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Geometry Chapter 3 Notes
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Notes #12: Section 3.1 and Algebra Review
Transversal: a line that intersects two or more coplanar lines in different points
Draw:
Example:
Special Angles: The angles formed by lines and their transversals are special:
Alternate Interior Angles: (
)
Same-side Interior Angles: (
2
1
)
2
6
7
8
Vertical Angles: (reminder)
(
)
2
1
3
4
6
8
6
5
7
3
5
3
4
2
4
6
5
7
)
2
1
Same-side Exterior Angles:
(
)
1
Alternate Exterior Angles: (
3
8
7
8
Corresponding Angles: (
5
6
5
7
8
4
3
4
6
5
1
2
1
3
4
)
7
8
Practice: Classify each pair of angles as alt. int., s-s int., corr, alt. ext., s-s ext, or vertical.
1.) 1, 5
2.) 2, 8
3.) 4, 2
5.) 1, 7
4.) 3, 8
6.) 4, 7
1
4
3
2
5 6
8 7
-1-
Identifying Lines and Transversals: Name the two lines and the transversal that form each pair of
angles. What type of special angles are they? (Hint: Trace the angles in two different colors – where they
overlap is the transversal, the leftovers are the two lines.)
7.) 4, 2
lines: ____, _____
transversal: ______
type: _______________
C
B
1
2
8.) B, BAD lines: ____, _____
transversal: ______
type: _______________
4
5
3
A
9.) BAD, 5 lines: ____, _____
transversal: ______
type: _______________
D
10.) BCD, 5
E
lines: ____, _____
transversal: ______
type: _______________
Algebra Practice: Solving Linear Systems by Addition/Subtraction/Elimination
- Rearrange each equation so that the variable expressions are on the left side of the equals sign and the
constant is on the right side of the equal sign (called Standard Form)
- Multiply whole equations so that one variable expression is equal but has the opposite sign.
-Add the equations together; watch one variable cancel out
-Solve for BOTH variables
-Write your answer as a point ( x, y ) (in alphabetical order)
11.) 2x – 3y = 8
4x + 3y = -2
12.)
13.) 3x – 5y = -11
2x – 4y = -9
14.)
x = -4y - 3
3x – 2y = 5
2x + 3y – 10 = x + y - 14
x – 2y + 5 = 2x – y + 6
Solve for x and y:
15.)
70
x - 15
3x - 35
2x - y
-2-
Notes #13: Review of Special Pairs of Angles and Linear Systems
A. Special Pairs of Angles: Name the two lines and the transversal that form each pair of angles. What
type of special angles are they?
Y
Z
1.) 4, 3
lines: ____, _____
5
4
6
transversal: ______
type: _______________
2.) X , ZWV lines: ____, _____
transversal: ______
type: _______________
3
X
3.) 6, 2 lines: ____, _____
transversal: ______
type: _______________
4.) 4, 5
2
W
1
V
lines: ____, _____
transversal: ______
type: _______________
B. Linear Systems:
Substitution Method
 Choose one equation - get _____ ____________ alone
 Substitute (______________) this variable with the new expression in the second equation – USE
PARENTHESES!!
 Solve
 Substitute back in the first equation to solve for the other variable
 Express your answer as a point (____, ____) in alphabetical order
Solve using the substitution method:
5.) y = -3
6.)
b=3–a
2x + 3y = 3
7.) y = -x + 15
4x + 3y = 38
b=a–1
8.)
2m – 3n = 4
m + 4n = -9
-3-
Possible “strange” answers: solve using either substitution or elimination.
y  2 x  1
y  3x  4
9.)
10.)
y  2 x  3
12 x  4 y  16
11.) 2x – 8y = 6
x – 4y = 8
12.) y + 2x = 4
2x + 3y = 12
C. Linear Systems with fractions and decimals.
To clear decimals:
- multiply both sides of the equation by a multiple of 10; scoot the decimal over
To clear fractions:
- multiply both sides of the equation by the common denominator; cross cancel
13.)
1
1
 x  y  4
4
3
3x  y  30
14.)
