interior angle

measure (regular polygon)

One exterior angle

measure (regular polygon

Sum of the

exterior angles

measures

3 3 0 1

4

5 5 2 3 540°

6

7

8

9

10

11

12 1800°

n

25

Date: ______________________

Section 8 – 1: Angles of Polygons Notes

Diagonal of a Polygon: A segment that ________________ any two

_____________________ vertices.

Theorem 8.1: Interior Angle Sum Theorem:

If a convex polygon has n sides and S is the sum of the

_________________ of its interior angles, then ____________________.

Example #1: Find the sum of the measures of the interior angles of the regular pentagon

below.

Example #2: The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.

Example #3: Find the measure of each interior angle.

26

Theorem 8.2: Exterior Angle Sum Theorem:

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

______________ angles, one at each _____________, is 360.

Example #4: Find the measures of an exterior angle and an interior angle of convex regular

nonagon ABCDEFGHJ.

27

Example #1: RSTU is a parallelogram. Find m URT∠ , m RST∠ , and y.

Example #2: Find x and y so that each quadrilateral is a parallelogram and justify your reasoning. a.)

b.)

28

Given: VZRQ and WQST Q

R S

W

Prove: ∠ Z ≅ ∠ T T

V Z Statements Reasons

1. VZRQ 1.

2. ∠ Z ≅ ∠ Q 2.

3. WQST 3.

4. ∠ Q ≅ ∠ T 4.

5. ∠ Z ≅ ∠ T 5. Example #3: Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x – 4, find x.

29

Example #4: Quadrilateral LMNP is a rectangle.

a.) Find x.

b.) Find y.

Example #5: Use rhombus LMNP and the given information to find the value of each

variable.

a.) Find y if 1m∠ = y – 54

b.) Find m PNL∠ if m MLP∠ = 64.

30

Median: The segment that joins __________________ of the _________ of

a trapezoid.

Theorem 8.20: The median of a trapezoid is _______________ to the

bases, and its measure is _______________ the sum of the measures of

the bases.

Ex:

Example #6: DEFG is an isosceles trapezoid with medianMN .

a.) Find DG if EF = 20 and MN = 30.

b.) Find 1m∠ , 2m∠ , 3m∠ , and 4m∠ if 1m∠ = 3x + 5 and 3m∠ = 6x – 5.

31

Type of Figure Draw an Example Special Characteristics (Some things to ask yourself: diagonals equal, diagonals bisect each other, diagonals bisect angle, diagonals perpendicular, angle measures equal, angle measures supplementary, sides equal, sides parallel, etc…) 1. Quadrilateral 2. Parallelogram 3. Square 4. Rectangle 5. Rhombus 6. Trapezoid 7. Isosceles Trapezoid

chapter 8 notes 2013-2014x

Documents