CHAPTER 3
Common Core State Standards
G.CO.1 Know precise definitions of angle, circle,
perpendicular line, parallel line, and line segment based on the
underfined notions of point, line, distance along a line, and
distance around a circular arc.
Student Learning Targets
1. Students will be able to identify the relationships between
two lines or two planes.
2. Students will be able to name angle pairs formed by parallel
lines and transversals.
Section 3.1 Notes: Parallel Lines and Transversals
Vocabulary
Parallel Lines
Definition
Perpendicular lines are coplanar lines that
______________________________.
Denoted by: Arrows are used to indicate parallel lines
Skew Lines
Parallel Planes
Example 1: Use the figure to the right.
a) Name all segments parallel to BC .
b) Name a segment skew to EH .
c) Name a plane parallel to plane ABG.
Skew lines are lines that
_______________________________.
Parallel planes are planes that
_________________________________.
Picture
Vocabulary
Transversal
Definition
Picture
A line that ____________ two or more lines at two
different points.
Transversal Angle Pair Relationships
Interior Angles
Exterior Angles
Consecutive Interior Angles
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Example 2: Classify the relationship between the given angles as alternate interior, alternate exterior, corresponding, or consecutive
interior angles.
a) 2 and 6
b) 1 and 7
c) 3 and 8
d) 3 and 5
Example 3: Classify the relationship between the given angles as alternate interior, alternate exterior, corresponding, or consecutive
interior angles.
a) 4 and 5
b) 7 and 9
c) 4 and 7
d) 2 and 11
Example 4: The driveways at a bus station are shown. Identify the transversal connecting the given angles. Then classify the
relationship between the pair of angles.
a) 1 and 2
b) 2 and 3
c) 4 and 5
Summary
2
1
5
8
3
4
6
7
13
14
16 15
9 10
12 11
Refer to the above figure and identify the special angle pair name.
1) 3 and 13 ____________________________________________________
2) 8 and 10 ____________________________________________________
3) 11 and 15 ___________________________________________________
4) 8 and 6 _____________________________________________________
5) 1 and 6 _____________________________________________________
6) 6 and 10 ____________________________________________________
7) 14 and 15 ___________________________________________________
8)
m1 = 3x - 17
m2 = x + 1
3
x = ________
9)
1
4
m3 = 20k + 11
m4 = 8k + 1
k = _________
10)
m1 = 95 + 7h
m2 = 55 - h
3 1
2
h = ________
11)
m3 = 5k + 12
m4 = 7k - 16
4
k = _________
12)
m1 = 7y + 16
m2 = 2x
m3 = 4x – 30
1
2
x = ______
3
y = ______
1
5
9
13
2
6
10
14
3 4
7 8
11
15
12
16
Refer to the above figure and identify the special angle pair name.
13) 7 and 2 ______________________________________________________
14) 6 and 14 _____________________________________________________
15) 13 and 12 ____________________________________________________
16) 7 and 11 _____________________________________________________
17) 4 and 8 ______________________________________________________
2
Mastery connect – section 3.1
1. Name the plane parallel to plane AEF.
a. plane EFH
b. plane CHF
c. plane DBA
d. plane DGH
̅̅̅̅
c. 𝐻𝐶
̅̅̅̅
d. 𝐸𝐺
̅̅̅̅ ?
2. Which segment is skew to 𝐺𝐻
̅̅̅̅
a. 𝐸𝐹
̅̅̅̅
b. 𝐴𝐷
3. Name a pair of consecutive interior angles.
a. ∠ 6 and ∠ 7
b. ∠ 4 and ∠ 6
c. ∠ 10 and ∠ 8
d. ∠ 10 and ∠ 9
4.∠8 and ∠13 are ___________________________.
a. corresponding angles
b. alternate exterior angles
c. alternate interior angles d. consecutive interior angles
5. Which statement is true about skew lines?
a. They do not intersect.
b. They intersect
c. They are perpendicular d. They lie in the same plane.
