Download If two lines and a transversal form ALTERNATING

Geometry
Chapter 2 Learning Targets!
By the unit of the chapter, you should be able to:
Identify the relationships between two lines or two planes
Name angle pairs formed by parallel lines and transversals
Use theorems to determine the relationships between specific pairs of angles
Use algebra to find angle measurements
Find slopes of lines
Use slope to identify parallel and perpendicular lines
Write an equation of a line given information about the graph
Solve problems by writing equations
Recognize angle pairs that occur with parallel lines, and prove that two lines are parallel using
angle relationships
Section 1 ~
Parallel and Skew lines!
L.T. #1: Be able to identify angle pairs
(corresponding, alternate interior, same-side interior,
alternate exterior, same-side exterior)!
Quick Definitions:
Parallel lines:
Parallel planes:
Skew lines:
A transversal is a _____ that
intersects two other ______!
In this picture, line ____ is
the transversal.
This transversal creates ____
angles!
Pairs of these angles have
special names, depending on
their positions.
1
3 42
5
7 8
t
n
6
m
Identify the transversal(s) in
each picture:
n
c
a
b
k
m
s
t
p
r
Special Angle Pairs in Parallel
Lines Cut by a Transversal:
Angle Pair:
Corresponding
Angles
Alternate Interior
Angles
Consecutive
Interior Angles
Alternate Exterior
Angles
Consecutive
Exterior Angles
Picture:
Relationship:
Practice Identifying the Special
Angle Pairs:
Corresponding
Angles
Alternate Interior
Angles
Consecutive
Interior Angles
Alternate
Exterior Angles
Consecutive
Exterior Angles
10
11
9
1 2
4 3
12
5 67
8
Use the picture to complete each statement:
If m5 = 130, then m8 =____
because they are…
If m4 = 125, then m6 =____
because they are…
If m4 = 125, then m8 =____
because they are…
10
12 11
9
1 2
4 3
5 67
8
If m2 = 45, then m7 =____ because they are…
If m3 = 50, then m6 =____ because they are…
If m7 = 42, then m1 =____ because they are…
MORE PRACTICE:
Use the picture to complete each statement.
If m5 = 130, then m4 =____
because they are…
If m5 = 25, then m3 =____
because they are…
If m5 = 125, then m1 =____
because they are…
10
12 11
9
1 2
4 3
5 67
8
If m1 = 50, then m8 =____ because they are…
If m3 = 50, then m2 =____ because they are…
If m2 = 51, then m8 =____ because they are…
Find the value of each variable.
2x
2x + 60
3x - 20 x + 60
Your Turn!
60°
3y
Section 3.2 ~
Angles and Parallel Lines
L.T.: Be able to determine relationships between
specific pairs of angles and use algebra to find
specific angle measurements.
Quick Review: Find the value of x and the measure of each angle. Justify each step!
3x
2x + 50
Postulate: When lines are parallel,
corresponding angles are ____!
Theorem: When lines are parallel, alternate
interior angles are ____!
Theorem: When lines are parallel, consecutive
interior angles are _________!
Theorem: When lines are parallel, alternate exterior
angles are _________!
Theorem: When lines are parallel, consecutive
exterior angles are _________!
Let’s use our theorems to find angle
measures
If m2  120 find the following angles and
state the theorem used.
m3  ____
m7  ____
m8  ____
12
3 4
5 6
7 8
Using the picture at the left, find the measure of each
angle and tell which postulate or theorem you used.
m 1 =
10
11
9
1 2
4 3
12
5 67
8
m 2 =
m 3 =
m 4 =
m 5 =
m 6 =
m 7 =
Find the value of the variables in each picture. Explain
your answer.
75°
3n -15 °
13 - x
68 +10x
Write a 2 column proof to solve for y.
Given: m 4 =
m 5 =
12
3 4
5 6
7 8
Section 3.5 ~
Proving Line are Parallel
L.T.: Be able to prove lines are parallel using the
properties of the special angle pairs
Quick Review: Find the value of x and justify each step.
Find each angle measure.
5x -25
3x + 45
Converse of Corresponding Angle Postulate:
If two lines and a transversal form CORRESPONDING
angles that are CONGRUENT, then the two lines are
________________!
Where are there parallel lines in the pictures?
90°
45°
89°
90°
37°
37°
45°
Converse of Alternating Interior Angle Theorem:
If two lines and a transversal form ALTERNATING
INTERIOR angles that are CONGRUENT, then the two
lines are ________________!
