Transcript LinesAnglesPlanesTriangles

Points

Points do not have actual size.

How to Sketch:
A
Using dots

B
C
A
How to label:
Use capital letters
Never name two points with the same letter
(in the same sketch).
Lesson 1-1 Point, Line, Plane
1
Lines


Lines extend indefinitely and have no thickness or width.
How to sketch : using arrows at both ends.
n
A
B


C
How to name: 2 ways
(1) small script letter – line n
(2) any two points on the line - AB , BC, AC , BA, CA, CB
Never name a line using three points - ABC
Lesson 1-1 Point, Line, Plane
2
Collinear Points


Collinear points are points that lie on the same line. (The line does
not have to be visible.)
A point lies on the line if the coordinates of the point satisfy the
equation of the line.
Ex: To find if A (1, 0) is collinear with
A
B
C
the points on the line y = -3x + 3.
Substitute x = 1 and y = 0 in the equation.
Collinear
0 = -3 (1) + 3
C
0 = -3 + 3
A
0=0
B
The point A satisfies the equation, therefore the point is collinear
Non collinear
with the points on the line.
Lesson 1-1 Point, Line, Plane
3
Postulate 1-5
Ruler Postulate
The distance between any two points is the absolute
value of the difference of the corresponding
numbers (on a number line or ruler)
Congruent Segments: two segments with the same
length
Midpoint: a point that divides the segment into two
equal parts
A
B
B is the midpoint, so AB = BC
C
Postulate 1-6
Segment Addition Postulate
If three points A, B, and C are collinear and B is
between A and C, then AB + BC = AC
C
B
A
Using the Segment Addition
Postulate
If DT = 60, find the value of x. Then find DS and ST.
2x - 8
D
3x - 12
S
T
Using the Segment Addition
Postulate
If EG = 100, find the value of x. Then find EF and FG.
4x - 20
E
2x + 30
F
G
Finding Lengths
C is the midpoint of AB. Find AC, CB, and AB.
3x – 4
2x + 1
A
C
B
Intersection of Figures
The intersection of two figures is the set of points that are
common in both figures.
The intersection of two lines is a point.
m
Line m and line n intersect at point P.
P
n
Lesson 1-1 Point, Line, Plane
Continued…….
10
Planes
A plane is a flat surface that extends indefinitely in all directions.
 How to sketch: Use a parallelogram (four sided figure)
 How to name: 2 ways
(1) Capital script letter – Plane M

(2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC /
BCA / CAB / CBA
A
B
M
C
Horizontal Plane
Vertical Plane
Lesson 1-1 Point, Line, Plane
Other
11
3 Possibilities of Intersection of a Line and a Plane
(1) Line passes through plane – intersection is a point.
(2) Line lies on the plane - intersection is a line.
(3) Line is parallel to the plane - no common points.
Lesson 1-1 Point, Line, Plane
12
Throughout New York City there are free movie nights in
the park. The movie screen in the middle of the park is an
example of a plane (2D) in space (3D).
Lesson 1-1 Point, Line, Plane
13
Intersection of Two Planes is a Line.
B
P
A
R
Plane P and Plane R intersect at the line AB
Lesson 1-1 Point, Line, Plane
14
Different planes in a figure:
A
D
B
C
E
H
Plane EFGH
F
G
Plane ABCD
Plane BCGF
Plane ADHE
Plane ABFE
Plane CDHG
Etc.
Lesson 1-1 Point, Line, Plane
15
Other planes in the same figure:
Any three non collinear points determine a plane!
A
D
B
Plane AFGD
Plane ACGE
C
Plane ACH
E
H
F
G
Plane AGF
Plane BDG
Etc.
Lesson 1-1 Point, Line, Plane
16
Coplanar Objects
Coplanar objects (points, lines, etc.) are objects that lie on
the same plane. The plane does not have to be visible.
A
D
B
C
E
H
F
G
Are the following points coplanar?
A, B, C ?
A, B, C, F ?
H, G, F, E ?
E, H, C, B ?
A, G, F ?
C, B, F, H ?
Lesson 1-1 Point, Line, Plane
Yes
No
Yes
Yes
Yes
No
17
Regents Questions on Planes