6. How many segments intersect ̅̅̅̅̅
𝐴𝑀?
a. 5
b. 3
c. 4
d. 6
7. Which of the following best describes the front and back covers of a notebook when close?
a. intersecting planes
b. a single plane
c. skew planes
d. parallel planes
Section 3.1 Worksheet
Name: _____________________________________
For numbers 1 – 4, refer to the figure at the right to identify each of the following.
1. all planes that intersect plane STX
3. all segments that are parallel to XY .
2. all segments that intersect QU
4. all segments that are skew to VW .
For numbers 5 – 10, classify the relationship between each pair of angles as alternate interior, alternate exterior, corresponding, or
consecutive interior angles.
5. 2 and 10
6. 7 and 13
7. 9 and 13
8. 6 and 16
9. 3 and 10
10. 8 and 14
For numbers 11 – 14, name the transversal that forms each pair of angles. Then identify the special name for the angle pair.
11. 2 and 12
12.  and 18
13. 13 and 19
14. 11 and 7
15. FIGHTERS Two fighter aircraft fly at the same speed and in the same direction leaving a
trail behind them. Describe the relationship between these two trails.
15. ESCALATORS An escalator at a shopping mall runs up several levels. The escalator
railing can be modeled by a straight line running past horizontal lines that represent the
floors. Describe the relationships of these lines.
16. NEIGHBORHOODS John, Georgia, and Phillip live nearby each other as shown in the
map. Describe how their corner angles relate to each other in terms of alternate interior,
alternate exterior, corresponding, consecutive interior, or vertical angles.
17. MAPPING Use the figure to the right.
a) Connor lives at the angle that forms an alternate interior angle with Georgia’s residence.
Add Connor to the map.
b) Quincy lives at the angle that forms a consecutive interior angle with Connors’ residence.
Add Quincy to the map.
Common Core State Standards
G.CO.1 Know precise definitions of angle, circle,
perpendicular line, parallel line, and line segment based on the
undefined notions of point, line, distance along a line, and
distance around a circular arc.
Student Learning Targets
1. Students will be able to use theorems to determine the
relationships between specific pairs of angles.
2. Students will be able to use algebra to find angle
measurements.
G.CO.9 Prove theorems about lines and angles.
Section 3.2 Notes: Angles and Parallel Lines
Vocabulary
Corresponding Angle Postulate
Definition
Picture
If two ______________________ are cut by a
transversal, then each ___________ of
corresponding angles is ________________.
Example 1: Use the diagram below to find the missing angle measure. Tell which postulates (or theorems) you used.
a) If m11 = 51, find m15.
because….
m11 = m________ because….
m15 = _________ because…..
b) If m11 = 51, find m16.
Vocabulary
Definition
Alternate Interior Angles Theorem
If _______________________________ are cut by a
transversal, then each pair of alternate interior angles
is _____________.
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal, then each
pair of ___________________________________ is
_______________________.
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then each
pair of
_______________________________________ is
______________________.
Picture
Example 2: The diagram represents the floor tiles in Michelle’s house.
a) If m2 = 125, find m3.
b) If m2 = 125, find m.
Example 3: In the figure below,
t
and t is a transversal. If
g
, find the measure of the other angles.
m1 _________ m2 _________
m3 _________ m5 _________
1
8
h
27
m6 _________ m7 _________
m8 _________
6
3
5
4
Example 3: Use the diagram below to determine the value of the variable.
a) m5 = (2x – 10)° and m7 = (x + 15)°
b) m4 = (4(y – 25))° and m8 = (4y)°
Vocabulary
Perpendicular Transversal Theorem
Definition
If a line is perpendicular to one of two
____________________________, then it is
perpendicular to the other.
