Chapter 1
Discovering Geometry
1.1 Basic Geometric Figures
I. Point
A. Geometric figure with no dimensions
B. Used to identify a point in space
C. Represented by a dot
•
1.1 Basic Geometric Figures
I. Point
A. Geometric figure with no dimensions
B. Used to identify a point in space
C. Represented by a dot
D. Labeled by a capital letter
A
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
A
•
B
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
E. Labeled by any two point that it contains
A
•
B
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
E. Labeled by any two point that it contains
A
•
AB
B
•
C
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
E. Labeled by any two point that it contains
A
AB
B
•
•
AC
BC
C
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
E. Labeled by any two point that it contains
F. The intersection of two lines is a _______
point
W
•
P
•
J
•
W
•
P
•
The intersection of WP and PJ is P.
J
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
E. Labeled by any two point that it contains
F. The intersection of two lines is a _______
point
G. Through any one point there are
infinitely many lines
•
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
E. Labeled by any two point that it contains
F. The intersection of two lines is a _______
point
G. Through any one point there are
infinitely many lines
H. Through any two points there is exactly
one line
III. Plane
A. Geometric figure having infinite
length and width but no height.
B. Represented by a flat rectangular surface
C. Planes consist of lines
D. Labeled by any three points on the plane
A
L
M
P
Q
D
K
H
B
C
Are Points L, K, and M COPLANAR?
Yes, they are COPLANAR because they LIE ON THE SAME PLANE P.
Is point H, coplanar with points L, K, and M?
No, because it lies on plane Q and points L, K, and M are in different plane, on plane P.
NON-COPLANAR points are points that lie in different planes.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
A
P
B
Q
C
On what planes does point D lie?
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
D
A
P
B
Q
C
On what planes does point D lie? It only lies on plane Q.
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
D
III.
Plane
A. Geometric figure having infinite
length and width but no height.
B. Represented by a flat rectangular surface
C. Planes consist of lines
D. Labeled by any three points on the plane
E. Through any two points there are
infinitely many planes
F. Through any three points, there is exactly
one plane
III. Plane
A. Geometric figure having infinite
length and width but no height.
B. Represented by a flat rectangular surface
C. Planes consist of lines
D. Labeled by any three points on the plane
E. Through any two points there are
infinitely many planes
F. Through any three points, there is exactly
one plane
G. The intersection of two planes is a_______
line
H. The intersection of three planes is a
_____________
or ___________
point
line
IV. Line Segment
A. A piece of a line
B. Has two endpoints
C. Labeled by its endpoints
∙
∙
S
T
ST
V.
Ray
A. Geometric figure with one endpoint
B. Labeled by it’s endpoint and one
other point
∙
∙
P
Q
PQ
1.2 Measuring Line Segments
I.
“Measure” of a Line Segment
A. The distance between its endpoints
B. Always positive
1.2 Measuring Line Segments
B
A
-5
•
-4
-3 -2
-1
0
1
a
2
•
3
b
coordinates
4
5
1.2 Measuring Line Segments
AB = “the measure of AB”
AB = _________
7 units
A
-5
•
-4
B
AB
-3 -2
-1
0
1
a
2
•
3
b
coordinates
4
5
1.2 Measuring Line Segments
I.
“Measure” of a Line Segment
A. The distance between its endpoints
B. Always positive
C. AB = b – a or
a-b
1.2 Measuring Line Segments
AB = 3 – (-4)
AB = 3 + (+4)
AB = 7
A
-5
•
-4
or
AB = -4 – 3
AB = -4 + -3
AB = -7
B
AB
-3 -2
-1
0
1
a
2
•
3
b
coordinates
4
5
1.2 Measuring Line Segments
AB = 3 – (-4)
AB = 3 + (+4)
AB = 7 units
A
-5
•
-4
or
AB = -4 – 3
AB = -4 + -3
AB = -7 = 7 units
B
AB
-3 -2
-1
0
1
a
2
•
3
b
coordinates
4
5
Examples
PQ = ________________
95 – 23 = 72 units
P
•
23
PQ
Q
•
95
Examples
46 – (-15) = 61 units
EF = ________________
OR
-15 – 46 = -61 = 61 units
EF = ________________
E
•
-15
EF
F
•
46
Examples
-18 – (-92)
= 74 units
|
RS = ________________
OR
|
-92 – (-18)
RS = ________________
= -74 = 74 units
R
•
-92
RS
S
•
-18
1.2 Measuring Line Segments
II.
Segment Addition
A. “collinear”= “on the same line”
B. If A, B, & C are collinear and B is
between A and C, then
AB + BC = AC
•
• •
A
B
AB
C
BC
AC
Examples of Segment Addition
A carpenter must cut a 54 inch board into two
pieces so that one piece is twice as long as the
other. What will be the length of the two board
after the cut?
AB + BC = AC
x + 2x = 54
3x = 54
3
3
A
B
• •
•
C
X
2x
54 in.
Examples of Segment Addition
A carpenter must cut a 54 inch board into two
pieces so that one piece is twice as long as the
other. What will be the length of the two board
after the cut?
18
AB + BC = AC
3 54
x + 2x = 54
3
3x = 54
24
3
3
24
x = 18 in.
0
A
B
• •
X
18 in.
54 in.
•
C
2x
= 2(18)
36 in.
Examples of Segment Addition
A 45 foot piece of pipe must be cut so that the
longer piece is 9 feet longer than the shorter.
What will be the lengths of the two pieces?
AB + BC = AC
x + x + 9 = 45
2x + 9 = 45
- 9 = -9
2x
= 36
2
2
A
B
• •
X
45 ft.