2.4 x  0.3 y  3
4x  5 y  6
-4-
0.2m  0.5n  1.4
15.)
1
1
1
m n  
2
3
3
0.2m  1.2n  8.8
16.) 1
1
1
m n 
4
6
6
D. Word Problems in 2 variables
 Write a let statement (often x = 1st number, y = 2nd number)
 Translate ________ _______________ into an equation
 Translate ________ ______________ into an equation
 Solve the system of equations using the substitution method
Write let statements, translate to two equations, and solve using substitution:
17.) The sum of two numbers is 27. The
difference of these two numbers is 3. Find the
numbers.
18.) The sum of two numbers is 82. One number
is twelve more than the other. Find the numbers.
-5-
Notes #14: Sections 3.1 and 3.2
(A) Parallel Lines Theorems & the Converse Theorems
Alternate Interior Angles Theorem
Converse of Alternate Interior Angles Theorem
If two ________________ lines are cut by a
__________, then alternate interior angles are
_________________.
If two lines cut by a ________________ form
______________________, then the lines are
________________.
2
1
3
4
l
2
1
6
5
8 7
3
4
6
5
m
l
8 7
m
Corresponding Angles Postulate
Converse of the Corresponding Angles Postulate
If two ________________ lines are cut by a
__________, then corresponding angles are
_________________.
If two lines cut by a ________________ form
________________________, then the lines are
________________.
2
1
3
4
l
8 7
If two ________________ lines are cut by a
__________, then same-side interior angles are
_________________.
2
3
4
5
l
8 7
8 7
m
m
Converse of the Same-Side Interior Angles
Theorem
If two lines cut by a ________________ form
__________________________, then the lines are
________________.
2
1
3
4
6
l
6
5
m
Same-Side Interior Angles Theorem
3
4
6
5
1
2
1
5
l
6
8 7
m
-6-
Alternate Exterior Angles Theorem
If two ________________ lines are cut by a
__________, then alternate exterior angles are
_________________.
2
1
3
4
l
Converse of the Alternate Exterior Angles
Theorem
If two lines cut by a ________________ form
__________________________ then the lines are
________________.
8 7
3
4
6
5
2
1
m
6
5
8 7
Same-Side Exterior Angles Theorem
If two ________________ lines are cut by a
__________, then same-side exterior angles are
_________________.
2
1
3
4
5
l
m
Converse of the Same-Side Exterior Angles
Theorem
If two lines cut by a ________________ form
_________________________, then the lines are
________________.
2
1
3
4
6
8 7
l
m
5
l
6
8 7
m
Complete the sentences and solve for x.
1.) The labeled angles are ____________ angles
and their measures are ___________ because of the
_________________________________________
2.) The labeled angles are ____________ angles
and their measures are ______________________
because of the _____________________________
3x + 10
3x
120
100
-7-
3.) The labeled angles are ____________ angles
and their measures are ___________________
because of the _____________________________
4.) The labeled angles are ____________ angles
and their measures are ___________________
because of the _____________________________
2x + 21
7x - 4
2x + 52
4x - 8
B. Identifying Parallel Lines: Use the given information to name the lines that must be parallel.
(Trace angles and look for special pairs of angles and special relationships.)
5.) 1  4
Type of angle pair:
S
Relationship:
T
4
3
5
Parallel lines?:
2
6.) m1  m2  m3  180
Type of angle pair:
Relationship:
X
10
6
1
7
U
8
Parallel lines?:
9
7.) 9  2
Type of angle pair:
W
V
Relationship:
Parallel lines?:
8.) 4  7
Type of angle pair:
Relationship:
9.) 2  10
Type of angle pair:
Relationship:
Parallel lines?:
Parallel lines?:
C. Special Pairs of Angles
Solve for all variables. All measurements are in degrees. (Hint: extend the parallel lines and look for
special pairs of angles)
10.)
11.)
y
x
2y
120
120
80
4x
-8-
12.)
13.)