Denoted by:
Summary
1
5
9
13
10
14
2
6
3 4
7 8
11
15
12
16
Let m1 = 115 and m12 = 110
1. m9 = ______
2. m4 = ______
3. m10 = ______
4. m11 = ______
5. m8 = ______
6. m5 = ______
7. m3 = ______
8. m14 = ______
Picture
Mastery connect – section 3.2
̅̅̅̅ ∥ ̅̅̅̅
1. If m ∥ n, 𝐴𝐶
𝐵𝐷 , and m ∠3 = 108°, what is the measure ∠7?
a. 72°
b. 180°
c. 108°
d. 76°
For questions 2 – 4, use the picture below.
2. Find the value of x.
a. 126
b. 54
c. 110
d. 145
b. 28
c. 24
d. 30
c. 8
d. 6
3. What is the value of y?
a. 26
4. Determine the value of z.
a. 20
b. 48
5. Assuming two parallel lines are cut by a transversal, which types of angles are not necessarily congruent?
a. consecutive interior angles
b. alternate interior angles
c. corresponding angles
d. vertical angles
6. Name the postulate or theorem that concludes ∠1 ≅ ∠2.
a. Corresponding Angles Postulate
b. Alternate Interior Angles Theorem
c. Alternate Exterior Angles Theorem
d. Consecutive Interior Angles Theorem
7. Find the measure of ∠2 if g ∥ h, ̅̅̅̅̅
𝑊𝑌 ∥ ̅̅̅̅
𝑋𝑍 , and m∠5 = 70°.
a. 110°
b. 65°
c. 70°
d. 60°
Section 3.2 Worksheet
Name: _____________________________________
For numbers 1 – 6, use the figure with m2 = 92 and m12 = 74. Find the measure of each angle. Tell which postulate(s) or
theorem(s) you used.
1. m10 =
2. m8 =
3. m9 =
4. m5 =
5. m11 =
6. m13 =
For numbers 7 and 8, find the value of the variable(s) in each figure. Explain your reasoning.
7.
8.
For numbers 9 and 10, solve for x. (Hint: Draw an auxiliary line.)
9.
10.
11. PROOF Write a paragraph proof of Theorem 3.3.
Given: ℓ || m, m || n
Prove: 1 ≅ 12
12. FENCING A diagonal brace strengthens the wire fence and prevents it from sagging.
The brace makes a 50° angle with the wire as shown. Find the value of the variable.
13. CITY ENGINEERING Seventh Avenue runs perpendicular to both 1st and 2nd Streets,
which are parallel. However, Maple Avenue makes a 115° angle with 2nd Street. What is
the measure of angle 1?
Common Core State Standards
G.GPE.5 Prove the slope criteria for parallel and
perpendicular lines and use them to solve geometric problems.
Student Learning Targets
1. Students will be able to find slopes of lines.
2. Students will be able to use slope to identify parallel and
perpendicular lines.
Section 3.3 Notes: Slopes of Lines
Vocabulary
Slope
Definition
The slope of a line is the _____________________
along the y-axis to the change along the x-axis
between ___________________________________
_____________________________.
Catchy Saying :
Formula :
Example 1: Find the slope of the given lines.
a)
b)
c)
d)
Picture
Positive Slope
Classifying Slopes
Negative Slope
Zero Slope
Undefined Slope
Example 2: Find the slope of the lines that contain the given points:
a) (–3, 4), (2, 1)
b) (–1, –3), (6, –3)
c) (2, –4), (5, 2)
d) (3, 5), (3, 2)
Vocabulary
Rate of Change
Definition
Can also be referred to as: ______________
Real-Life Example
Describes the change in x in relation to the change in
y.
Example 3: In 2000, the annual sales for one manufacturer of camping equipment were $48.9 million. In 2005, the annual sales were
$85.9 million. If sales increase at the same rate, what will be the total sales in 2015?
Example 4: Between 1994 and 2000, the number of cellular telephone subscribers increased by an average rate of 14.2 million per
year. In 2000, the total subscribers were 109.5 million. If the number of subscribers increases at the same rate, how many subscribers
will there be in 2010?
