Geometry Chapter 3 Review

1
Geometry Chapter 3 Review

• Parallel Lines Transversals
• Angles and Parallel Lines
• Slopes of Lines
• Writing Linear Equations
• Proving Lines Parallel
• Parallels Distances

2
Transversals, Lines and their Angle Relationships
1
1
Corresponding
2
2
3
4
3
4
5
5
6
6
7
7
8
8
Transversal
Alternate Interior Angles - Angles that are on
opposite sides of the transversal and inside the
parallel lines. Examples 3 6, 4 5. Alternate
Exterior Angles - Angles on opposite sides of the
transversal and outside the parallel lines.
Examples 1 8, 2 7. Corresponding Angles -
Angles that are in the same position on two
different parallel lines cut by a transversal.
Ex 1 5, 3 7, 4 8 etc. Consecutive Angles -
Angles that are on the same side of the
transversal. 4 6, 3 5 are consecutive
interior angles, 1 7, 2 8 are consecutive
exterior angles.
3
Angles and Parallel Lines

• If two parallel lines are cut by a transversal,
  then
• Each pair of corresponding angles is congruent
  (Corresponding Angles Postulate)
• Each pair of alternate interior angles is
  congruent (Alternate Interior Angle Theorem)
• Each pair of consecutive interior angles is
  supplementary (Consecutive Interior Angle
  Theorem)
• Each pair of alternate exterior angles is
  congruent (Alternate Exterior Angle Theorem)
• In a plane, if a line is perpendicular to one of
  two lines, then it is perpendicular to the other
  (Perpendicular Transversal Theorem)

4
Parallel Lines and their Angle Relationships
Angles are supplementary
1
99
81
2
Angles are vertical
3
4
81
99
99
81
5
6
7
81
8
99
Corresponding Angles
Alternate Interior Angles are congruent, as are
Alternate exterior, as well as Corresponding
angles.
5
Find the values of x, y z
z
3x 5
6x 14
y 8
The angles 3x 5 6x 14 form a linear pair.
Thus, 3x 5 6x 14 180, 9x 9 180,
9x 189, x 21 Angles 3x 5 y 8 are
corresponding angles, therefore they are
congruent. 3(21) 5 y 8, 68 y 8, y
60 Angles of a triangle sum to 180, therefore x
90 68 180, z 22
6
1) Solve for each angle
p
q
47
133
9
8
47
7
10
31
2
6
47
133
102
3
78
1
102
4
5
78
In the figure above, lines p q are parallel,
and the measures of angle 1 is 78, and angle 2
is 47. Find the measure of each numbered angle.
2 7 are vertical. 2 6 form a linear pair
are supplementary. 8 7 are corresponding. 9 is
supplementary with 8. 3 is supplementary with 1.
4 5 are corresponding with 1 3 10 is the 3rd
angle in triangle with 102 47.
7
2) Find each angle measure
A
B
C
8
6
42
96
5
42
7
1
42
42
G
42
4
18
3
2
120
E
D
7 is the other half of a bisected angle. 6, 7,
8 form a straight line. 7 4 are alternate
interior. 1 4 are alternate interior. 5 is 3rd
angle of triangle. 2 is 3rd angle of triangle
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3) Find the value of x, y z
z
4x 8
2y 8
142
4x 8 142 4x 134 x 33.5 2y 8 142
180 2y 30 y 15 z 142
9
4) Find the value of x, y z
4z 2
x
3y - 11 y 19 2y 30, y 15 4z 2 34
90 4z 54 z 13.5 x 90
3y 11
Y 19
10
5) Find the value of x, y z
z
7x 9
11x 1 2y 5 180 11x 2y 176 7x 9 7y
4 180 7x 7y 175
7y 4
11x 1
2y 5
7(11x 2y 176) 77x 14y 1232 11(7x 7y
175) 77x 77y 1925 -63y -693 y 11 11x
2(11) 176 11x 154 x 14 Z 7(11) 4
73
11
6) Find the value of x, y
7y
8x 40
6x
3y 10
8x 40 6x 180 14x 140 x 10 3y - 10 7y
180 10y 190 y 19
12
Finding Slopes (m) of lines
Always read graph form Left to Right
If a line rises from left to right, the slope of
the line is positive. If it falls, then the slope
is negative.
Rise 6
Run 4
Slope (m) 6/4 3/2
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Finding Slopes (m) of lines
Always read graph form Left to Right
Rise 2
If a line rises from left to right, the slope of
the line is positive. If it falls, then the slope
is negative.
Run 4
Slope (m) 2/4 -1/2
14
Slopes (m) of Special lines
Always read graph form Left to Right
Vertical lines have no slope or undefined
Horizontal lines have slope 0
15
Find the slope of the line below
Slope is 1
16
Find the slope of the line below
Slope is -4/3
17
Find the slope of the line below
Slope is no slope
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Finding Slopes from points
When given two points we can use the slope
formula to find slope.
Example Find the slope of the line that passes
though points (3, 2) (4, 0).
Find the slope from the set of points (3, 9)
(4, 3) (4, 2) (4, 3) (7, 3) (3,
4) 12 No slope 7/10
19
Find the slope of the following lines
The line through the points (4, 7) (8,
3) -10/4 -5/2 Line through the points (-2, 4)
(3, -1) -5/5 -1 Line through the points (8,
7) (8, -7) Undefined or No slope
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Slope of Special Lines

