Ch. 3 Notes

3-1 Lines and Angles

Objectives

Identify parallel, perpendicular, and skew lines. Identify the angles formed by two lines and a transversal.

Vocabulary

Example 1

parallel lines corresponding angles

perpendicular lines alternate interior angles

skew lines alternate exterior angles

parallel planes same-side interior angles

transversal

.

Example 2: Give an example of each angle pair.

A. corresponding angles

B. alternate interior angles

C. alternate exterior angles

D. same-side interior angles

Example 3: Identifying Angle Pairs and Transversals

A. Ð1 and Ð3

B. Ð2 and Ð6

C. Ð4 and Ð6

Identify the transversal and classify the angle pair Ð2 and Ð5 in the diagram.

3-2 Angles Formed by Parallel Lines and Transversals

Objectives

Prove and use theorems about the angles formed by parallel lines and a transversal.

Example 1: Using the Corresponding Angles Postulate

A. mÐECF B. mÐDCE

--Find mÐQRS

Sketch this situation

Example 2: Finding Angle Measures

Find each angle measure.

Find mÐABD

A. mÐEDG

B. mÐBDG

Example 3

Find x and y in the diagram.

If a transversal is perpendicular to two parallel lines, all eight angles are congruent.

Helpful

Hint

Pg. 158 (Let’s do 38-40 together!)

Homework Due Wednesday 10-213-1 Pg. 149 (14-25, 35-40 odd, 47, 48) 3-2 Pg. 158 (5-11 odd, 12, 13-23odd, 25, 37)There will be a short HW quiz on Wednesday where you can use your HW on the quiz!

PLEASE PRINT Ch. 3 Notes starting with page 5!

Don’t forget we are doing LOGIC PROJECT PRESENTATIONS!

3-3 Proving Lines Parallel

Objectives

Use the angles formed by a transversal to prove two lines are parallel.

Example 1A: Using the Converse of the Corresponding Angles Postulate

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

Ð4 @ Ð8

Example 1B: Using the Converse of the Corresponding Angles Postulate

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

mÐ3 = (4x – 80)°,

mÐ7 = (3x – 50)°, x = 30

Check It Out! Example 1a

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

mÐ1 = mÐ3

Use the Converse of the Corresponding Angles Postulate and the given information to show that ℓ || m.

mÐ7 = (4x + 25)°, mÐ5 = (5x + 12)°, x = 13

Example 2A: Determining Whether Lines are Parallel

Use the given information and the theorems you have learned to show that r || s.

Ð4 @ Ð8

Example 2B: Determining Whether Lines are Parallel

Use the given information and the theorems you have learned to show that r || s.

mÐ2 = (10x + 8)°,

mÐ3 = (25x – 3)°, x = 5

Example 3: Proving Lines Parallel

Given: p || r , Ð1 @ Ð3

Prove: ℓ || m

Statements Reasons

Example 4: Carpentry Application

A carpenter is creating a woodwork pattern and wants two long pieces to be parallel. mÐ1= (8x + 20)° and mÐ2 = (2x + 10)°. If x = 15, show that pieces A and B are parallel.

What if…? Suppose the corresponding angles on the opposite side of the boat measure (4y – 2)° and (3y + 6)°, where

y = 8. Show that the oars are parallel.

3-4 Perpendicular Lines

Objectives: -Prove and apply theorems about perpendicular lines.

Vocabulary: - perpendicular bisector and -distance from a point to a line

The perpendicular bisector of a segment is a line _______________ to a segment at the segment’s ____________________.

The shortest segment from a point to a line is perpendicular to the line. This fact is used to define the distance from a point to a line as the length of the perpendicular segment from the point to the line.

Example 1: Distance From a Point to a Line

A. Name the shortest segment from point A to BC.

B. Write and solve an inequality for x.

Example 2: Proving Properties of Lines Write a two-column proof.

Given: r || s, Ð1 @ Ð2 Prove: r ^ t

Statements Reasons

Write a two-column proof. Given:

Prove:

Statements Reasons

HYPOTHESISCONCLUSION

Example 3: Real- Life Word Problems

A carpenter’s square forms a right angle. A carpenter places the square so that one side isparallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the square to the right and draws a second line. Why must the two lines be parallel?

A swimmer who gets caught in a rip current should swim in a direction perpendicular to the current. Why should the path of the swimmer be parallel to the shoreline?

3-5 Slopes of Lines

Warm-Up (Remember how to find the slope between 2 points)

Find the value of m.

1. 2. 3. 4.

Objectives

Find the slope of a line. Use slopes to identify parallel and perpendicular lines.

Vocabulary rise, run, slope

The slope of a line in a coordinate plane is a number that describes the ________________ of the line. Any two ________________ on a line can be used to determine the slope.

Use the slope formula to determine the slope of each line.

Summary: Slope of a Line

Positive Slope Negative Slope Zero Slope Undefined Slope

AB AC AD

One interpretation of slope is a ______ __ _______. If y represents miles traveled and x represents time in hours, the slope gives the rate of change in miles per hour.

Example 2: Transportation Application

Justin is driving from home to his college dormitory. At 4:00 p.m., he is 260 miles from home. At 7:00 p.m., he is 455 miles from home. Graph the line that represents Justin’s distance from home at a given time. Find and interpret the slope of the line.

If a line has a slope of ____ , then the slope of a perpendicular line is ____ .

The ratios are called _________________ _________________ .

Example 3A: Determining Whether Lines Are Parallel, Perpendicular, or Neither

Graph each pair of lines. Use their slopes to determine whether they are parallel, perpendicular, or neither.

WX and YZ for W(3, 1), X(3, –2), Y(–2, 3), and Z(4, 3)

GH and IJ for G(–3, –2), H(1, 2), I(–2, 4), and J(2, –4)

CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)

UV and XY for U(0, 2), V(–1, –1), X(3, 1), and Y(–3, 3)

3-6 Lines in the Coordinate Plane

Warm Up

Substitute the given values of m, x, and y into the equation y = mx + b and solve for b.1. m = 2, x = 3, and y = 0 2. m = –1, x = 5, and y = –4

Solve each equation for y.

3. y – 6x = 9 4. 4x – 2y = 8

Objectives

Graph lines and write their equations in slope-intercept and point-slope form. Classify lines as parallel, intersecting, or coinciding.

CD and EF for C(–1, –3), D(1, 1), E(–1, 1), and F(0, 3)

Examples: Writing Equations In Lines and Graphing Lines

1. the line with slope 6 through (3, –4) in point-slope form

2. line through (–1, 0) and (1, 2) in slope-intercept form

3. the line with the x-intercept 3 and y-intercept –5 in point slope form

4. the line with slope 0 through (4, 6) in slope-intercept form

5. the line through (–3, 2) and (1, 2) in point-slope form

Examples : Classifying Pairs of Lines

Determine whether the lines are parallel, intersect, or coincide.

1. y = 3x + 7, y = –3x – 4

2.

Parallel Lines Intersecting Lines Coinciding Lines

3. 2y – 4x = 16, y – 10 = 2(x - 1)

4. 3x + 5y = 2 , 3x + 6 = -5y

Example 4: Problem-Solving Application

Erica is trying to decide between two car rental plans. For how many miles will the plans cost the same?

What if…? Suppose the rate for Plan B was also $35 per month. What would be true about the lines that represent the cost of each plan?