On Geometric Permutations Induced by Lines Transversal through a Fixed Point
Chapter 1 Discovering Geometry. 1.1 Basic Geometric Figures I. Point A. Geometric figure with no...
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Transcript of Chapter 1 Discovering Geometry. 1.1 Basic Geometric Figures I. Point A. Geometric figure with no...
![Page 1: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/1.jpg)
Chapter 1
Discovering Geometry
![Page 2: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/2.jpg)
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
•
![Page 3: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/3.jpg)
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
![Page 4: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/4.jpg)
II. Line
A. Geometric figure having infinite length
B. No width or height
C. Consists of points
D. Represented by a double pointed arrow
![Page 5: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/5.jpg)
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
![Page 6: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/6.jpg)
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
![Page 7: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/7.jpg)
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
AB
•C
![Page 8: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/8.jpg)
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
AB
•C
AC BC
![Page 9: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/9.jpg)
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
![Page 10: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/10.jpg)
•
••
W
P J
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•
••
W
P J
The intersection of WP and PJ is P.
![Page 12: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/12.jpg)
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
![Page 13: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/13.jpg)
•
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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
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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
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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?
P
Q
A
B
LK
M
H
C
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.
D
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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On what planes does point D lie?
P
Q
A
B
C
D
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
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On what planes does point D lie? It only lies on plane Q.
P
Q
A
C
B D
PRESENTATION CREATED BY SIMON PEREZ. All rights reserved
![Page 19: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/19.jpg)
III. PlaneA. Geometric figure having infinite length and width but no height. B. Represented by a flat rectangular surfaceC. Planes consist of linesD. 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
![Page 20: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/20.jpg)
III. PlaneA. Geometric figure having infinite
length and width but no height. B. Represented by a flat rectangular surfaceC. Planes consist of linesD. Labeled by any three points on the planeE. Through any two points there are
infinitely many planesF. Through any three points, there is exactly
one plane G. The intersection of two planes is a_______lineH. The intersection of three planes is a _____________ or ___________linepoint
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IV. Line Segment
A. A piece of a line
B. Has two endpoints
C. Labeled by its endpoints
∙ ∙S T
ST
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V. Ray
A. Geometric figure with one endpoint
B. Labeled by it’s endpoint and one other point
∙ ∙P Q
PQ
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1.2 Measuring Line Segments
I. “Measure” of a Line Segment
A. The distance between its endpoints
B. Always positive
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1.2 Measuring Line Segments
0-1-2-3-4-5 2 3 4 51•A
•B
coordinates
a b
![Page 26: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/26.jpg)
1.2 Measuring Line Segments
0-1-2-3-4-5 2 3 4 51•A
•B
coordinates
a b
AB
AB = “the measure of AB”AB = _________7 units
![Page 27: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/27.jpg)
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
![Page 28: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/28.jpg)
1.2 Measuring Line Segments
0-1-2-3-4-5 2 3 4 51•A
•B
coordinates
a b
AB
AB = 3 – (-4)
AB = 3 + (+4) AB = 7
orAB = -4 – 3 AB = -4 + -3 AB = -7
![Page 29: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/29.jpg)
1.2 Measuring Line Segments
0-1-2-3-4-5 2 3 4 51•A
•B
coordinates
a b
AB
AB = 3 – (-4)
AB = 3 + (+4) AB = 7 units
orAB = -4 – 3 AB = -4 + -3 AB = -7 = 7 units
![Page 30: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/30.jpg)
Examples
•P
•QPQ
23 95
PQ = ________________95 – 23 = 72 units
![Page 31: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/31.jpg)
Examples
•E
•FEF
-15 46
EF = ________________46 – (-15) = 61 units
OR
EF = ________________-15 – 46 = -61 = 61 units
![Page 32: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/32.jpg)
Examples
•R
•SRS
-92 -18
RS = ________________-18 – (-92) = 74 units
OR
RS = ________________-92 – (-18) = -74 = 74 units
|
|
![Page 33: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/33.jpg)
1.2 Measuring Line SegmentsII. 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 C
AB BCAC
![Page 34: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/34.jpg)
Examples of Segment AdditionA 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?
• • •A B C
X 2x54 in.
AB + BC = AC x + 2x = 54 3x = 54 3 3
![Page 35: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/35.jpg)
Examples of Segment AdditionA 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?
• • •A B C
X 2x
54 in.
AB + BC = AC x + 2x = 54 3x = 54 3 3
3 541
324
8
240x = 18 in.
18 in.= 2(18)
36 in.
![Page 36: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/36.jpg)
Examples of Segment AdditionA 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?
• • •A B C
X X + 945 ft.
AB + BC = AC x + x + 9 = 45 2x + 9 = 45
2 2
- 9 = -9 2x = 36
![Page 37: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/37.jpg)
Examples of Segment AdditionA 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?
• • •A B C
X X + 9
45 ft.
AB + BC = AC x + x + 9 = 45
2x = 36 2 2
2 361
216
8
160 x = 18 ft.
