GEOMETRYGEOMETRY

Chapter 1Chapter 1

CONTENTSCONTENTSNaming Figures

Describing Figures

Distance on a number line

Distance on a grid

Segment Addition Postulate

Angles and Their Measures

Measuring Angles

Angle Addition Postulate

Classify Angles

Segment Bisectors and Midpoints

Angle Bisectors

NAMING FIGURESNAMING FIGURESFIGURE DESCRIPTION NAME IT

A POINT A

BD

C3 POINTS B, C, D

A line containing 3 known points FE FG

EF GE

OR

OR

OR.....

H

J

A segment with 2 end points HJ JHOR

EF GG

NAMING FIGURESNAMING FIGURESFIGURE DESCRIPTION NAME IT

Ray with endpoint K

A plane containing 3 known points NOP

Collinear points

KM

L

ORKL KM

NO

P

QOR Q

R, S, & TR

TS

U Noncollinear points U, R, S, & T

NO

P

Q

R

yDescribe the figure:

Plane Q contains Line NP, Line PR,and Points N, P, R, and O.

Line NP and Line PR intersect at Point P.

Line y intersects plane Q at point O.

DESCRIBING FIGURESDESCRIBING FIGURES

DISTANCEDISTANCEOn a Number LineOn a Number Line

EF G

-15 -2 7

Finding the length of a segment isthe same as finding the distance between its endpoints. When we measure a segment and attach a number to it we drop the bar in the symbol:

Since the length of AB is 12, we write AB = 12.

The length of FG is | F – G |.FG = | F – G |

YOU TRY: Find GE and FE.

= | -15 – - 2 |

= | -15 + 2 |

= | -13|

= 13

DISTANCEDISTANCEOn a Number LineOn a Number Line

EF G

-15 -2 7

The length of GE is | G – E |.GE = | G – E | = | - 2 – 7 | = | - 9|

= 9

Find GE and FE.

The length of FE is | F – E |.FE = | F – E | = | - 15 – 7 | = | - 22|

= 22

DISTANCEDISTANCEOn a Number LineOn a Number Line

The length of PQ is | P – Q |.PQ = | P – Q | = | 16 – - 4 | = | 20|

= 20

Find the length of the segment that has

endpoints with coordinates P(16) and Q(- 4).

DISTANCEDISTANCEOn a GridOn a Grid

Subtract x-values

To find the distance between two points on a

Grid, use the Distance Formula:

2122

12 yyxx Subtract y-values

Square the result Square the result

Add the results

Take the SQUARE ROOT and simplify

DISTANCEDISTANCEOn a GridOn a Grid

Example: Find the distance between

A( - 10, 4) and B( - 6, 1)

2122

12 yyxxAB

22 41106

22 34

525916 AB = 5

DISTANCEDISTANCEOn a GridOn a Grid

Find the distance between C( 7, - 3) and D( - 5, 2)

2CD2

CD yyxxCD

223275

22512

1316925144

CD = 13

Segment Addition PostulateSegment Addition Postulate

If B is between A and C,If B is between A and C,

A

C

B

Then Then ABAB + + BC BC = = ACAC

Segment Addition PostulateSegment Addition PostulateIf W is between X and Z, If W is between X and Z,

X

Z

W XWXW + + WZWZ = = XZXZ

2424

2424

+ + 5353

5353

= = XZXZ

7777 = = XZXZ

XW = 24 ,XW = 24 , WZ = 53 ,WZ = 53 ,

Find XZ .Find XZ .

Segment Addition PostulateSegment Addition PostulateIf W is between X and Z, If W is between X and Z,

X

Z

W XWXW + + WZWZ = = XZXZ

6969

6969

+ + WZWZ

142142

= = 142142

7373WZ WZ ==

XW = 69 ,XW = 69 ,

Find WZ .Find WZ .

XZ = 142,XZ = 142,

– – 6969 – – 6969

Segment Addition PostulateSegment Addition Postulate

== MPMP

Find all three segment measures .Find all three segment measures .

