Chapter 1 Introduction to Geometry. Slide 2 1.1 Getting Started Points – To name a point always...

Chapter 1

Introduction to Geometry

Slide 2

1.1 Getting StartedPoints – To name a point always use

Lines – All lines are and extend in both directions. To name a line use on the line.

Line Segment – Has a definite and , called . To name a segment use .

Ray – Begins at and then extends in one direction. To name a ray you must name the first and then on the ray.

m

CBA

FED

JHG

Slide 3

1.1 Getting StartedAngle – Two with the same form an angle. The common is called the , and the two are called the .

Triangle –To name a triangle use of the triangle.

Union ( ) – What do the objects ?

Intersection (∩) – What do the objects ?

1

M

LK

U

TS

32

R

QPO

N

Slide 4

1.1 Example

R O

PS

T

1) SO U RO =

2) TP U TR =

3) PO SP =

4) SR PO =

5) TS U TO =

6) PT SR =

7) SP U TP U ST =

Slide 5

1.2 Measurement of Segments and Angles

Measuring Segments

Find AB.

Classifying Angles

Acute: Angle measures

Right: Angle measures

Obtuse: Angle measures

Straight: Angle measures

BA

10-1-2-3-4

Slide 6

1.2 Measurement of Segments and Angles

Measuring Angles

60 minutes =

60 seconds =

Congruent ( )

Two angles with the

Two segments with the

On diagrams we use to indicate congruentparts.

Slide 7

1.2 Examples

71) Change 35 to degrees, minutes and seconds.

18

2) ABC is a right , m DBC 25 70'13" Find m ABD.

3) Find m formed by the hands of a clock at 6:15.

D

CB

A

Slide 8

1.2 Examples

4) ABD DBC Find m CBE.

5) D is acute What are the restrictions on x?

55(x + 35)

(x + 5)

E

D

CB

A

D(2x - 30)

Slide 9

1.3 Collinearity, Betweenness, and Assumptions

Collinear – Points that lie on

Noncollinear – Points that

Betweenness of Points – All three points must be

Triangle Inequality – The sum of the lengths of any is always than the length of the

You should assume from a diagram…

1) Straight lines and angles

2) Collinearity of points

3) Betweenness of points

4) Relative positions of points

Slide 10

1.3 Examples

1) BD must be smaller than what number? 2) BD must be larger than what number?

1710

D

E

B

Slide 11

1.3 Examples

3) Name three collinear points. 4) Name three noncollinear points. 5) Is A E? 6) Is B D? 7) Name a straight angle. 8) What two points is F b

etween? 9) Is A right of C? 10) Is AB DE? 11) What two segments do the tick marks indicate are . 12) Is AC BF?

F

E

D

C

B

A

Slide 12

1.4 Beginning Proofs

Given: D is a right E is a right Prove: D E

Statements Reasons

E D

C

B

A

Theorem – A mathematical model that can be .

2 s rt. s

2 s st. s

Slide 13

1.4 Examples

Given: Diagram as shownConclusion: ROP NOQ

Statements ReasonsQ

P

N

O

R

Slide 14

1.4 Examples

Given: AC 12 BC 7 DE 5

Prove: AB DEStatements Reasons

ED

CBA

Slide 15

1.5 Division of Segments and Angles

Bisect – Divide a segment or angle into parts

•On a segment the bisection point is called the .

•In an angle, the dividing ray is called the .

Trisect – Divide a segment or angle into parts

•On a segment, the two points that divide the segment are called .

•In an angle, the two dividing rays are called .

Slide 16

1.5 Examples

1) Given: XY bisects AC at B

Prove: AB BC

Statements Reasons

Y

CB

X

A

Slide 17

1.5 Examples

����������������������������2) Given: EB & EC trisect AED Conclusion: AEB BEC CED

Statements ReasonsE

D

C

BA

Slide 18

1.5 Examples

3) Given: MI IA

Prove: I is the midpoint of MA

Statements Reasons

AIM

Slide 19

1.7 Deductive Structure & 1.8 Statements of Logic

Conditional Statement: If p, then q. Hypothesis: Conclusion: Negation:

Every conditional statement has three other statements. 1. Converse – hypothesis and conclusion2. Inverse – hypothesis and conclusion3. Contrapositive – hypothesis and

conclusion

Theorem 3: If a conditional statement is true, then the of the statement is also true.

p q

Slide 20

1.7 & 1.8 Examples

If Joe is a member of the RB soccer team, then he is a student at RB.

1) Write the converse.

2) Write the inverse.

3) Write the contrapositive.

4) Are the above statements true?

Slide 21

1.7 & 1.8 Examples

What conclusion can you draw given:

~ ~

g et wt e

Slide 22

1.9 Probability

# of winnersProbability =

# of possibilities

1) If one of the three points is picked at random, what is the probability that the point lies on BD?

2) If two of the four points are selected at random, what is

the probability that both lie on AT?

DEB

T

R

A

C

Slide 23

1.9 Example3) A point P is randomly chosen on AT. What is the probability that it is within four units of R?

14-4-6

TRA