56
a
d
c
60
27
110
2x - 3y
2x - 5y
e
b
130
D. Proofs with Parallel Lines:
14.) Prove the alternate exterior angles theorem:
1
If a transversal intersects two parallel lines, then
alternate exterior angles are congruent.
2
3
Given: k l
Prove: 1  3
Statements
Reasons
1.)
1.)
2.)
2.) Corresponding Angle Postulate
3.)
3.) 3   ____
4.)
4.)
15.) Prove the converse of the alternate exterior
angles theorem:
1
2
If two lines and a transversal form alternate
exterior angles that are congruent, then the two
lines are parallel.
Given: 1  3
Prove: m n
3
Statements
Reasons
1.)
1.)
2.) 1  _____
2.)
3.)
3.) Substitution
4.)
4.) Converse of the _____________________
-9-
Notes #15: Sections 3.3 and 3.4
A. Parallel and Perpendicular Lines
Prove the converse of the same-side interior
angles theorem:
If two lines and a transversal form same-side
interior angles that are supplementary, then the
two lines are parallel.
1
2
3
Given: 2 and 3 are supplementary
Prove: m n
Statements
1.)
1.)
2.) m2  m3  _____
2.) Definition of __________________ angles
3.) m2  m ____  _____
3.)
4.)
4.) Substitution
5.)
m2  m2
Reasons
5.)
6.)
6.) Subtraction
7.)
7.) Converse of the ________________________
If two lines are parallel to the same line, then
they are ___________ to each other.
In a plane, if two lines are perpendicular to the
same line, then they are _____________ to each
other.
In a plane, if a line is perpendicular to one of two
parallel lines, then it is also _____________ to
the other.
- 10 -
Examples:
For #1-3, consider coplanar lines j, k, l, and m. Given each of the following statements, what more, if
anything, can you conclude about the lines?
1.) j k , k m
2.) j k , l  k
3.) j  k , k  l , l m
B. Classifying Triangles Triangles are described based on the lengths of their sides and the measures of
their angles
Names Based on Side Lengths
Scalene
Isosceles
Equilateral
Names Based on Angle Measures
Acute
Obtuse
Right
Equiangular
Examples: For #5-7, classify each triangle (drawn to scale) by its angles and sides.
5.)
6.)
7.)
For #8-10, draw a triangle, if possible, to fit each description.
8.) obtuse scalene
9.) acute isosceles
10.) right equilateral
- 11 -
11.) The perimeter of ∆ABC is 32m. AB = 4x – 2, BC = 3x + 1, AC = 2x + 6. Write an equation and
solve for x. Then, classify the triangle as scalene, isosceles, or equilateral.
(Hint: draw a picture first and label what you know)
C. The angles of a triangle:
** The sum of the interior angles of a triangle is always ______**
12.)
13.) Find the missing values and classify
2
M
4x + 11
1
3
2x + 8
m1  m2  m3  _____
L
x
N
LMN :
x  ____
mL  ____
mM  ____
mN  ____
LMN is __________ and __________
14.) Find the values of each variable and the measure of each angle. Then classify each triangle by its
angles. (All measurements shown are in degrees)
w  ____
A
B
w
x
ABC is _________
x  ____
61
ABD is _________
y  ____
134
z
DBC is _________
z  ____
D
y
C
- 12 -
Notes #16: Sections 3.4 and 5.5
A. Triangle Exterior Angle Theorem (Section 3.4)
** The measure of an exterior angle of a triangle equals the sum of the
measures of the two _________________ __________________ angles.**
Explore:
85
40
1.) Complete:
2
7
m1  m2  _____
1
m2  m7  _____
m7  m3 _____
3
6
m4  m3  m5  m6 _____
4
m2  m3  m7 _____
5
Diagram for #1
3.) m2  103 , m3  156 , m1  _____
2.) Solve for x and y:
95
2
y
50
1
x
3
B. Inequalities in Triangles:
The largest angle of a triangle is opposite the ________________ side of the triangle.
The smallest angle of a triangle is opposite the _______________ side of the triangle.