PGPG ++ MGMG

P MG4x + 64x + 6

9x + 129x + 12

3x + 263x + 26

4x + 64x + 6 3x + 263x + 26 9x + 129x + 12++ ==

If G is between P and M, If G is between P and M, PG = 4x + 6 ,PG = 4x + 6 ,MP = 9x + 12,MP = 9x + 12, and MG = 3x + 26,and MG = 3x + 26,

Segment Addition PostulateSegment Addition Postulate

4x + 64x + 6 3x + 263x + 26 9x + 129x + 12++ ==

7x7x

9x + 129x + 12==

9x + 129x + 12==

2x + 122x + 12==3232- 7x- 7x - 7x- 7x

2020 2x2x==xx==1010

PG = 4x + 6 = 4PG = 4x + 6 = 4(10)(10) + 6 = 46 + 6 = 46MG = 3x + 26 = 3MG = 3x + 26 = 3(10)(10) +26 = 56 +26 = 56

MP = 9x + 12 = 9MP = 9x + 12 = 9(10)(10) + 12 = 102+ 12 = 102

+ 32+ 32

- 12- 12 - 12- 12

PG = 46PG = 46

MG = 56MG = 56

MP = 102MP = 102

Angles and Their MeasuresAngles and Their Measures

L

J

KJ and KL form JKL

sides

Angle symbol

Naming angles (4 ways)1) JKL2) LKJ3) K (only if 1 angle)4) 1

1

Vertex is K

K

KK

Naming AnglesNaming Angles

MNP orPNM

ONP orPNO

MNO or ONM

Interior of an AngleInterior of an Angle

Adjacent AnglesAdjacent Angles

Common vertex

Common ray

No interior points in common

Measuring AnglesMeasuring Angles

Congruent angles are angles with the same measure.

If m ABC = 50 and m JKL = 50 Then ABC JKL

Angles are congruentAngle Measures are equal!!

Angle Addition PostulateAngle Addition Postulate

If P is in the interior of RST, then m RSP + m PST = m RST .

TS

RP

Angle Addition PostulateAngle Addition Postulate

Suppose that the angle at the right measures 60° and that there is a point K in the interior of the angle such that m GHK = 25 . Find m KHI .

m GHK + m KHI = m GHI

K 60°?°

25°

25 + x = 60X = 60 – 25 = 35 m KHI = 35

Classify AnglesClassify Angles

Right Angle90

Obtuse Angle90 < x < 180Straight Angle

180

Acute Angle0 < x < 90

Segment BisectorSegment Bisector

Bisect means to cut into 2 congruent pieces.

The midpoint of a segment is the point that bisects the segment.

A segment bisector is a segment, ray, line or plane that intersects the segment at its

midpoint.

Construct the Midpoint of a Construct the Midpoint of a SegmentSegment

MidpointsMidpoints

If X is the midpoint of AB,If X is the midpoint of AB,

A

X

B

Then, Then, AXAX = = XBXB..

Midpoints on number linesMidpoints on number lines

To find the midpoint of a segment To find the midpoint of a segment on a number line, just on a number line, just averageaverage

the coordinates of the endpoints.the coordinates of the endpoints.

- 23- 23 4747

-23 + 4722

2424 22

= 12= 12

1212

==

Endpoints on number linesEndpoints on number linesTo find the To find the endpointendpoint of a segment on of a segment on

a number line with one endpoint a number line with one endpoint and the midpoint: Midpoint x 2, and the midpoint: Midpoint x 2,

then subtract the known endpoint.then subtract the known endpoint.

- 14- 14 2323

46 - -14 =46 - -14 = 23 x 2 - -14 =

6060

46 +14 =46 +14 = 6060

Midpoint FormulaMidpoint FormulaMidpoints on a GridMidpoints on a Grid

The Midpoint Formula: The midpoint of a segment

with endpoints (x1 , y1) and (x2 , y2)

has coordinates

( )

Midpoint on a GridMidpoint on a Grid

A is (-3, 4)

B is ( 2, 1)

Midpoint is -3 +2, 4 + 1 2 2

(- .5, 2.5)

Endpoint on a GridEndpoint on a Grid

A segment has endpoint J(-7, 8) and midpoint P(2, -1). Find the other endpoint.

Double the midpoint P(2, -1). Then subtract the endpoint you know J(-7, 8).

P(2, -1) x 2 gives (4, -2). (4, -2) - (-7, 8)(4 - -7, -2 – 8)

(11, -10)

The other endpoint is (11, -10).

Angle BisectorsAngle Bisectors

An angle bisector is a ray that dividesan angle into two adjacent congruent angles.

Angle bisector

Construct an Angle BisectorConstruct an Angle Bisector