GeometryChapter 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 ______!
tm
n
In this picture, line ____ is the transversal.
This transversal creates ____ angles!
Pairs of these angles have special names, depending on their positions.
12
5 67 8
3 4
Special Angle Pairs in Parallel Lines Cut by a Transversal:
Corresponding Angles
Alternate Interior Angles
Consecutive Interior Angles
Alternate Exterior Angles
Consecutive Exterior Angles
Angle Pair: Picture: Relationship:
Practice Identifying the Special Angle Pairs:
1 234
5 678
9 10
1112
Corresponding Angles
Alternate Interior Angles
Consecutive Interior Angles
Alternate Exterior Angles
Consecutive Exterior Angles
Use the picture to complete each statement:
1 234
5 678
9 10
1112
If m5 = 130, then m8 =____ because they are…
If m4 = 125, then m6 =____ because they are…
If m4 = 125, then m8 =____ because they are…
If m2 = 45, then m7 =____ because they are…
If m3 = 50, then m6 =____ because they are…
If m7 = 42, then m1 =____ because they are…
MORE PRACTICE:Use the picture to complete each statement.
1 234
5 678
9 10
1112
If m5 = 130, then m4 =____ because they are…
If m5 = 25, then m3 =____ because they are…
If m5 = 125, then m1 =____ because they are…
If m1 = 50, then m8 =____ because they are…
If m3 = 50, then m2 =____ because they are…
If m2 = 51, then m8 =____ because they are…
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 find the following angles and state the theorem used.
3 ____m
8 ____m
1 243
6587
2 120m
7 ____m
Using the picture at the left, find the measure of each angle and tell which postulate or theorem you used.
1 234
5 678
9 10
1112
m 1 =
m 2 =
m 3 =
m 4 =
m 5 =
m 6 =
m 7 =
L.T.: Be able to prove lines are parallel using the properties of the special angle pairs
Section 3.5 ~ Proving Line are Parallel
Quick Review: Find the value of x and justify each step.
Find each angle measure.
3x + 45
5x -25
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?
45°
45°
37°
37°
90° 90°
89°
Converse of Alternating Interior Angle Theorem:
If two lines and a transversal form ALTERNATING INTERIOR angles that are CONGRUENT, then the two
lines are ________________!
100°
100°
Where are there parallel lines in the pictures?
50°
130°
Where are there parallel lines in the pictures?
70°
110° 100°
80°
75°
115°
Converse of Consecutive Interior Angle Thm:
If two lines and a transversal form CONSECUTIVE INTERIOR angles that are SUPPLEMENTARY, then the
two lines are ________________!
Converse of ALTERNATE EXTERIOR Angle Thm:
If two lines and a transversal form ALTERNATE EXTERIOR angles that are CONGRUENT, then the two
lines are ________________!
6x - 24
x +116
Solve for x so the lines m and n are //
m
n
1 234
5 678
9 101112
1315
1416
r
s
v
t
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
12 + 13 = 180
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 - 30
80°
2x + 40
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!
3.3 Slope and rate of change3.3 Slope and rate of change
Objective: We are going to find the rate of change and Objective: We are going to find the rate of change and determine the slope of a line.determine the slope of a line.What does each of the following look like?What does each of the following look like?
Positive SlopePositive Slope Negative Slope Zero Slope Undefined Negative Slope Zero Slope Undefined SlopeSlope
How to find Slope…How to find Slope…Anyone remember?Anyone remember?
When given 2 points (x1, y1) and (x2, y2) plug them into our slope formula:
1: ( 4,3) (2,5)Ex and
Try these Try these Ex 2: Find the slope of the line that passes Ex 2: Find the slope of the line that passes
through (–1, 4) and (1, –2).through (–1, 4) and (1, –2).
Ex 3: Find the slope of the line that passes Ex 3: Find the slope of the line that passes through (9, –3) and (2, 7).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)
Rate of ChangeRate of ChangeCOLLEGE 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 MoreLet’s try One More• Find the rate of change for the data in the table.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 = ______
What about if m = ?
// m = _____ m = ______
7
3
More Parallel and Perpendicular lines
Determine whether and 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)
ABGHHHHHHHHHHHHH H
CDGHHHHHHHHHHHHH H
L.T.#1: Be able to graph lines from equations in slope-intercept form!
L.T.#2: Be able to write the equation of a line using point-slope form!
Section 3.4 ~ Equations of Lines!
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).
Write an equation of a line that passes through (2, -3) and has a slope of –4.
Point-Slope Form:y = m (x – x1) + y1
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.
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.
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