List the angles of the triangle in order from smallest to largest: (not necessarily drawn to scale)
4.)
5.)
B
a+1
X
11
Y
12
a
a-1
A
C
10
Z
- 13 -
List the sides of the triangle in order from shortest to longest: (not necessarily drawn to scale)
6.)
7.)
X
B
A
46
42
Y
63
27
C
Z
Name the longest side: (not necessarily drawn to scale)
8.)
9.)
B
S
48
50
A
55 T
R
72
42
C
52 65
60
U
Longest: ________
D
Longest: ________
The Triangle Inequality
The sum of the lengths of _____________________ of a triangle is
greater than the length of the _______________________.
When given three side lengths of a possible triangle, they can form a triangle if:
(short side) + (short side) > (long side)
Is it possible for a triangle to have sides with length: (Yes/No)
10.) 8, 7, 6
11.) 9, 9, 19
12.) 2, 3, 4
When given two side lengths of a triangle, the third side of the triangle must be between their
difference and their sum.
(length – length) < third side < (length + length)
The lengths of two sides of a triangle are given. The length of the third side must be greater than ____ but
less than ____.
13.) 3, 7
14.) 12, 19
15.) 3a, 4a + 2
- 14 -
Notes #17: Section 3.5
A. Polygons: ( _________ sided figures)
Convex Polygon
Concave Polygon
Explore:
Triangle
Quadrilateral
# of Sides = ______
# of Triangles =________
Sum of Interior Angles = ______
Sum of Exterior Angles = ______
# of Sides = ______
# of Triangles =________
Sum of Interior Angles = ______
Sum of Exterior Angles = ______
Pentagon
Hexagon
# of Sides = ______
# of Triangles =________
Sum of Interior Angles = ______
Sum of Exterior Angles = ______
# of Sides = ______
# of Triangles =________
Sum of Interior Angles = ______
Sum of Exterior Angles = ______
Other Common Polygons:
Polygon Name
# Sides
Octagon
Nonagon
Decagon
Dodecagon
18-gon
20-gon
n-gon
# Triangles
Sum of Int. Angles
Sum of Ext. Angles
- 15 -
Patterns for polygonal angle sums:
Sum of Interior Angles
Sum of Exterior Angles
each interior angle + each exterior angle = ___________
Find the sum of the measures of the interior angles of each polygon:
1.) 7-gon
2.) 16-gon
3.) 22-gon
Find the missing angle measures: (all measures shown are in degrees)
4.)
5.)
b
130
131
a
107
85
28
160
114
123
B. Regular Polygons (where n is the number of sides in the polygon)
Regular Polygons:
all sides ________________
all angles ________________
Find the sum of the exterior angles for each regular polygon & then find the measure of each
exterior angle in the regular polygon.
6.) Regular Pentagon
7.) Regular Hexagon
8.) Regular Octagon
9.) Regular Decagon
10.) Regular Dodecagon
12.) Regular 15-gon
- 16 -
Notes #18—Section 3.5 Continued
Regular Polygon Formulas:
Interior Angles
Sum of Interior Angles =
Exterior Angles
Sum of Exterior Angles =
Each Exterior Angle =
(each interior angle) + (each exterior angle) = ________
Examples:
1) Find all of the following values for a regular
dodecagon (12 sides):
Number of sides =
2.) Find the following information for a regular
octagon:
Number of sides =
Sum of exterior angles =
Sum of exterior angles =
Each exterior angle =
Each exterior angle =
Each interior angle =
Each interior angle =
Sum of interior angles =
Sum of interior angles =
Find the following information for the regular polygon:
3.)
4.)
Number of sides =
Number of sides =
Sum of exterior angles =
Sum of exterior angles =
Each exterior angle = 120
Each exterior angle = 72
Each interior angle =
Each interior angle =
Sum of interior angles =
Sum of interior angles =
5.)
Number of sides =
6.)
Number of sides =
Sum of exterior angles =
Sum of exterior angles =
Each exterior angle =
Each exterior angle =
Each interior angle = 144
Each interior angle =
Sum of interior angles =
Sum of interior angles = 2340
- 17 -
Complete the chart for the regular polygons:
7.