•
C
X+9
Examples of Segment Addition
A 45 foot piece of pipe must be cut so that the
longer piece is 9 feet longer than the shorter.
What will be the lengths of the two pieces?
AB + BC = AC
x + x + 9 = 45
18
2x + 9 = 45
2 36
- 9 = -9
2
2x
= 36
16
16
2
2
0
x = 18 ft.
A
B
• •
X
18 ft.
45 ft.
•
C
X + 9 = 18 + 9
27 ft.
1.2 Measuring Line Segments
III. Midpoint of a Segment
A. If A, B, and C are collinear and
AC = CB, then C is the midpoint of AB.
1.2 Measuring Line Segments
III. Midpoint of a Segment
A. If A, B, and C are collinear and
AC = CB, then C is the midpoint of AB.
•
A
•
C
•
B
1.2 Measuring Line Segments
III. Midpoint of a Segment
A. If A, B, and C are collinear and
AC = CB, then C is the midpoint of AB.
B. Midpoint Formula
•
A
12
a
•
C
•
B
58
b
1.2 Measuring Line Segments
III. Midpoint of a Segment
A. If A, B, and C are collinear and
AC = CB, then C is the midpoint of AB.
B. Midpoint Formula
The midpoint of AB = a + b = 12 + 58
2
2
•
•
•
A
C
B
12
35
58
a
b
= 70 = 35
2
1.2 Measuring Line Segments
III. Midpoint of a Segment
A. If A, B, and C are collinear and
AC = CB, then C is the midpoint of AB.
B. Midpoint Formula
The midpoint of AB = a + b = -15 + 35
2
2
•
•
•
A
C
B
-15
10
35
a
b
= 20 = 10
2
1.2 Measuring Line Segments
III. Midpoint of a Segment
A. If A, B, and C are collinear and
AC = CB, then C is the midpoint of AB.
B. Midpoint Formula
The midpoint of AB = a + b = -84 + -12
2
•
•
•
A
C
B
-84
-48
-12
a
b
2
= -96 = -48
2
Examples of Segment Addition
A carpenter must cut a 65 inch board into two
pieces so that one piece is five inches more than
twice the length of the other. What will be the
length of the two board after the cut?
AB + BC = AC
x + 2x+5 = 65
A
B
• •
•
C
X
2x+5
65 in.
1.3 Measuring Angles
A. Using a Protractor
60°
ACUTE Angle
less than 90 °
1.3 Measuring Angles
A. Using a Protractor
RIGHT Angle
90°
1.3 Measuring Angles
A. Using a Protractor
140°
OBTUSE Angle
Greater than 90 °
1.3 Measuring Angles
A. Using a Protractor
120°
1.3 Measuring Angles
A. Using a Protractor
•B
•C •D
•F
•
E
•
A
O
1.3 Measuring Angles
A. Using a Protractor
B. Angle Addition
•
A
•
D
B
•
•
C
1.3 Measuring Angles
A. Using a Protractor
m
B. Angle Addition
ABD + m DBC = m ABC
•
A
•
D
B
•
•
C
A 70 ° angle is divided into two smaller angles
such that the larger angle is two more than
three times the smaller.
m ABD + m DBC = m ABC
x + 3x +2 = 70
4x + 2 = 70
–2 –2
___
4x = ___
68
4
4
x = 17
B
•
•
A
3(17) + 2 70°
3x + 2
D
53°
17°
•
x
•
C
1.3 Measuring Angles
A. Using a Protractor
B. Angle Addition
C. Vertical Angle Conjecture
“ the vertical angles formed by intersecting
lines have equal measure”
1.4 Special Angles
A. Complementary Angles
A pair of angles whose sum is 90
1
2
º
1.4 Special Angles
A. Complementary Angles
A pair of angles whose sum is 90
1
2
º
1.4 Special Angles
B. Supplementary Angles
A pair of angles whose sum is 180°
1
2
1.4 Special Angles
B. Supplementary Angles
1
2
An angle is four times it’s compliment. Find
both angles.
x + 4x = 90
5x = 90
x = 18
4(18) = 72°
4x
x
18°
1.5 Parallel and Perpendicular Lines
A. Parallel Lines
Lines on the same plane that do not
l
intersect
m
l || m
1.5 Parallel and Perpendicular Lines
A. Parallel Lines
Lines on the same plane that do not
intersect
B. Perpendicular Lines
Two lines that intersect at a right angle
1.5 Parallel and Perpendicular
Lines
k
k
j
j
1.5 Parallel and Perpendicular
Lines
C. Corresponding Angles
1
2
m<1 = m<2 = m<3
3
1.5 Parallel and Perpendicular
Lines
C. Corresponding Angles
3x + 20 = 5x – 10
-3x
-3x
20 = 2x – 10
+10 = + 10
30 = 2x
2
2
15 = x
4
m<1 = 53(15)
+20
75
m<2 = 5(15) –
10
65º
(3x+20)
1
(5x -10) 65º
2
m<1 = m<2 = m<3
1.5 Parallel and Perpendicular
Lines
C. Corresponding Angles
6x + 30 = 3x + 57
-3x
-3x
3x + 30 =
57
- 30 = - 30
3x = 27
3
3
x= 9
5
m<1 = 46(9)
+30
27
m<2 = 3(9) +
57
84º
1 (6x+30)
84º
2 (3x
+57)
m<1 = m<2 = m<3
1.5 Parallel and Perpendicular
Lines
C. Corresponding Angles
9x + 50 + 4x + 39 =
13x +89 = 180
180
- 89 - 89
13x = 91
13
13
x= 7
6
m<1 = 39(7)
+50
28
m<2 = 4(7) +
39
113º
(9x+50) (4x
1 2 +39)
m<1 = m<2 = m<3
67º