Vocabulary
Definition
Slopes of Parallel Lines
Parallel lines have ____________________________
_______________.
Slopes of Perpendicular Lines
Picture
Perpendicular lines have slopes that are
_________________________________________.
Example 5: Determine whether FG and HJ are parallel,
perpendicular, or neither for F(1, –3), G(–2, –1), H(5, 0), and
J(6, 3). Graph each line to verify your answer.
Example 6: Determine whether AB and CD are parallel,
perpendicular, or neither for A(–2, –1), B(4, 5), C(6, 1), and
D(9, –2).
Example 7: Graph the line that contains Q(5, 1) and is parallel to MN with M(–2, 4) and N(2, 1).
Example 8: Determine which graph represents the line that contains R(2, –1) and is parallel to OP with O(1, 6) and P(–3, 1).
a)
b)
c)
d) None of these
Summary
Given the following lines find their slopes.
Line 1
Line 3
Line 2
Line 4
Line 1________________
Line 3_______________
Line 2________________
Line 4_______________
Line 5
Line 7
Line 6
Line 8
Line 5________________
Line 7__________________
Line 6________________
Line 8__________________
Mastery connect – section 3.3
1. Find the slope of the line.
a. −
3
4
c. 4
b.−
4
3
d. 0
2. Determine the slope of the line parallel to the line shown.
a.
1
4
c. 0
b. -1
d. 1
3. Which graph shows a line with an undefined slope?
a.
b.
c.
d.
4. Determine the slope of any line perpendicular to ̅̅̅̅
𝐺𝐻 .
a.
1
3
c. – 3
b. 3
d. −
1
3
1
5. Graph the line that is perpendicular to the graph of the equation 𝑦 = − 𝑥 − 3 that passes through the point (5, 0).
2
a.
b.
c.
d.
Section 3.3 Worksheet
Name: _____________________________________
For numbers 1 and 2, determine the slope of the line that contains the given points.
1. B(–4, 4), R(0, 2)
2. I(–2, –9), P(2, 4)
For numbers 3 – 6, find the slope of each line.
3. LM
4. GR
5. a line parallel to GR
6. a line perpendicular to PS
For numbers 7 – 10, determine whether KM and ST are parallel, perpendicular, or neither.
7. K(–1, –8), M(1, 6), S(–2, –6), T(2, 10)
8. K(–5, –2), M(5, 4), S(–3, 6), T(3, –4)
9. K(–4, 10), M(2, –8), S(1, 2), T(4, –7)
10. K(–3, –7), M(3, –3), S(0, 4), T(6, –5)
For numbers 11 – 14, graph the line that satisfies each condition.
11. slope = 
1
, contains U(2, –2)
2
12. slope =
4
, contains P(–3, –3)
3
13. Contains B(–4, 2), parallel to FG
with F(0, –3) and G(4, –2)
14. Contains Z(–3, 0), perpendicular to EK with E(–2, 4) and K(2, –2)
15. CITY BLOCKS The figure shows a map of part of a city consisting of two pairs
of parallel roads. If a coordinate grid is applied to this map, Ford Street would have a
slope of –3.
a) The intersection of B Street and Ford Street is 150 yards east of the intersection
of Ford Street and Clover Street. How many yards south is it?
b) What is the slope of 6th Street? Explain.
c) What are the slopes of Clover and B Streets? Explain.
d) The intersection of B Street and 6th Street is 600 yards east of the intersection of B Street and Ford Street. How many yards north is
it?
Common Core State Standards
G.GPE.5 Prove the slope criteria for parallel and
perpendicular lines and use them to solve geometric problems.
Student Learning Targets
1. Students will be able to write an equation of a line given
information about the graph.
2. Students will be able to solve problems by writing
equations.
3.4 Notes: Equations of Lines
Vocabulary
Definition
Equation
m:__________________________
Slope-Intercept Form
b:___________________________
(x1,y1):_______________________
Point-Slope Form
m:__________________________
Example 1:
a) Write an equation in slope-intercept form of the line with slope of 6 and
y-intercept of –3. Then graph the line.
b) Write an equation in slope-intercept form of the line with slope of –1 and y-intercept of 4.