8.
# of sides (n)
6
9.
Sum of Exterior Angles
Each Exterior Angle
90˚
Each Interior Angle
2880˚
Sum of Interior Angles
10.)
Given: 1  3
Prove: m n
1
2
3
Statements
Reasons
1.)
1.)
2.)
2.)
3.)
3.)
4.)
4.)
- 18 -
Notes #19: Quiz Review
1.) Solve for x and y:
2.) a b, c d Solve for w, x, y, and z.
c
3x  5 y  1
d
a
4 x  2 y  36
y
x
112
b
27
z
w
w = ____, x = ____, y = ____, z = ____
3.) m n . Solve for x and y:
4.) The perimeter of ABC is 82ft. Solve for x
and y and classify the triangle based on its sides and
angles.
B
m
5y + 35 10x - 20
7y + 9
60
n
x- 3
A
x+ 6
3y - 1
5y - 23
C
3x - 11
x = ___, y = ____, classify: _________, _________
5.) Solve for x and y.
6.) Name the sides in order from smallest to largest:
B
4x + 13
72 65
A
2y + 15
5x
28
70
C
x + 17
80 45
x = _____, y = _____
D
7.) Complete for the regular decagon:
8.) Complete for the regular polygon:
Number of sides =
Number of sides =
Sum of exterior angles =
Sum of exterior angles =
Each exterior angle =
Each exterior angle =
Each interior angle =
Each interior angle = 120
Sum of interior angles =
Sum of interior angles =
- 19 -
Notes #20: Section 3.6
A. Slope:
Slope is used to describe the ________________ and _________________ of lines.
Sketch a line with:
a) positive slope
b) negative slope
c) zero slope
d) undefined slope
A line is shown. Use two marked points and count “rise over run” to find the slope of the line.
1.)
2.)
3.)
Slope =
Slope =
Slope =
Without using a graph and given two points:
 x1, y1  and  x2 , y2 
Slope = m =
0
0
n
y2  y1
x2  x1
n
 undefined
0
For #4-5, find the slope of the line passing through the two given points:
4.) (-4, 1), (3, 2)
5.) (6, -3) and (2, -1)
6.) A line with slope
7
passes through the
3
points (1, 2) and (-2, y). Find y.
- 20 -
B. Graphing Lines
There are many ways to graph a line. You need to know how to graph a line:
(i) given a point and a slope
(ii) by finding the y-intercept and the slope of the line
(i) Graphing lines using a point and a slope
Point P which lies on the line and the slope of the line are given. Sketch the line and find the coordinates
of two other points on the line
7.) P (-2, 1); slope =
4
5
1st point:
8.) P (0, -3); slope = -2
1st point:
2nd point:
2nd point:
9.) P (2, 0); slope = 
2
3
1st point:
2nd point:
- 21 -
(ii) Graphing Lines using the slope and y-intercept:
- Get y alone so the equation is in y = mx + b form (m = _________, b = _________)
- Graph b first. This point goes on the ____ axis.
- Use slope and count rise over run to the next point(s). When you have at least three points, then
connect the points to make a line.
- Label your graphed line with the original equation
Most common errors:
 Graphing b on the x-axis instead of the y-axis
 Graphing the slope in the wrong direction (e.g. forgetting a negative)
10.)
1
y   x5
2
(  I’m already in slope-intercept form!)
m = ___ ( graph me second! Watch the
negative!)
b = ___ ( graph me first! I go on the y-axis!)
11.) x – 2y =2
(  Get me in slope-intercept form first)
m = ______
b = ______
- 22 -
12.) x + 3y = -6
(  Get me in slope-intercept form first)
m = ______
b = ______
Special Cases: Graphing Horizontal and Vertical Lines
13.) x = 4
(This equation describes the line for which ALL
points have an x-coordinate of 4. There are no
restrictions on the value of y).
14.) y = -2
(This equation describes the line for which ALL
points have an y-coordinate of -2. There are no
restrictions on the value of x).
