Chapter 3 Notes
3.1 Lines and Angles/3.2 Properties of Parallel Lines
Not all lines and planes intersect!
Parallel lines: ___________________________________________________________________
Real world example:
Symbol for parallel:
Skew lines: ____________________________________________________________________
Real world example:
Parallel planes: _________________________________________________________________
Real world example:
Use the cube to name the following:
1.) All lines that are parallel to line BC
2.) A pair of parallel planes
3.) Two lines that are skew to line AE
Lines Intersecting Other Lines: When lines intersect two or more other lines, special angles are
formed. When lines are parallel, these angles will have special relationships.
Transversal: ____________________________________________________________________
Label the interior, exterior, and transversal in the diagram:
Label the 8 angles formed in the drawing above. Which ones would be interior angles? Which
ones would be exterior?
Angles Formed by Transversals:
Alternate interior angles: _________________________________________________________
______________________________________________________________________________
Same-side interior angles: ________________________________________________________
______________________________________________________________________________
Corresponding angles: ___________________________________________________________
______________________________________________________________________________
Alternate exterior angles: _________________________________________________________
______________________________________________________________________________
Identify all pairs of each type of angles in the diagram. Name the two lines and the transversal
that form each pair.
4.) Corresponding angles
5.) Alternate interior angles
6.) Same-side interior angles
7.) Alternate exterior angles
Questions to Ponder:
Why is the word coplanar included in the definition for parallel lines?
How does the phrase alternate interior angles describe the position of two angles?
The 4 special types of angles formed by transversals and parallel lines are either congruent,
supplementary, or both.
Corresponding angles postulate: ___________________________________________________
______________________________________________________________________________
Alternate interior angle theorem: __________________________________________________
______________________________________________________________________________
Same-side interior angle theorem: _________________________________________________
______________________________________________________________________________
Alternate exterior angle theorem: __________________________________________________
______________________________________________________________________________
What must you know in order for the angles to be congruent or supplementary?
How do you know that lines are parallel?
Find m<1 and m<2. State the theorems or postulates that justify your answers.
1.)
2.)
3.)
4.)
Find the value of each variable. Then find the measure of each labelled angle.
5.)
6.)
7.) Complete the proof. Given:
Statements
1.)
2.)
3.)
4.)
5.)
6.)
7.)
8.)
<1  <2
<2 and <3 are supplementary
<3 and < 4 are supplementary
<2  <4
<1  <4
<4  <5
<1  <5
, Prove: <1  <5.
Reasons
1.) Given
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
_________________________
3.3 Proving Lines Parallel
What was special about the angles formed by parallel lines?
How can we use those angles pairs to prove that lines are parallel?
Converse of Corresponding Angles Postulate: _________________________________________
______________________________________________________________________________
Diagram:
Converse of Alternate Interior Angles Theorem: _______________________________________
______________________________________________________________________________
Diagram:
Converse of Same-side Interior Angles Theorem: ______________________________________
______________________________________________________________________________
Diagram:
Converse of Alternate Exterior Angles Theorem: ______________________________________
______________________________________________________________________________
Diagram:
Use the given information to determine which lines, if any, must be parallel. If any lines are
parallel, use a theorem or postulate to tell why.
1.) <9  <11
a
2.) <1  <14
1
5
b
2
6
3
7
4
c
8
3) <2 is supplementary to <3
4)
M<6 = 60, m<10 = 120
5) <7  <12
9 10
13 14
11 12
15 16
d
Determining the Value of a Variable



