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
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Lesson 1.3 Collinearity, Betweenness, and Assumptions
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. - PowerPoint PPT Presentation
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Lesson 1.3Collinearity, 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
Def. Points that lie on the same line are called collinear.
Def. Points that do not lie on the same line are called noncollinear.
Definitions…
U
A
N
S
H
P
NoncollinearCollinear
Name as many sets of points as you can that are collinear and
noncollinear
Example #1
YX
SR
O
M
P
T
In order for us to say that a point is between two other points, all three points MUST be collinear.
Definitions…
U
A
NS
H
P
P is NOT between H and SA is between N and U
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)
2.They are noncollinear (the 3 points determine a triangle)
Triangle Inequality
5.5A 12.5
BC
18
11 14
24A
B
C
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”
Triangle Inequality
11 14
24A
B
C
When given a diagram, sometimes we need to assume certain information, but you know what they
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.
Definitions
Note:
Only segments have midpoints!
X
Y
X is not a midpoint
Y is not a midpoint
Why can’t a ray or line have a midpoint?
A
Y
B
X
M
Conclusions:
Example
F
G
ED
If D is the midpoint of segment FE, what conclusions can we draw?
Point D bisects
bisects
FD DE
FE
DG FE
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.
Definitions
Note:
One again, only segments have trisection points!
If , what conclusions can we draw?
Examples
A
S
C
R
AR RS SC
If E and F are trisection points of segment DG, what conclusions can we draw?
H
D E F G
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.
Definitions
Examples
DA
B C
If , then is the bisector of ABCABC DBC BD ��������������
40°40°
D
A
B E
35°35°
C
35°
If ,
then and trisect ABE
ABC CBD DBE
BC BD
����������������������������
Example #1
MO P
x + 8 2x - 6
44
Does M bisect segment OP?
Example #2 A
BGiven: B is a midpoint of
Prove:
Statement Reason
AC
AB BCC
D
Example #3
FE G
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