Use the pattern you found above to complete these sentences:
• Any line in the form x = _____ is a ________________ line because it intersects the ___ _________
• Any line in the form y = _____ is a ________________ line because it intersects the ___ _________
- 23 -
Use this pattern to graph these lines without a table of solutions.
15.) y = 3
16.) x = -2
17.) y = -4
- 24 -
Notes #21: Writing Linear Equations
A. Converting equations of lines:
Lines can be written in either Slope-Intercept form (y = mx + b) or Standard Form (Ax + By = C). You
need to know how to convert from one to the other.
Converting to Slope-Intercept Form
Converting to Standard Form
Goal: y = mx + b
(where m and b are integers or fractions)


Goal: Ax + By = C
(where A, B, and C are integers and
where A is positive)
Get y alone
Reduce all fractions



1.) Convert to slope-intercept form:
4x – 12y = 8
Get x and y terms on the left side and
the constant term on the right side of
the equation
Multiply ALL terms by the common
denominator to eliminate the fractions
If necessary, change ALL signs so
that the x term is positive
2.) Convert to standard form:
y
2
x 5
3
B. Writing linear equations given the slope and y-intercept
- Find the slope (m) and y-intercept (b) [If the given information is a graph, then you will have to count by
hand to find these values.]
- Fill in m and b so you have an equation of the line in y = mx + b form.
y = ________ x + ____________
(  Put m here!)
(  Put b here!)
3.) Find the equation of the line
with slope of 5 and y-intercept
of -2. Write in standard form.
4.) Find the equation of the
given line in slope-intercept
form.
5.) Write the equation of a line
that has the same slope
4
as y  x  3 and has a y-intercept
5
of 1. Write in standard form.
- 25 -
C. Writing linear equations given the slope and a point





plug slope = m into y = mx + b
name your point (x, y) and plug these values in for x and y
solve for b
plug m and b back into y = mx + b
convert to standard form, if necessary
** Remember to leave x and y as variables! **
6.) Find the equation of the line in slope-intercept
form with slope of -2 and going through (-1, 3) in.
7.) Find the equation of the line in standard form with
1
slope of and going through (6, -2).
3
8.) Find the equation of the line in slope-intercept
2
form with slope
and passing through the point
5
(-3, 7).
9.) Find the equation of the line in standard form with
3
slope
and going through (-8, 6).
4
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Notes 22—Writing Linear Equations Continued
Writing linear equations given two points
 find the slope
 pick one of your points to be x and y
 plug m, x, y into y = mx + b
 solve for b; plug m and b into y = mx + b
 convert to standard form, if necessary
** Remember to leave x and y as variables! **
1.) Find the equation of the line in slope-intercept
form going through (-3, 1) and (4, 8).
2.) Find the equation of the line in standard form with
x-intercept 3 and y-intercept -2.
3.) Find the equation of the line going through
(5, 2) and (-1, 3) in standard form.
4.) Find the equation of the line with x-intercept 5 and
y-intercept -4 in slope-intercept form.
5.) Find the equation of the line in standard form
going through (-8, -2) and (-6, 4).
6.) Find the equation of the line in standard form with
x-intercept -8 and y-intercept -2.
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7.) Find the equation of the line in slope-intercept
1
form with slope and passing through the point
4
(-12, 3).
8.) Find the equation of the line in standard form with
8
slope  and passing through the point (27, 5)
9
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Notes #23: Section 3.7
A. Review - Writing Linear Equations:
1.) Find the equation of the line with
x-intercept -1 and y-intercept 2 in standard
form.
2.) Find the equation of the line going through
(4, 3) with x-intercept 6 in standard form.
B. Parallel and Perpendicular Lines
For #3-4, a pair of parallel lines and a pair of perpendicular lines are graphed below. Use the
graphs to find the slope of each of the four lines and to complete the sentences.
3.)
Slope of l1 :
Slope of l2 :
Parallel lines have _________ slopes.
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4.)
