chapter 1 introduction to geometry. slide 2 1.1 getting started points – to name a point always...
TRANSCRIPT
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