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

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

Identify the transversal(s) in each picture:

a b

c

km

n

r

s

t

p

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…






MORE PRACTICE:Use the picture to complete each statement.

1 234

5 678

9 10

1112







Find the value of each variable.

2x

2x + 60

x + 603x - 20

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 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 =

75°

3n -15 °

68 +10x

13 - x

Find the value of the variables in each picture. Explain your answer.

Write a 2 column proof to solve for y.Given: m 4 =

m 5 =

1 243

6587

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°


50°

130°


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!

A

DC

B

1

2 3

Last one, YOUR TURN …Are the lines parallel? Explain.

50°

50°

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)

From a graph!From a graph!Find the slope of the line.

Blue Line:

Red Line:

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).

43

1 xy

Slope-Intercept Form:

y = m x + b

xy 21

Let’s practice s’more!842 yx 2434 xy

Write the equation of the graph shown:

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.

Graph the equations35 yx 1236 yx

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?_______

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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