Example 2:
10
3
a) Write an equation in point-slope form of the line whose slope is  that
5
contains (–10, 8). Then graph the line.
8
6
4
2
-10 -8
-6
-4
-2
2
-2
-4
1
b) Write an equation in point-slope form of the line whose slope is that contains (6, –3).
3
-6
-8
-10
4
6
8
10
Example 3:
a) Write an equation in slope-intercept form for a line
containing (4, 9) and (–2, 0).
b) Write an equation in slope-intercept form for a line
containing (–3, –7) and (–1, 3).
Example 4: Write an equation of the line through (5, –2) and (0, –2) in slope-intercept form.
Vocabulary
Definition
Picture
The equation of a horizontal line is:
Horizontal Line
b is the:
The equation of a vertical line is:
Vertical Line
a is the:
Example 5:
a) Write an equation in slope-intercept form for a line
1
perpendicular to the line y  x  2 through (2, 0).
5
b) Write an equation in slope-intercept form for a line
1
perpendicular to the line y  x  2 through (0, 8).
3
Example 6: An apartment complex charges $525 per month plus a $750 annual maintenance fee.
a) Write an equation to represent the total first year’s cost, A,
for r months of rent.
b) Compare this rental cost to a complex which charges a $200
annual maintenance fee but $600 per month to rent. If a person
expects to stay in an apartment for one year, which complex
offers the better rate?
Example 7: A car rental company charges $25 per day plus a $100 deposit.
a) Write an equation to represent to the total cost, C, for d days
of use.
b) Compare this rental cost to a company which charges a $50
deposit but $35 per day for use. If a person expects to rent a
car for 9 days, which company offers the better rate?
Summary
Write the equation in slope intercept form of the line parallel and line perpendicular to given line through given point.
Parallel
Perpendicular
1) y = 4x + 7
(─2, ─9)
____________________
2) 2x ─ 5y = 10
(3, ─7)
_____________________
3) 3x + 4y = 16
(12, ─5)
__________________
__________________
_____________________ ___________________
Write the equation of the perpendicular bisector of AB .
4) A (3, ─6) B (7, 2)
5) A (2, 5)
B (6, ─7)
_________________________
______________________________
6) A (8, 1) B (0, ─1)
7) A (7, 9) B (1, 5)
_________________________
_______________________________
State if the lines are Parallel, Perpendicular, or Oblique
8) 6x ─ 12y = 24
4x + 2y = 8
9) 4x + y = 5
3x +12y = ─6
10) ─2x + 7y = 14
4x = 14y
Mastery connect – section 3.4
1. Which is the equation of a line parallel to the line with equation 𝑦 = 3𝑥 − 4?
a. 𝑥 + 3𝑦 = 4
b.3𝑥 − 𝑦 = −2
c. 3𝑥 + 𝑦 = 7
d. 𝑥 − 3𝑦 = 1
2. What is the equation of a line perpendicular to the line with equation
2𝑥 − 5𝑦 = 5?
a. 2𝑥 + 5𝑦 = −3
b. 2𝑥 − 5𝑦 = 11
c. 2𝑥 − 𝑦 = −1
d. 10𝑥 + 4𝑦 = 3
3. Which is an equation of the line in the graph shown?
a. 𝑦 = 2𝑥 + 3
b. 𝑦 = −2𝑥 − 3
c. 𝑦 = −2𝑥 + 3
d. 𝑦 = 2𝑥 − 3
4. Which is an equation of the line with slope 0 and passing through the point at (–2, 3)?
b. y = –2
a. y = 0
c. y = 3
d. y = –2x + 3
3
5. Write the equation of the line perpendicular to 𝑦 = 𝑥 + 4 that passes through the point (3, -13).
5
3
3
4
5
5
5
5
5
3
3
a. 𝑦 = − 𝑥 + 8 b. 𝑦 = 𝑥 − 14 c. 𝑦 = − 𝑥 − 8 d. 𝑦 = − 𝑥 + 12
1
6. Which is an equation of the line with slope and passing through the point at (1, 2)?
2
1
3
2
2
a. 𝑦 = 𝑥 +
b. 𝑦 = 2𝑥 + 1
1
5
2
2
c. 𝑦 = − 𝑥 +
1
d. 𝑦 = 𝑥 + 2
2
Section 3.4 Worksheet
Name: _____________________________________
For numbers 1 – 3, write an equation in slope-intercept form of the line having the given slope and y-intercept or given points.