Slope of l3 :
Slope of l4 :
Perpendicular lines have
____________, _____________ slopes.
The slope of a line is given. Find the slope of a line parallel to it and the slope of a line
perpendicular to it:
2
5.) m = 
6.) m = 7
7.) m = 0
3
Parallel:
Parallel:
Parallel:
Perp.:
Perp.:
Perp.:
Are the lines with these slopes parallel, perpendicular, or neither?
2, 4
3 6
8.)
9.) -4, 4
10.) -1, 1
11.) Find the slope of a line parallel and perpendicular to AB where A(-3, 1) and B (2, 4)
Parallel:_______
Perp:______
For #12-14, state whether the given pair of lines is parallel, perpendicular, or neither:
12.)
3
y   x 1
4
8 x  6 y  12
13.)
1
y   x5
2
2x  4 y  9
14.)
y  5 x  4
x  5 y  4
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C. Writing linear equations given a point and another line (parallel or perpendicular to your line)
 find m from the given line
 if the line is parallel, this is your m;
if the line is perpendicular, find its ______________ _______________
 plug m, x, y into y = mx + b
 solve for b; plug m and b into y = mx + b
 convert to standard form, if necessary
** Remember to leave x and y as variables! **
15.) Find the equation of the line going through
16.) Find the equation of the line going through
(1, 2) and parallel to y = 3x + 4 in slope-intercept (3, -2) and perpendicular to x – 4y = 3 in standard
form.
form.
17.) Find the equation of the line going through
(-1, 5) and perpendicular to y = 3x + 4 in slopeintercept form.
18.) Find the equation of the line going through
(9, -3) and parallel to 2x – 3y = 3 in standard
form.
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Notes #24: Review
You can now solve linear systems (a set of 2 lines) using algebra (substitution/elimination) AND using
coordinate Geometry (graphing). You should get the same answer for both methods.
- solve the equations using substitution or elimination; write your answer as a point ( , )
- graph the two lines using either the intercept method OR slope-intercept method
- confirm that the two lines intersect (meet) at your solution point
1.)
y=x–2
x+y=4
2nd Method: graphing (graph both lines on the
coordinate plane below)
1st Method: substitution or elimination
solution: (
2.)
,
)
2x – y = -3
x + 2y = 6
2nd Method: graphing (graph both lines on the
coordinate plane below)
1st Method: substitution or elimination
solution: (
,
)
- 32 -
Chapter 3 Study Guide
For #1-4, identify whether the angles are vertical angles, same side interior angles, corresponding
angles, alternate interior angles, same-side exterior angles, or alternate exterior angles.
1.) 2 and 6
2.) 1 and 6
3
7
4
3.) 5 and 2
2
4.) 4 and 7
5
1
6
For #5-6, find the slope of the line passing through the two points
5.) (-3, 2) and (4, 1)
6.) (-9, 2) and (2, 2)
7.) The slope of line l is
given. Find the slope of the
line parallel to it and the slope
of the line perpendicular to it:
a) -2
b) 3/2
Parallel:____
Parallel:___
Perp.: _____
Perp.:_____
For #8-10, name the two lines and transversal that form each pair of angles:
1
2
9.) BAD, CDA
lines: ____, ____ trans: ______
10.) BAD, 5
lines: ____, ____ trans: ______
C
B
8.) 1, 3
lines: ____, ____ trans: ______
4
5
3
A
E
D
In the diagrams, the lines shown are parallel. Write an equation and solve for x and y. Justify your
work with a postulate or theorem.
12.)
11.)
2y + 20
2x+40
2x + 10
5y+20
x+80
3y + 65
Find the values of x and y.
13.)
40
14.)
5y
y
20
15x-20
x
5x
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Define your variables, write an equation, and solve:
15.) The sum of two numbers is 10. The difference of the first number and twice the second number is 1.
Find the numbers.
In each exercise, some information is given. Use this information to name the segments that must
be parallel. If there are no such segments, write none.
