Transcript Example #1

Lesson 1.3
Collinearity, Betweenness, and
Assumptions
Objective:
Recognize collinear, and non-collinear points, recognize
when a point is between two others, recognize that each side
of a triangle is shorter than the sum of the other two sides,
and correctly interpret geometric diagrams
Definitions…
Def. Points that lie on the same line are called
collinear.
Def. Points that do not lie on the same line are
called noncollinear.
P
U
A
H
S
N
Collinear
Noncollinear
Example #1
Name as many sets of points as you
can that are collinear and
noncollinear
O
R
S
M
X
Y
P
T
Definitions…
In order for us to say that a point is between two
other points, all three points MUST be collinear.
U
A
P
H
S
N
A is between N and U
P is NOT between H and S
Triangle Inequality
For any 3 points there are only 2 possibilities:
1. They are collinear (one point is between the other
two and two of the distances add up to the 3rd)
A
C
B 5.5
12.5
18
2. They are noncollinear (the 3 points determine a
B
triangle)
14
11
A
24
C
Triangle Inequality
Notice in this triangle, 14 + 11 > 24.
This is extra super important!
“The sum of the lengths of any 2 sides of a triangle is
always greater than the length of the third”
B
14
11
A
24
C
Assumptions
When given a diagram, sometimes we need to
assume certain information, but you know what they
say about assuming….
There are do’s and don’ts!
You should Assume
*Straight lines and angles
*Collinearity of points
*Betweenness of points
*Relative positions of points
You should NOT
Assume
*Right angles
*Congruent segments
*Congruent angles
*Relative sizes of segments
and angles
Homework
Lesson 1.3 Worksheet
Lesson 1.4
Beginning Proofs
Objective:
Write simple two-column proofs
Introducing…
The Two-Column Proof!
The two-column proof is the major type of proof we
use throughout our studies.
Def. A theorem is a mathematical statement that
can be proved.
Theorem Procedure…
1. We present a theorem(s).
2. We prove the theorem(s).
3. We use the theorems to help prove sample
problems.
4. You use the theorems to prove homework
problems.
Note:
The sooner you learn
the theorems, the
easier your homework
will be! 
Theorem 1
If two angles are right angles,
then they are congruent.
Given: <A is a right <.
B
<B is a right <.
A
Prove: A  B
Statement
Reason
1.
2.
3.
4.
<A is a right <
m<A = 90°
<B is a right <
m<B = 90°
1.
2.
3.
4.
5.
A  B
5. If 2 <‘s have the same measure then
they are congruent.
Given
If an < is a right < then its measure is 90°
Given
If an < is a right < then its measure is 90°
Theorem 2
If two angles are straight angles,
then they are congruent.
U
A
Given: <NAU is a straight <.
<PHS is a straight <.
Prove:
NAU  PHS
Statement
Reason
N
P
H
S
Practice Makes Perfect…
Now that we know the two theorems (and have
proved them), we apply what we know to sample
problems.
about what we can and cannot assume from a
diagram! This is important with proofs!
Example #1
Given: <RST = 50°
<TSV = 40°
<X is a right angle
Prove:
R
T
RSV  X
X
S
Statement
Reason
V
Example #2
Y
X
Given: <ABD = 30°
D
A
<ABC = 90°
E
<EFY = 50° 20’
B
<XFY = 9° 40’
Prove: DBC  XFE
Statement
Reason
C
F
Homework
Lesson 1.4 Worksheet
Lesson 1.5
Division of Segments and Angles
Objective:
Identify midpoints and bisectors of segments, trisection
points and trisectors of segments, angle bisectors and
trisectors.
Definitions
Def. A point (or segment, ray, or line) that divides
a segment into two congruent segments
bisects the segment. The bisection point is
called the midpoint of the segment.
Y
A
M
Note:
Only segments
have midpoints!
B
X
Why can’t a ray or line have a midpoint?
X
X is not a midpoint
Y
Y is not a midpoint
Example
If D is the midpoint of segment FE, what
conclusions can we draw?
G
F
Conclusions:
FD  DE
Point D bisects FE
DG bisects FE
D
E
Definitions
A segment divided into three congruent parts is said
to be trisected.
Def. Two points (or segments, rays, or lines) that
divides a segment into 3 congruent segments trisect
the segment. The 2 points at which the segment is
divided are called trisections points.
Note:
One again, only
segments have
trisection points!
Examples
If
AR  RS  SC , what conclusions can we draw?
C
S
R
A
If E and F are trisection points of segment DG, what
conclusions can we draw?
H
D
E
F
G
Definitions
Like a segment, angles can also be bisected and
trisected.
Def. A ray that divides an angle into 2 congruent
angles bisects the angle. The dividing ray is called
the angle bisector.
Def. Two ray that divide an angle into 3 congruent
angles trisects the angle. The 2 dividing rays are
called angle trisectors.
Examples
If ABC  DBC , then BD is the bisector of ABC
A
D
40°
40°
B
C
If ABC  CBD  DBE ,
then BC and BD trisect ABE
C
A
35°
B
35°
35°
D
E
Example #1
Does M bisect segment OP?
2x - 6
x+8
O
M
44
P
Example #2
A
Given: B is a midpoint of AC
Prove: AB  BC
B
C
Statement
Reason
D
Example #3
Segment EH is divided by F and G in the ratio 5:3:2 from left
to right. If EH = 30, find FG and name the midpoint of
segment EH.
E
F
G
H
Classwork
1.1-1.3 Review Worksheet
Homework
Lesson 1.5 Worksheet