• Parallel lines have equal slopes
• Perpendicular lies have slopes that opposite
  (change sign) and inverses (flip)
• Find the slope of line that is parallel to the
  line through the points (0,7) (4, 9)
• Slope of given line is 2/4 1/2, parallel 1/2
• Find the slope of line that is perpendicular to
  the line through the points (0,7) (4, 9)
• Slope of given line is 2/4 1/2, perpendicular
  -2

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Writing Equations of Lines
Point-slope equation y - y1 m(x x1) Standard
Linear form Ax By C Where A, B C are
integers (no fractions or decimals). And A 0
(coefficient of x is positive). Slope Intercept
Form y mx b (x,y) m slope b y-intercept
22
A line runs through points (3,2) (-7,5)
Find the point slope of the line above. Y 2
-3/10(x 3) or y 5 -3/10(x 7) Find the
Stand Form of the Line above. 3x 10y 29 Find
the slope intercept form 0f the line parallel to
line above and passing through point (8,1). Y
-3/10x 17/5 Find the Standard Form of the line
perpendicular to the line above ad contains the
point (-1,9)
23
Proving Lines Parallel

• Postulate 3-4 If two lines is a plane are cut by
  a transversal so that corresponding angles are
  congruent, the the lines are parallel.
• Parallel Postulate If there is a line and a
  point not on the line, then there exists exactly
  one line through the pint that is parallel to the
  given line.
• Theorem 3-5 If two lines are cut by a
  transversal so that a pair of alternate exterior
  angles are congruent, then the two lines are
  parallel.

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Proving Lines Parallel

• Theorem 3-6 If two lines in a plane are cut by a
  transversal so that a pair of consecutive
  interior angles is supplementary, then the lines
  are parallel.
• Theorem 3-7 If two lines in a plane are cut by a
  transversal so that a pair of alternate interior
  angles is congruent, then the lines are parallel.
• Theorem 3-8 In a plane, if two lines are
  perpendicular to the same line, then they are
  parallel.

25
Find the value of x and angle ABC so that p q
are parallel
A
5x 90
B
C
14x 9
E
q
F
p
The two angles given are corresponding. 14x 9
5x 90 9x 81 x 9 Angle ABC 5(9) 90 135
26
Given the following information, determine which
lines are parallel and why.
10
11
9
12
5
15
16
6
13
1
14
2
7
q
8
3
m
4
l
p
Alt. Int are congruent, l m are parallel
Alt. Ext are congruent, p q are parallel
Corresponding angle are congruent, l m are
parallel
27
Determine the value of x so that a b are
parallel
a
80
b
4x 10
The corresponding angle of 80 will be
supplementary with the 4x 10 angle. 4x 10
80 180 4x 90 x 22.5
28
Determine the value of x so that a b are
parallel
a
3x 50
b
2x 5
The corresponding angles will be supplementary
3x 50 2x 5 180 5x 235 x 47
29
Determine the value of x so that a b are
parallel
a
6x 12
b
2x
The angles will be supplementary. 6x 12 2x
180 8x 168 x 21
30
Determine the value of x so that a b are
parallel
a
57
b
3x 9
The corresponding angles each will be
supplementary . 3x 9 57 180 3x 132 x 43
31
Determine the value of x so that a b are
parallel
a
x2 10
b
4x 11
These are alternate interior angle. x2 10 4x
11 x2 4x 21 This factors (x 7)(x 3) X
7 -3, but -3 would not work.
32
Determine the value of x so that a b are
parallel
a
x2 9
b
These are corresponding angles. x2 9 90 x2
81 0 This factors (x 9)(x 9) X 9 -9,
both work
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Theorem 3-6 states that if two lines are cut by
a transversal so that consecutive interior angles
are supplementary, then the lines are parallel.
Write a proof of this theorem using the diagram
below. You cannot use the theorem as a reason.
a
1
2
b
3
Given that 1 2 are supplementary, prove that a
b are parallel.
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(No Transcript)
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Parallels and Distances

• The distance from a line and a point not on the
  line is the length of the perpendicular segment
  to the line from the point.
• The distance between two parallel lines is the
  distance between one of the lines and any point
  on the other line.

36
Show the distance between the segments
37
Draw Determine the distance between the given
point and line.
Use the Pythagorean Theorem
3
Or use the distance formula
5
m 5/3
38
Draw Determine the distance between the given
point and line.
39
Draw Determine the distance between the point
(5, 3) and the line y 4x.
40
Draw Determine the distance between the point
(5, 3) and the line x 2
41
Draw Determine the distance between the point
(2, 5) and the line 3x 4y 1
42
Draw Determine the distance between the lines y
2x 2 and the line y 2x 3