Go to jmap.org -> Resources by Topics>Geometry->Planes
Groups assigned G.G.1 through G.G.9
to complete and present whole class
Lesson 1-1 Point, Line, Plane
18
Angle Relationships
Adjacent angles are “side by side”
and share a common ray.
45º
15º
These are examples of adjacent
angles.
80º
45º
35º
55º
85º
20º
130
º
50º
These angles are NOT adjacent.
100
º
50º
35º
35º
55º
45º
Complementary Angles
sum to 90°
50
40° °
Complementary angles add up to 90º.
30º
40º
60º
Adjacent and
Complementary Angles
50º
Complementary Angles
but not Adjacent
Supplementary Angles
sum to 180°
30°
150
°
Supplementary angles add up to 180º.
40º
120
º
60º
Adjacent and
Supplementary Angles
140
º
Supplementary Angles
but not Adjacent
Vertical Angles
are opposite one another.
Vertical angles are congruent.
100
°
100
°
Vertical Angles
are opposite one another.
Vertical angles are congruent.
80
°
80°
Lines l and m are parallel.
l||m
Note the 4
angles that
measure
120°.
120
°
120
°
Line n is a
transversal.
120
°
120
°
n
l
m
Lines l and m are parallel.
l||m
Note the 4
angles that
measure
60°.
60
°
60
°
60
°
l
m
60
°
Line n is a
transversal.
n
Lines l and m are parallel.
l||m
There are many
pairs of angles
that are
supplementary.
Line n is a
transversal.
There are 4
pairs of angles
that are vertical.
60
°
120
°
60
°
120
°
120
°
60
°
120
°
60
°
n
l
m
If two lines are intersected by a transversal and
any of the angle pairs shown below are congruent,
then the lines are parallel. This fact is used in the
construction of parallel lines.
Let’s Practice
1201 2 60
°
°
60 3 4
120
°
120 5 660°
°7 8 °
120
60
°
°
m<1=120°
Find all the remaining
angle measures.
Practice Time!
1) Find the missing angle.
?°
36°
1) Find the missing angle.
?°
36°
2) Find the missing angle.
?°
64°
2) Find the missing angle.
?°
64°
3) Solve for x.
2x
3x° °
3) Solve for x.
2x
3x° °
4) Solve for x.
2x + 5
4) Solve for x.
2x + 5
5) Find the missing angle.
?°
168°
5) Find the missing angle.
?°
168°
6) Find the missing angle.
58°
?°
6) Find the missing angle.
58°
?°
7) Solve for x.
4x
5x
7) Solve for x.
4x
5x
8) Solve for x.
2x + 10
3x + 20
8) Solve for x.
2x + 10
3x + 20
9) Lines l and m are parallel.
l||m
Find the missing angles.
42
°
c°
a
°
b°
d°
e°
g°
f°
l
m
10) Lines l and m are parallel.
l||m
Find the missing angles.
81
°
c°
a
°
b°
d°
e°
g°
f°
l
m
In the figure a || b.
13. Name the angles congruent to 3.
14. Name all the angles supplementary to 6.
15. If m1 = 105° what is m3?
16. If m5 = 120° what is m2?
Find the missing angles.
70
°
Hint: The 3 angles
in a triangle sum to
180°.
d
°
70
b° °
65
°
Find the missing angles.
70
°
Hint: The 3 angles
in a triangle sum to
180°.
75
°
40
°
70
°
65
°
Find the missing angles.
45
°
Hint: The 3 angles
in a triangle sum to
180°.
d
°
50
b° °
75
°
Find the missing angles.
45
°
Hint: The 3 angles
in a triangle sum to
180°.
20
°
85
°
50
°
75
°
Another practice problem
40
°
120
°
Find all the missing
angle measures,
and name the
postulate or
theorem that
gives us
permission to
make our
statements.
Ex. : Interior Angles of a
Quadrilateral
P

80°
Find the measure of
each angle in the
quadrilateral?
70°
x°
Q
2x°
R
S
Find the measure of the
missing angle in the figure
below
135
100
70
quadrilateral
135 + 100 + 70 + x = 360
305 + x = 360
-305
-305
x
x = 55
The End