3-2 Angles Formed by Parallel Lines and Transversals
Objectives

Prove and use theorems about the angles formed by parallel lines and a transversal.
Example 1: Using the Corresponding Angles Postulate
A.
B.
mECF
mDCE
--Find mQRS
Sketch this situation
Example 2: Finding Angle Measures
Find each angle measure.
Find mABD
A. mEDG
B. mBDG
Example 3: Music Application
Find x and y in the diagram.
Find the measures of the acute angles in the diagram.
3-3 Proving Lines Parallel
Objectives
 Use the angles formed by a transversal to prove two lines are parallel.
Example 1A: Using the Converse of the Corresponding Angles Postulate
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
4  8
Example 1B: Using the Converse of the Corresponding Angles Postulate
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
m3 = (4x – 80)°,
m7 = (3x – 50)°, x = 30
Check It Out! Example 1a
Use the Converse of the Corresponding Angles Postulate and the given information to
show that ℓ || m.
m1 = m3
Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.
m7 = (4x + 25)°,
m5 = (5x + 12)°, x = 13
Example 2A: Determining Whether Lines are Parallel
Use the given information and the theorems you have learned to show
that r || s.
4  8
Example 2B: Determining Whether Lines are Parallel
Use the given information and the theorems you have learned to show
that r || s.
m2 = (10x + 8)°,
m3 = (25x – 3)°, x = 5
Example 3: Proving Lines Parallel
Given: p || r , 1  3
Prove: ℓ || m
Statements
Reasons
Example 4: Carpentry Application
A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. m1=
(8x + 20)° and m2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.
What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y –
2)° and (3y + 6)°, where
y = 8. Show that the oars are parallel.
3-4
Perpendicular Lines
Objectives:
-Prove and apply theorems about perpendicular lines.
Vocabulary: - perpendicular bisector and -distance from a point to a line
The perpendicular bisector of a segment is a line _______________ to a segment at the
segment’s ____________________.
The shortest segment from a point to a line is perpendicular to the line. This fact is used to
define the distance from a point to a line as the length of the perpendicular segment from
the point to the line.
Example 1: Distance From a Point to a Line
A. Name the shortest segment from point A to BC.
B. Write and solve an inequality for x.
Example 2: Proving Properties of Lines
Given: r || s, 1  2
Write a two-column proof.
Prove: r  t
Statements
Reasons
Write a two-column proof.
Given:
Prove:
Statements
Reasons
Example 3: Real- Life Word Problems
A carpenter’s square forms a right angle. A carpenter places the square so that one side is
parallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the
square to the right and draws a second line. Why must the two lines be parallel?
A swimmer who gets caught in a rip current should swim in a direction perpendicular to the
current. Why should the path of the swimmer be parallel to the shoreline?
3-5
Slopes of Lines
Warm-Up (Remember how to find the slope between 2 points)
Find the value of m.
1.
2.
3.
4.
Objectives


Find the slope of a line.
Use slopes to identify parallel and perpendicular lines.
Vocabulary rise, run, slope
The slope of a line in a coordinate plane is a number that describes the ________________ of the line.
Any two ________________ on a line can be used to determine the slope.
Use the slope formula to determine the slope of each line.
Summary: Slope of a Line
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
One interpretation of slope is a ______ __ _______. If y represents miles traveled and x
represents time in hours, the slope gives the rate of change in miles per hour.
Example 2: Transportation Application
Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he
is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and
interpret the slope of the line.
If a line has a slope of ____ , then the slope of a perpendicular line is ____ .
The ratios are called _________________ _________________ .
Example 3A: Determining Whether Lines Are Parallel,
Perpendicular, or Neither
Graph each pair of lines. Use their slopes to determine
whether they are parallel, perpendicular, or neither.
WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3)
CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)
UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3)
3-6
Lines in the Coordinate Plane
Warm Up
Substitute the given values of m, x, and y into the equation y = mx + b and solve for b.
1. m = 2, x = 3, and y = 0
2. m = –1, x = 5, and y = –4
Solve each equation for y.
3. y – 6x = 9
4. 4x – 2y = 8
Objectives


Graph lines and write their equations in slope-intercept and point-slope form.
Classify lines as parallel, intersecting, or coinciding.
Examples: Writing Equations In Lines and Graphing Lines
1. the line with slope 6 through (3, –4) in point-slope form
2. line through (–1, 0) and (1, 2) in slope-intercept form
3. the line with the x-intercept 3 and y-intercept –5 in point slope form
4. the line with slope 0 through (4, 6) in slope-intercept
form
5. the line through (–3, 2) and (1, 2) in point-slope form
Parallel Lines
Intersecting Lines
Coinciding Lines
Examples : Classifying Pairs of Lines
Determine whether the lines are parallel, intersect, or coincide.
1. y = 3x + 7, y = –3x – 4
2.
3. 2y – 4x = 16, y – 10 = 2(x - 1)
4. 3x + 5y = 2 , 3x + 6 = -5y
Example 4: Problem-Solving Application
Erica is trying to decide between two car rental plans. For how many miles will the plans cost
the same?
What if…? Suppose the rate for Plan B was also $35 per month. What would be true about
the lines that represent the cost of each plan?