Then graph the line.
1. m:
2
, b: –10
3
2. m: 
7 
1
,  0,  
9 
2
3. m: 4.5, (0, 0.25)
For numbers 4 – 7, write equations in point-slope form of the line having the given slope that contains the given point.
Then graph the line.
4. m:
3
, (4, 6)
2
5. m: 
6
, (–5, –2)
5
6. m: 0.5, (7, –3)
For numbers 8 – 17, write an equation in slope-intercept form for each line shown or described.
8. b
10. parallel to line b, contains (3, –2)
9. c
7. m: –1.3, (–4, 4)
11. perpendicular to line c, contains (–2, –4)
12. m = 
4
,b=2
9
13. m = 3, contains (2, –3)
14. x-intercept is –6, y-intercept is 2
15. x-intercept is 2, y-intercept is –5
16. passes through (2, –4) and (5, 8)
17. contains (–4, 2) and (8, –1)
Common Core State Standards
G.CO.9 Prove theorems about lines and angles.
G.CO.12 Make formal geometric constructions with a variety
of tools and methods.
Student Learning Targets
1. Students will be able to recognize angle pairs that occur
with parallel lines.
2. Prove that two lines are parallel.
Section 3.5 Notes: Proving Lines Parallel
Vocabulary
Converse
Definition
Picture
Real-Life Example :
Switching the hypothesis and the conclusion of a
statement to make another statement, called the
converse.
Usually “If, then” statements.
Converse of Corresponding Angles
Postulate
If two lines are cut by a transversal so that
__________________________________________
are congruent, then the lines are
_______________________.
Parallel Postulate
Given a line and a point _______________________,
there exists _____________________ through the
point that is parallel to the given line.
Vocabulary
Definition
Alternate Exterior Angles Converse
Consecutive Interior Angles Converse
Alternate Interior Angles Converse
Perpendicular Transversal Theorem
If two lines are cut by a transversal so that a pair of
______________________________________
_______________________, then the two lines are
_______________________.
If two lines are cut by a transversal so that a pair of
__________________________________________
_____________________, then the lines are
_____________________.
If two lines are cut by a transversal so that a pair of
__________________________________________
_____________________, then the lines are
_____________________.
If two lines are _____________________ to the same
line, then they are _____________________.
Picture
Example 1: Use the diagram to the right.
a) Given 1  3, is it possible to prove that any of the lines
shown are parallel? If so, state the postulate or theorem that
justifies your answer.
b) Given m1 = 103 and m4 = 100, is it possible to prove
that any of the lines shown are parallel? If so, state the
postulate or theorem that justifies your answer.
Example 2: Given 1  5, is it possible to prove that any of the lines shown are parallel?
Example 3:
a) Find mZYN so that PQ || MN . Show your work.
b) Find x so that GH || RS
The angle pair relationship formed by a transversal can be used to prove that two lines are parallel.
Example 4: In the window shown, the diamond grid pattern is constructed by hand.
Is it possible to ensure that the wood pieces that run the same direction are parallel?
If so, explain how. If not, explain why not.
Example 5: In the game Tic-Tac-Toe, four lines intersect to form a square with four right angles in the middle of the grid. Is it
possible to prove any of the lines parallel or perpendicular? Choose the best answer.
a) The two horizontal lines
are parallel.
b) The two vertical lines
are parallel.
c) The vertical lines are
perpendicular to the
horizontal lines.
d) All of these statements
are true.