A
B
16.) 3  10
17.) 7  10
2
1
18.) 2  3
3
10
F
9
11 8
19.) 9  5
4 C
5
6
7
D
E
Solve for x and y:
20.)
x+ y
130
120
Solve for x and y:
1
1
x  y  1
2
4
21.)
2
1
x  y  8
3
2
6x - 2y
22.) Prove the converse of the alt ext angles theorem:
If two lines and a transversal form alternate exterior
angles that are congruent, then the two lines are parallel.
Given: 1  3
Prove: m n
Statements
1.)
1.)
2.)
2.)
3.)
3.)
4.)
4.)
1
2
3
Reasons
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23.) Given: l m
4
Prove: m2  m4  180
3
l
2
m
1
Statements
Reasons
1.)
1.)
2.)
2.)
3.)
3.)
4.)
4.)
24.) Given: AB = CD
AE = FD
A
Prove: EB = CF
E
B
C
F
Statements
Reasons
1.)
1.)
2.)
2.)
3.)
3.)
4.)
AE = FD
D
4.)
5.)
5.)
For #25-26, classify the triangles based on their sides and their angles:
25.)
26.)
8x - 20
3x + 5
2x
60
60
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For #27-29, use the diagram for reference. Show all equations and work.
27.)
If m6  42 and m8  61,
then m10  ____
11
8
6
7
10
9
If m6  7 x, m7  2 x  5,
28.) and m11  6 x + 35, then
x = ___.
If m8  7 x  2, m7  4 x  7,
29.) and m9  10 x + 3, then
x = ___.
For #30-31, a, b, c, and d are distinct coplanar lines. How are a and d related?
30.) a b, b c, c  d
31.) a  b, b  c, c d
For #32-33, find the measure of each interior angle and each exterior angle of each regular polygon.
32.) octagon
33.) pentagon
Complete the table for regular polygons
34.)
Number of Sides
Sum of exterior angles
Measure of each exterior angle
Measure of each interior angle
Sum of interior angles
6
20
162
- 36 -
Graphing Linear Equations:
35.) Graph each line using the slope and y-intercept:
3
a) y =  x + 1
b) 2x + y = 4
c) 3x – 2y = 8
2
36. Find the slope and y-intercept of each line:
a) y = 3x – 5
b) y = 3
c) 3x – 2y = 4
37. Find the intersection of the two lines using
the substitution or elimination method.
x + 2y = 8 and 2x + 3y = 10
38. Explain why these two lines will not intersect
y = 2x – 1 and 8x – 4y = 16
Writing Linear Equations: Write an equation of the line with:
39. y-intercept -2 and slope -4 in standard form
40. x-intercept 4 and y-intercept -2 in slopeintercept form
- 37 -
42. through (6, 2) and parallel to x – 2y = 5 in
standard form
41. through (1, -2) with slope -3 in slopeintercept form
43. through (2, -1) and perpendicular to
x + 3y = 7 in standard form
44. through (3, 2) and (4, 7) in slope-intercept
form
45. through (3, -2) and (7, -2) in slope-intercept
form
46. through (4, -3) and with x-intercept -2 in
standard form
47. x-intercept -3 and y-intercept 5 in standard
form
49. through (3, -3) and perpendicular to
2x – y = 1 in slope-intercept form
For #50-52, are the given lines parallel, perpendicular, or neither?
1
y  x2
50.)
3
2 x  6 y  10
51.)
1
y  x2
3
2 x  6 y  10
1
y  x2
52.)
3
6 x  2 y  10
- 38 -
For questions 53-54, can a triangle have sides with the given lengths? Write yes or no and show
work to support your answer.
53. 4m, 7m, and 8m
54. 18ft, 20ft, and 40ft.
55. The lengths of two sides of a triangle are 6 km and 8 km. Describe the lengths possible for the third
side.
For questions 56-57, list the sides in order from shortest to longest.
56.
57.
B
B
C
60 61
47
A 65
59 C
55 60
41
D
A
For questions 58-59, list the angles from smallest to greatest.
58. In ∆ABC, AB = 11, BC = 12, and AC = 10.
59.
R
b
Q
b +1
b -1
P
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