Assume that lines are parallel when asked what value makes them parallel.
Solve it the same way.
Remember the special relationships between these angles.
Find the value of x for which a//b.
6.)
7.)
8.)
9) Complete the flow proof below.
Given: <1 is supplementary to <2
Prove: l//m
10) Prove the following.
a
1
b
2
3
3.4/3.5 Parallel and Perpendicular Lines and Parallel Lines and Triangles
Parallel lines: ___________________________________________________________________
What is true of the slopes of parallel lines?
Perpendicular Lines: _____________________________________________________________
What is true of the slopes of perpendicular lines?
Theorem 3-7: If two lines are parallel to the same line, then they are parallel to each other.
Diagram and Symbols:
Theorem 3-8: In a plane, if two lines are perpendicular to the same line, then they are parallel
to each other.
Diagram and Symbols:
Proof of 3-8
Given: r is perpendicular to t, s is perpendicular to t
Prove r is parallel to s
Perpendicular Transversal Theorem: If a line is perpendicular to one of two parallel lines, then it
is also perpendicular to the other.
Diagram and Symbols:
Triangle Sum Activity:
Draw a triangle on a piece of paper and cut it out. Label the angles in the interior as 1, 2, and 3.
Tear off the angles from the triangle. Put all three together. What special shape do you form?
What is the degree measure? Make a conjecture about the angle sum of a triangle.
Parallel Postulate: Through a point on a line, there is one and only one line parallel to the given
line.
Diagram:
Triangle Sum Theorem: The sum of the angle of a triangle is always _______________________
Diagram:
Exterior Angle of a Polygon: an angle formed by a side and an extension of an adjacent side
Remote Interior Angle: the two nonadjacent interior angles to each exterior angle
Diagram:
Triangle Exterior Angle Theorem: The measure of each exterior angle of a triangle equals the
______________________________________________________________________________
Examples: Find m<1 or find the value of the variable.
1.)
2.)
3.)
4.)
5.)
3.7/3.8 Equations of Lines and Slopes of Parallel and Perpendicular Lines
Slope of Lines: m =
y2  y1
x2  x1
Given point, rise/run
Slope intercept form: y = mx + b
Given a slope and y-intercept; m = slope, b = y-intercept
Point Slope form: y –y1 = m(x – x1) Given a point and slope or given two points; m = slope,
(x1,y1) = point given
Find the slope of the line that passes through the given pair of points
1.) (-2,3) and (4,5)
2.) (1,5) and (-6,-9)
Slopes Can Be One of Four Types
Positive
Negative
Zero
Undefined
Graphing Lines Using Slope Intercept Form



Equation must be in y = mx + b form
Remember that m = slope and b = y intercept
Plot the y intercept first, then use slope to move to next point
y
Graph the following lines:
2
1.) Y = − 3 𝑥 + 4
2.) Y – 3 = -2(x + 3)
x
Writing Equations of Lines: All equations should end up in slope intercept form!
Use the table to determine the best route. Work down based on what you have.
Given slope and y-intercept
Replace the m with the slope
and the b with the y
intercept in y = mx + b
Finished!
Given point and slope
Plug into y – y1 = m(x – x1)
The point is (x1,y1) and m is
the slope.
Distribute
Given 2 points
Find the slope between the
points and label it m.
Plug into y – y1 = m(x – x1)
The point is (x1,y1) and m is
the slope.
Get y by itself
Distribute
Finished!
Get y by itself
Finished!
Use the given information to write an equation of each line. Write all answers in slope
intercept form.
5) slope = -4, y-intercept = 6
6.) Slope = 7, passes through (1,-2)
7.) (4,2) and (6,-3)
8.) (3,-5) and (-5,3)
What form of an equation will tell you the slope?
What does it mean for a number to be an opposite (negative) reciprocal?
Give an example of two numbers that are opposite reciprocals.
Comparing Parallel and Perpendicular Slopes


Equations must be in y = mx + b form
Compare the two slopes
o Slopes that are the same represent parallel lines
o Slopes that are opposite reciprocals are perpendicular lines.
Without graphing, tell whether the lines are parallel, perpendicular, or neither. Explain.
1.) Y = 4x – 8
Y = 4x - 2
2.) 13y – x = 7
𝑦
7 -2=𝑥
4
Y = −3𝑥 + 2
3)
4
𝑦 =𝑥−1
3
4) 2x + 3y = 5
5x – 10y = 30
Writing Equations of Lines




Identify the slope. You must find it if not given.
Determine if you will use the same slope or the opposite reciprocal.
Plug the slope you are using and the point you are using into y – y1 = m(x – x1)
Write the final answer in slope intercept, unless told otherwise.
Write an equation for the line parallel to the given line through the given point.
5.) y = x – 7; (0,4)
1
6.) y = 2x + 3; (6,3)
Write an equation for the line perpendicular to the given line through the given point.
7) y = x + 2; (3,2)
1
2
8) y = 3 𝑥 − 5; (5, -1)