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Chapter 3 Goals – Perpendicular and Parallel Lines
3.1 Lines and Angles
 Identify relationships between lines such as parallel, perpendicular, or skew
 Identify the four types of angles formed by 2 lines and a transversal
3.2 Proof and Perpendicular Lines
 Compare the three different kinds of proof (2-column, paragraph, or flow)
 Prove results about perpendicular lines
3.3 Parallel Lines and Transversals
 Given that lines are parallel, prove congruent angles using theorems
 Using parallel line theorems, set up equations to solve for missing variables
and angles (don’t forget vertical angles and linear pair!)
3.4 Proving Lines are Parallel
 Prove that two lines are parallel using parallel converse theorems
3.5 Using Properties of Parallel Lines
 Use parallel and perpendicular line theorems to determine if two lines are
parallel or not.
3.6 Parallel Lines in the Coordinate Plane
 Rewrite equations in slope-intercept form and identify the slope of the line.
 Using slopes, determine if two lines are parallel.
 Write the equation of a line parallel to another line given a point it passes
through.
3.7 Perpendicular Lines in the Coordinate Plane
 Identify the slopes of lines and determine if two lines are perpendicular.
 Write the equation of a line perpendicular to another line given a point it
passes through.
EXTRA: Finding the distance from a point to a line using slopes & equations
Vocabulary for Chapter 3
3.1
Parallel Lines
Perpendicular Lines
Skew Lines
Parallel Planes
Transversal
Corresponding angles
Alternate exterior angles
Alternate interior angles
Consecutive interior angles (same side interior angles)
3.2
Two-column proof
Paragraph proof
Flow proof
3.6
Slope
Slope-intercept form
3.7
opposites
reciprocals
Ohio Academic Content Standards
G.CO.1 Experiment with transformations in the plane.
Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a line, and arc
length.
G.CO.9 Prove geometric theorems.
Apply multiple proof methods, such as narrative paragraphs, flow diagrams, coordinate
proofs, two-column proofs, diagrams without words, and the use of dynamic software.
Prove and apply theorems about lines and angles. Theorems include but are not
restricted to: vertical angles are congruent; when a transversal crosses parallel lines,
alternate interior angles are congruent and corresponding angles are congruent; points
on a perpendicular bisector of a line segment are exactly those equidistant from the
segment’s endpoints.
C.GO.12 Make geometric constructions.
Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folder, dynamic geometric software, etc.)
Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a
line parallel to a given line through a point not on the line.
3.1 Lines & Angles
Define the following words and draw a diagram:
Parallel Lines – p.129
Diagram:
Perpendicular Lines – p.79
Diagram:
Skew Lines – p.129
Diagram:
Parallel Planes – p.129
Diagram:
Transversal – p. 131
Diagram:
p.129
Angle Relationships
Vertical Angles: - p. 44
Linear Pair – p. 44
Corresponding Angles – p. 131
Alternate Exterior Angles – p. 131
Alternate Interior Angles – p. 131
Consecutive Interior Angles – p. 131
Practice:
Q: Compare and Contrast parallel lines and skew lines.
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3.2 Types of Proof
Read Example 1 on page 136 in your textbook.
Notice the same proof is shown using 3 different formats.
In the diagram, 𝑚∠8 = 𝑚∠5 and 𝑚∠5 = 125°.
Prove that 𝑚∠7 = 55°.
TWO – COLUMN PROOF:
PARAGRAPH (NARRATIVE) PROOF:
FLOW PROOF:
3.3 Parallel Lines & Transversals
GOAL: SOLVE FOR MISSING ANGLE MEASURES
Example 1:
Example 2:
Use properties of parallel lines to find the value of x. Tell which postulate or theorem you use.
(a)
(b)
(c)
Example 3:
(d)
3.4&3.5 Proving Lines Parallel
We will learn six ways to prove that lines are parallel.
Transitive Property of Parallel Lines:
Lines Perpendicular to a Transversal Theorem:
Example 1: Is it possible to prove the lines parallel? If so, explain what theorem/postulate you
used.
Example 2:
Find the value of x that makes line r parallel to line s.
Example 3:
Example 4: Determine which lines, if any, are parallel.