18 ft.= 18 + 9
27 ft.
2x + 9 = 45 - 9 = -9
![Page 38: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/38.jpg)
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.
![Page 39: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/39.jpg)
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 BC
![Page 40: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/40.jpg)
1.2 Measuring Line SegmentsIII. Midpoint of a Segment
A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB.
• ••A BC
12 58a b
B. Midpoint Formula
![Page 41: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/41.jpg)
1.2 Measuring Line SegmentsIII. Midpoint of a Segment
A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB.
• ••A BC
12 58a b
B. Midpoint Formula
The midpoint of AB = a + b2
12 + 58 2
=
= 70 2
= 35
35
![Page 42: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/42.jpg)
1.2 Measuring Line SegmentsIII. Midpoint of a Segment
A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB.
• ••A BC
-15 35a b
B. Midpoint Formula
The midpoint of AB = a + b2
-15 + 35 2
=
= 20 2
= 10
10
![Page 43: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/43.jpg)
1.2 Measuring Line SegmentsIII. Midpoint of a Segment
A. If A, B, and C are collinear and AC = CB, then C is the midpoint of AB.
• ••A BC
-84 -12a b
B. Midpoint Formula
The midpoint of AB = a + b2
-84 + -12 2
=
= -96 2
= -48
-48
![Page 44: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/44.jpg)
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Examples of Segment AdditionA 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?
• • •A B C
X 2x+565 in.
AB + BC = AC x + 2x+5 = 65
![Page 46: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/46.jpg)
1.3 Measuring Angles
A. Using a Protractor
ACUTE Angle less than 90
60°
°
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1.3 Measuring Angles
A. Using a Protractor
RIGHT Angle
90°
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1.3 Measuring Angles
A. Using a Protractor
OBTUSE Angle
Greater than 90 °
140°
![Page 49: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/49.jpg)
1.3 Measuring Angles
A. Using a Protractor
°120
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1.3 Measuring Angles
A. Using a Protractor
•
•
•
• •
•A
BC
D
E
FO
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1.3 Measuring Angles
A. Using a ProtractorB. Angle Addition
•
••
•
A
B
C
D
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1.3 Measuring Angles
A. Using a ProtractorB. Angle Addition
•
••
•
A
B
C
D
m ABD + m DBC = m ABC
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•
••
•
A
B
C
D
m ABD + m DBC = m ABC
A 70 angle is divided into two smaller angles such that the larger angle is two more than three times the smaller.
70°
°
x
3x + 2
x + 3x +2 = 704x + 2 = 70
–2 –24x = 68___ ___
4 4
x = 17
17°
3(17) + 2
53°
![Page 54: 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.](https://reader035.fdocuments.net/reader035/viewer/2022062404/551bd3dd550346b9588b5638/html5/thumbnails/54.jpg)
1.3 Measuring Angles
A. Using a ProtractorB. Angle AdditionC. Vertical Angle Conjecture
“ the vertical angles formed by intersecting lines have equal measure”
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1.4 Special AnglesA. Complementary Angles
A pair of angles whose sum is 90
º
12
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1.4 Special AnglesA. Complementary Angles
A pair of angles whose sum is 90
º
12
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1.4 Special Angles
B. Supplementary Angles
A pair of angles whose sum is 180°
1 2
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1.4 Special Angles
B. Supplementary Angles
1 2
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An angle is four times it’s compliment. Find both angles.
x4x
x + 4x = 90
5x = 90
x = 18
18°
4(18) = 72°
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1.5 Parallel and Perpendicular Lines
A. Parallel Lines
Lines on the same plane that do not intersect
l
m
l || m
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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
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1.5 Parallel and Perpendicular Linesk
j
k j
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1.5 Parallel and Perpendicular Lines
C. Corresponding Angles
1 2 3
m<1 = m<2 = m<3
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1.5 Parallel and Perpendicular Lines
C. Corresponding Angles
1 2
m<1 = m<2 = m<3
(3x+20) (5x -10)
3x + 20 = 5x – 10 -3x -3x 20 = 2x – 10 +10 = + 10 30 = 2x 2 2 15 = x
m<1 = 3(15) +20
45
65º
m<2 = 5(15) – 10
75
65º
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1.5 Parallel and Perpendicular Lines
C. Corresponding Angles
1 2
m<1 = m<2 = m<3
(6x+30) (3x +57)
6x + 30 = 3x + 57 -3x -3x3x + 30 = 57 - 30 = - 30 3x = 27 3 3 x = 9
m<1 = 6(9) +30
54
84º
m<2 = 3(9) + 57
27
84º
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1.5 Parallel and Perpendicular Lines
C. Corresponding Angles
1 2
m<1 = m<2 = m<3
(9x+50) (4x +39)
9x + 50 + 4x + 39 = 180 13x +89 = 180
- 89 - 89 13x = 91 13 13
x = 7
m<1 = 9(7) +50
63
113º
m<2 = 4(7) + 39
28
67º
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