Where are there parallel lines in the pictures?
50°
130°
100°
100°
Converse of Consecutive Interior Angle Thm:
If two lines and a transversal form CONSECUTIVE
INTERIOR angles that are SUPPLEMENTARY, then the
two lines are ________________!
Where are there parallel lines in the pictures?
70°
80°
110° 100°
75°
115°
Converse of ALTERNATE EXTERIOR Angle Thm:
If two lines and a transversal form ALTERNATE
EXTERIOR angles that are CONGRUENT, then the two
lines are ________________!
Solve for x so the lines m and n are //
x +116
m
n
6x - 24
Given the following information, is it possible to prove
that any of the lines shown are parallel? If so state the
postulate or theorem that justifies your answer.
2 =  8
t
v
1 2
4 3
9 10
12 11
r
13 14
16
6
15
5
8 7
12 + 13 = 180
s
4 = 6
14 = 15
9 = 13
Find the value of the variable that would make the
lines parallel. State which postulate or theorem
justifies your answer.
2x + 40
80°
2x - 30
4x - 40°
Two More Theorems:
Theorem:
If two lines are parallel to
the same line, then they are
parallel to each other!
Theorem:
If two lines are
perpendicular to the same
line, then they are parallel
to each other!
A
B
1
2
C
3
D
Last one, YOUR TURN …
Are the lines parallel? Explain.
50°
50°
3.3 Slope and rate of change
Objective: We are going to find the rate of change and
determine the slope of a line.
What does each of the following look like?
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
How to find Slope…
Anyone remember?
When given 2 points (x1, y1) and (x2, y2)
plug them into our slope formula:
Ex1: (4,3)and (2,5)
Try these 
Ex 2: Find the slope of the line that passes
through (–1, 4) and (1, –2).
Ex 3: Find the slope of the line that passes
through (9, –3) and (2, 7).
Find the slope of the line that passes
through the following points.
Ex. 4: (2, 3) and (2, 6)
Ex. 5: (-5, 7) and (4, 7)
From a graph!
Find the slope of the line.
Blue Line:
Red Line:
Rate of Change
COLLEGE ADMISSIONS In 2004, 56,878 students applied to
UCLA. In 2006, 60,291 students applied. Find the rate of
change in the number of students applying for admission from
2004 to 2006.
X – independent variable
Y – Dependent variable
Let’s try One More
• Find the rate of change for the data in the table.
Parallel and Perpendicular
Lines
If 2 lines are parallel, there slopes are __________
If 2 lines are perpendicular, there slopes are
_________
If m=-4 what is the slope of a line perpendicular and
parallel to the line?
// m = _____
 m = ______
7
What about if m =
?
3
// m = _____
 m = ______
More Parallel and Perpendicular
lines
Determine whether AB and CD are parallel,
perpendicular, or neither for the given set of
points.
Ex 1: A(1, -3) B(-2, -1) C(5, 0) and D(6, 3)
Ex 2: A(3, 6) B(-9, 2) C(5, 4) and D(2, 3)
Section 3.4 ~
Equations of Lines!
L.T.#1: Be able to graph lines from equations in slopeintercept form!
L.T.#2: Be able to write the equation of a line using
point-slope form!
Recall: Coordinate pairs: (x1, y1) and (x2, y2)
SLOPE of a line:
Example: Find the slope
of the line that passes
through (4, 5) and (-1, 2).
Slope-Intercept Form:
y = mx + b
1
y  x4
3
y  1  2 x
Let’s practice s’more!
2x  4 y  8
4 y  3 x  24
Write the
equation of the
graph shown:
Point-Slope Form:
y = m (x – x1) + y1
Write an equation of a line that passes through (2, -3) and has
a slope of –4.
Write an equation of a line that passes through (-3, 4) and has
a slope of 2/3. Write your final answer in slope-intercept form.
Write an equation of a line that passes through (-2, -1) and (3, 0).
Write an equation of a line that passes through (1, 5) and (4, 2).
Write your final answer in slope-intercept form.
Graph the equations
 5 x  y  3
 6 x  3 y  12
Write an equation of a line that passes through (-1, 4) and has a
slope of 3.
Write an equation of a line that passes through (4, -9) and
(-1, 1). Write your final answer in slope-intercept form.
One Last thing… Do we know how to
graph horizontal and vertical lines???
y=#
Graph of a ______________ line
x=#
Graph of a ______________ line
What is the equation
Graph y = 2
Graph x = -3
of the line?_______