chapter 1 basics of geometry by: carly overleese, karmen spiker and lindsey lewis

Chapter 1Chapter 1Basics of GeometryBasics of Geometry

By: Carly Overleese, Karmen By: Carly Overleese, Karmen Spiker and Lindsey LewisSpiker and Lindsey Lewis

1.11.1

•Identify the pattern in the picture.

•What is the next figure?

You Try It Now…You Try It Now…

• What is the next three numbers.

15,30,45,60...

• What is the Pattern?

225,45,11.25,3.75

75,90,105 225,45,11.25,3.75 5 4 3 2

1.21.2

Points, Lines, and Points, Lines, and PlanesPlanes

1.21.2

• Line-Line-

• PlanePlane

1.21.2

• Line Segment Line Segment

• Ray Initial PointRay Initial Point

• Opposite raysOpposite rays

Now You TryNow You Try

• Draw four noncollinear Draw four noncollinear points. points.

• Label A,B,C,D.Label A,B,C,D.

• Draw a Segment from Draw a Segment from AB.AB.

• Draw a line through BC.Draw a line through BC.

• Through CD draw a ray.Through CD draw a ray.

• Draw a segment Draw a segment through AD.through AD.

A

B

C

D

1.21.2

• Collinear Points-Collinear Points-

• CoplanarCoplanar

Points-Points-

1.31.3

Segments and Their Segments and Their MeasuresMeasures

1.31.3Postulate 1- The Ruler PostulatePostulate 1- The Ruler Postulate

The points on a line can be The points on a line can be matched one to one with matched one to one with the real numbers. The real the real numbers. The real number that corresponds number that corresponds to a point is the to a point is the coordinate coordinate of the point.of the point.

The The DistanceDistance between between points A and B, written as points A and B, written as AB, is the absolute value of AB, is the absolute value of the difference between the the difference between the coordinates of A and B.coordinates of A and B.

AB is also called the AB is also called the length length of AB.of AB.

A B

1.31.3Postulate 2: Segment Addition Postulate 2: Segment Addition

PostulatePostulate

If B is between A and C, then If B is between A and C, then AB+BC=AC. AB+BC=AC.

If AB+BC=AC, then B is between A If AB+BC=AC, then B is between A and C. and C.

A B C

Distance FormulaDistance Formula• If A(xIf A(x11,y,y11) and B(x) and B(x22,y,y22) are points in a ) are points in a

coordinate plane, then the distance coordinate plane, then the distance between A and B isbetween A and B is

AB= (x2-x1) 2 + (y2-y1)2

Using the Distance FormulaUsing the Distance Formula

• Use the Distance Formula to find the Use the Distance Formula to find the lengths between the two points.lengths between the two points.

•A(-1,1) B(-4,3)A(-1,1) B(-4,3)• Try and then we will check it..Try and then we will check it..

ANSWERANSWER

((-4)- (-1))2 +(3-1) 2

(-3)2 +22

9+4

13

AB= (x2-x1) 2 + (y2-y1)2

DID YOU GET IT CORRECT?

Now your turn to try it…Now your turn to try it…

• In the picture of collinear In the picture of collinear points,points,

AE=20AE=20

BD=6BD=6

AB=BC=CDAB=BC=CDFind Each Length…

•BC

•AB

•AC

•AD

3

3

6

9

A

B

C

D

E

1.41.4

Angles and Their Angles and Their MeasuresMeasures

AngleAngle

Naming AnglesNaming Angles

•What are the two names of What are the two names of the angle?the angle?

A

B

C

L ABC and L CBA

1.41.4Postulate 3: Protractor Postulate 3: Protractor PostulatePostulate

Consider a point A on one side of OB. The rays of the Consider a point A on one side of OB. The rays of the form OA can be matched one to one with the real form OA can be matched one to one with the real numbers from 0 to 180. The measure of numbers from 0 to 180. The measure of L L AOB is AOB is equal to the absolute value of the difference equal to the absolute value of the difference between the real numbers for OA and OB.between the real numbers for OA and OB.

A

B O

1.41.4Postulate 4: Angle Addition Postulate 4: Angle Addition

PostulatePostulate

If P is in the interior of If P is in the interior of L L RST, thenRST, then

mmLRSP+LRSP+mmLPST=LPST=mmLRSTLRST

mLRSTmLRSP

mLPST

S

R

P

T

Acute AngleAcute AngleAn angle with a measure between An angle with a measure between

0 degrees and 90 degrees0 degrees and 90 degrees

Right AngleRight Angle

An angle with a measure of An angle with a measure of 90 degrees.90 degrees.

Obtuse AngleObtuse AngleAn angle with a measure An angle with a measure

between 90 degrees and 180 between 90 degrees and 180 degrees.degrees.

Straight AngleStraight Angle

An angle with a measure of An angle with a measure of 180 degrees.180 degrees.

Adjacent AnglesAdjacent Angles

Two angles with a common Two angles with a common vertex and side, but no vertex and side, but no common interior points.common interior points.

Now your TurnNow your Turn

• Using the Angle Addition Postulate.Using the Angle Addition Postulate.

• What is mWhat is mLLDBC?DBC?

60°

A

BC

D

Answer:30°

1.51.5

Segment and Angle Segment and Angle BisectorsBisectors

Midpoint FormulaMidpoint Formula

• If A(xIf A(x11,y,y11) and ) and B(xB(x22,y,y22) are ) are points in a points in a coordinate coordinate plane, then the plane, then the midpoint of AB midpoint of AB has coordinates.has coordinates.

(x(x11,y,y11))

(x(x22,y,y22))

Your turn Applying Midpoint Your turn Applying Midpoint FormulaFormula

A(-A(-2,32,3) and B(5,-2)) and B(5,-2)

•Find the midpoint of AB.Find the midpoint of AB.

ANSWER 3, 1

2 2

1.61.6

Angle Pair Angle Pair RelationshipsRelationships

Vertical Angles/ Linear PairVertical Angles/ Linear Pair

• Consists of two Consists of two angles whose sides angles whose sides form two pairs of form two pairs of opposite rays.opposite rays.

Consists of two adjacent angles whose non-common sides are opposite rays.

5

6

L1 and L3 are vertical angles.

L2 and L4 are vertical angles. L5 and L6 are linear pairs

Finding the Angle Measure…Finding the Angle Measure…

Find the Measurement of Find the Measurement of LL1.1.

30° 1

1

45°

Answer: 150°

Answer: 45°

That concludes That concludes Chapter 1.Chapter 1.

Basics of Geometry.Basics of Geometry.