Summary
USE THE DIAGRAM.
1
2
4
3
a
5
6
7
8
a || b
c || d
9
13 14
c
10
11
15
12
b
16
d
SUPPOSE m2 = 71. FILL IN THE REMAINING BLANKS.
1. m1 = _______ BECAUSE __linear pair_________________ WITH  ______
2. m3 = _______ BECAUSE ____________________________ WITH  ______
3. m4 = _______ BECAUSE __corresponding______________ WITH  ______
4. m5 = _______ BECAUSE ____________________________ WITH  ______
5. m6 = _______ BECAUSE _alternate-interior_____________ WITH  ______
6. m7 = _______ BECAUSE ____________________________ WITH  ______
7. m8 = _______ BECAUSE _co-exterior__________________ WITH  ______
8. m9 = _______ BECAUSE ____________________________ WITH  ______
9. m10 = _______ BECAUSE _co-interior__________________ WITH  ______
10. m11 = _______ BECAUSE ____________________________ WITH  ______
11. m12 = _______ BECAUSE ____________________________ WITH  ______
12. m13 = _______ BECAUSE _alternate-exterior____________ WITH  ______
13. m14 = _______ BECAUSE ____________________________ WITH  ______
14. m15 = _______ BECAUSE ____________________________ WITH  ______
15. m16 = _______ BECAUSE _vertical_____________________WITH  ______
ANSWER THE FOLLOWING QUESTIONS USING THE DIAGRAM BELOW.
x
x || y
13
9
8
1
14
y
10
15
7
11
2
16
12
6
3
5
4
16. Name 4 pair of alternate-interior angles.
17. Name 4 pair of alternate-exterior angles.
18. Name 8 pair of corresponding angles.
19. Name 8 pair of vertical angles.
20. How many linear pair angles are there in this diagram?
22. Name 2 pair of nonvertical, equal angles from the diagram.
23. Name 2 pair of angles that are supplementary but are NOT linear pairs.
mastery connect – section 3.5
1. If ∠1 ≅ ∠2, which lines must be parallel?
a. ⃡𝐸𝐵 , ⃡𝐶𝐹
⃡ , ⃡𝐵𝐹
b.𝐴𝐸
c. ⃡𝐴𝐷 , ⃡𝐸𝐵
d. ⃡𝐶𝐹 , ⃡𝐴𝐷
2. Find the value of x so that 𝑗 ∥ 𝑘.
a. 7
b. 12
c. 3
d. 23
3. Which statement cannot be used to prove ⃡𝐸𝐹 ∥ ⃡𝑅𝑆
⃡
⃡ ≅ 𝑅𝑆
a. 𝐸𝐹
b. 𝑚∠3 = 𝑚∠4
c. ∠1 ≅ ∠2
⃡
⃡ = Slope 𝑅𝑆
d. Slope 𝐸𝐹
4. Which statement needs to be true to prove 𝑝 ∥ 𝑞?
a. 𝑚∠1 + 𝑚∠3 = 180°
b. ∠1 ≅ ∠2
c. ∠1 ≅ ∠3
d. ∠1 ≅ ∠4
5. If two lines are cut by a transversal and ____________ angles are congruent, then the lines are parallel.
a. adjacent
b. consecutive interior
6. Find the value of x so that ̅̅̅̅̅
𝑀𝑁 ∥ ̅̅̅̅
𝑃𝑄 .
c. vertical
d. corresponding
Section 3.5 Worksheet
Name: _____________________________________
For numbers 1 – 4, use the given the following information to determine which lines, if any, are parallel. State the postulate or
theorem that justifies your answer.
1. mBCG + mFGC = 180°
2. CBF ≅ GFH
3. EFB ≅ FBC
4. ACD ≅ KBF
For numbers 5 – 7, solve for x so that l || m. Identify the postulate or theorem you used.
5.
6.
7.
8. LANDSCAPING The head gardener at a botanical garden wants to plant rosebushes in parallel rows on either side of an existing
footpath. How can the gardener ensure that the rows are parallel?