Example 5:
Example 6:
Example 7: Proving the Transitive Property of Parallel Lines
Example 8:
You are building a bookshelf. You cut the sides, bottom, and top so that each corner is
composed of two 45° angles. Prove that the top and bottom front edges of the bookshelf are
parallel.
3.6 & 3.7 Slopes of Parallel and Perpendicular Lines
Slope =
𝒓𝒊𝒔𝒆 ↑ ↓
𝒓𝒖𝒏 →
𝒚𝟏 −𝒚𝟐
=
𝒙𝟏 −𝒙𝟐
Slope-Intercept Equation 𝒚 = 𝒎𝒙 + 𝒃
m = slope
(0, b) = y-intercept
(x, y) = any point on the line
Example 1: Find the slope of the line that passes through the points (-4, -3) and (5, 3)
Method 1: Use slope formula
Method 2: Count rise and run
Example 2: Identify the slopes of the lines given by the equations.
3
(a) 𝑦 = 4 𝑥 + 2
(b) 𝑦 − 6𝑥 = 12
(c) 4𝑦 + 3𝑥 = −12
Example 3: A slope of a line is given. Find the slopes of a parallel and perpendicular line.
2
(a) 𝑚 = 3
∥ slope = ___________
⊥ slope = _____________
∥ slope = ___________
⊥ slope = _____________
(b) 𝑚 = −6
Example 4: Find both slopes. Are the lines parallel, perpendicular, or neither?
(a)
(c)
1
𝑦 = −2𝑥 + 9
𝑦 = −2𝑥 − 4
(b) 𝑦 = −3𝑥 − 4
6𝑥 + 2𝑦 = 8
(d)
3.6 & 3.7 Equations of Parallel and Perpendicular Lines
Slope-Intercept Equation 𝒚 = 𝒎𝒙 + 𝒃
m = slope
(0, b) = y-intercept
(x, y) = any point on the line
∥ Parallel lines have _______________ slopes.
⊥ Perpendicular lines have slopes that multiply to _________.
The slopes are ___________________ and _______________________.
______________________________________________________________________________
Example 1: Finding equations of lines given a y-intercept.
Line l has the equation 𝑦 = 𝑥 + 4.
(a) Find a line parallel to line l with a y-intercept of (0, -3).
(b) Find a line perpendicular to line l with a y-intercept of (0, 1).
Solution:
(a) First, identify the slope of the original equation. Then identify the slope of a parallel line.
Use the given y-intercept and slope you found to write your equation.
(b) First, identify the slope of the original equation. Then identify the slope of a perpendicular
line. Use the given y-intercept and slope you found to write your equation.
Check using a Graph
Example 2: Finding equations of lines given a point on the line.
Line l has the equation 𝑦 = −2𝑥 + 5.
(a) Find a line parallel to line l that passes through the point (-4, 3).
(b) Find a line perpendicular to line l that passes through the point (2, 6).
Solution:
(a) First, identify the slope of the original equation. Identify the slope of a parallel line.
Then, use your new slope and coordinate point to solve for b. Write new equation.
(b) First, identify the slope of the original equation. Identify the slope of a perpendicular line.
Then, use your new slope and coordinate point to solve for b. Write new equation.
Check using a Graph
Distances to Lines
The distance from a point to a line is the length of the perpendicular segment from the point to
the line. This perpendicular segment is the shortest distance between the point and the line. For
example, the distance between point A and line k is AB.
Recall:
𝑑 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
Example 1:
⃡ .
Find the distance from point A to 𝐵𝐷
You Try:
⃡ .
Find the distance from point E to 𝐹𝐻
Example 2:
Find the distance from the point (1, 0) to the line 𝑦 = −𝑥 + 3.
Step 1: Find the equation of the line perpendicular to
𝑦 = −𝑥 + 3 that passes through the point (1, 0).
Step 2: Use the two equations to write and solve a system
of equations to find the point where the two lines intersect.
Step 3: Use the distance formula to find the distance from
(1, 0) to (2, 1).
You Try:
Find the distance from the point (6, 4) to the line 𝑦 = 𝑥 + 4.
Constructing a Perpendicular Line through a Point not on the Line
Constructing a Parallel Line through a Point not on the Line