9. BOOKS The two gray books on the bookshelf each make a 70° angle with the
base of the shelf. What more can you say about these two gray books?
Common Core State Standards
G.CO.12 Make formal geometric constructions with a variety
of tools and methods.
Student Learning Targets
1. Students will be able to find the distance between a point
and a line.
2. Students will be able to find the distance between parallel
lines.
G.MG.3 Apply geometric methods to solve problems.
3.6 Notes: Perpendiculars and Distance
Vocabulary
Distance Between a Point and a Line
Perpendicular Postulate
Definition
Picture
The distance between a line and a point not on the
line is the _____________ of the segment
_______________________ to the line from the
point.
Given a line and a point not on the line, there exists
________________________________________
through the point that is _______________________
to the given line.
Example 1:
a) A certain roof truss is designed so that the center post extends from the
peak of the roof (point A) to the main beam. Construct and name the segment whose length
represents the shortest length of wood that will be needed to connect the peak of the roof to
the main beam.
b) Which segment represents the shortest distance from point A to DB.
Example 2:
a) Line s contains points at (0, 0) and (–5, 5). Find the distance between line s and point V(1, 5).
(–5, 5)
V(1, 5)
(0, 0)
b) Line n contains points (2, 4) and (–4, –2). Find the distance between line n and point B(3, 1).
(2, 4)
B(3, 1)
(–4, –2)
Vocabulary
Definition
Picture
Equidistant
Distance Between Parallel Lines
Two Lines Equidistant from a Third
The distance between two parallel lines is the
___________________________ between one of the
lines and ___________________________________.
If two lines are __________________ from a third
line, then the two lines are __________________.
Example 3:
a) Find the distance between the parallel lines a and b whose equations are
y = 2x + 3 and y = 2x – 1, respectively.
b) Find the distance between the parallel lines a and b whose equations are y 
1
1
x  1 and y  x  2 , respectively.
3
3
Mastery connect – section 3.6
̅̅̅̅̅?
1. Which segment represents the distance from N to 𝑀𝑃
̅̅̅̅
a. 𝐶𝑁
̅̅̅̅̅
b. 𝑀𝐴
c. ̅̅̅̅
𝑁𝐷
d. ̅̅̅̅
𝑁𝐵
2. The distance from a line to a point not on the line is the length of the segment _____________ to the line from the point.
a. drawn
b. parallel
c. between
d. perpendicular
3. The distance between two parallel lines is the distance between one of the lines and ________________.
a. any point not on the line
c. any point on the other line
b. the other line
d. any point farther than the other line.
Section 3.6 Worksheet
Name: _____________________________________
For numbers 1 – 3, construct the segment that represents the distance indicated.
1. O to MN
2. A to DC
3. T to VU
For numbers 4 – 7, find the distance from P to l.
4. Line l contains points (−2, 0) and (4, 8). Point P has coordinates (5, 1).
5. Line l contains points (3, 5) and (7, 9). Point P has coordinates (2, 10).
6. Line l contains points (5, 18) and (9, 10). Point P has coordinates (−4, 26).
7. Line l contains points (−2, 4) and (1, −9). Point P has coordinates (14, −6).
For numbers 8 – 10, find the distance between each pair of parallel lines with the given equation.
8. y = –x
y = –x – 4
9. y = 2x + 7
y = 2x – 3
10. y = 3x + 12
y = 3x – 18
11. Graph the line y = –x + 1. Construct a perpendicular segment through the point at (–2, –3). Then find the distance from the point to
the line.
12. CANOEING Bronson and a friend are going to carry a canoe across a flat field to the bank of a straight canal. Describe the
shortest path they can use.
13. DISTANCE What does it mean if the distance between a point P and a line ℓ is zero? What does it mean if the distance between
two lines is zero?
14. DISTANCE Paul is standing in the schoolyard. The figure shows his distance from various
classroom doors lined up along the same wall. How far is Paul from the wall itself?