1. Draw and Label a Pair of Parallel Lines

Name ______

Geometry 1

Unit 3:

Perpendicular and

Parallel Lines

1. Draw and label a pair of parallel lines.

2. Draw and label a pair of skew lines

3. Draw and label parallel planes

Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.1 Lines and Angles

Example 1

Example 2

1. What do you know about parallel?

2. What do you know about perpendicular?

3. Describe the Parallel Postulate in your own words

4. Describe the Perpendicular Postulate in your own words

Constructing Perpendicular Lines

How do alternate exterior angles differ from alternate interior angles?

If two lines are cut by a transversal, how many pairs of corresponding angles are formed?
Example 3

Label the intersections in the figure below

List all pairs of angles that fit the descriptions. There will be more than one answer for each set of angles.

a.  Transversal

b.  Corresponding Angles

c.  Alternate Exterior Angles

d.  Alternate Interior Angles

e.  Consecutive Interior Angles

1.  How is a flow proof different from a two-column proof?

2.  In a two column proof, what can be written under the reasons column?

3. In your own words describe parallel lines

4. In your own words describe perpendicular lines

5. Can two lines intersect and not be perpendicular?

6. In your own words describe a pair of vertical angles

7. In your own words describe a pair of supplementary angles

8. In your own words describe a pair of complimentary angles

Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.2 Proof and Perpendicular Lines

Review of 2.5:

Using a two-column proof

A two-column proof has numbered ______on one side, and ______that show the logical order of an argument on the other.

In the two-column proof, the reasons must use one of the following:

______;

a ______;

a ______;

a ______;

Or a ______

Digits displayed on a digital clock form a series of perpendicular segments.

Determine which digit is being described.

1) I am constructed from more than 4 distinct segments. ) I contain four different pairs of parallel segments.

2)I contain six different pairs of perpendicular segments. I am an odd number.

Example 1-Method 1

Given: AB = CD

Prove: AC = BD

Example 1-Method 2

Given: AB = CD

Prove: AC = BD

Two-Column Proof

1. In the two column proof above, show an example of what a statement might look like.

2. In the two column proof above, show an example of what a reason might look like.

3. Show, by writing, how you would write the two-column proof above as a paragraph proof

(it will only be one or two sentences).

Example 2- Method 1

Given: BA perpendicular to BC

Prove: Ð1 and Ð2 are complementary

Example 2- Method 1

Given: BA perpendicular to BC

Prove: Ð1 and Ð2 are complementary

Ann has a full length mirror resting against the wall of her room. Ann notices that the floor and its reflection do not form a straight angle. She concludes that the mirror is not perpendicular to the floor. Explain her reasoning.
Example 3- Method 1 Two Column Proof

Given: Ð5 and Ð6 are a linear pair

Ð6 and Ð7 are a linear pair

Prove: Ð5 z Ð7

Statements Reasons

Method 2 Paragraph Proof

Method 3 Flow Proof

Activity: Measuring angles of parallel lines and their transversals

Objective: Discover the relationships between the angles of parallel lines and their transversals

Question: What is the relationship between the angles and the lines?

Step 1: Construct a segment

Step 2: Construct 2 parallel lines crossing that Segment

Step 3: Number the angles 1 – 8

Step 4: Measure each angle with a protractor, write that measure on the figure

Step 5: Write, in paragraph form, the relationships you see

______

Draw here:

Write here

Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.3 Parallel Lines and Transversals

Example 1

Given: p || q

Prove: mÐ1 + mÐ2 = 180°

Practice your algebra!

1. 75 = 5x

2. 2x = 150

3. 12x = 54

4.  2x + 1 = 151

5.  (2x + 1) = 151

6.  (7x + 15) = 81

Example 2

q

p

9

6 8

7 5

Example 3

How many other angles have a measure of 100°?

AB || CD, AC || BD

Example 4

Example 5

You notice that when your windshield wipers in your car stopped they each mad a 30° angle with the bottom of the windshield. If the wipers were long enough, would they ever cross each other? Explain.

State the hypothesis and the conclusion of the Alternate Exterior Angles Theorem.
Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.4: Proving Parallel Lines

Example 1

Given: m p, m q

Prove: p || q

Example 2

Given: Ð5 z Ð6, Ð6 z Ð4

Prove: AD || BC

Fill in the following statements.

·  The sum of the interior degrees of a triangle is ______.

·  The sum of the degrees of a pair of complementary angles is ______.

·  The sum of the degrees of a pair of supplementary angles is ______.

·  The sum of the degrees of consecutive interior angles if transversal crosses parallel lines is ______.

·  Parallel lines have slopes that are ______.

Example 3

Find the value of x that makes m || n.

Example 4

Is AB || DC?

Is BC || AD?

Example 5

When the lines r and s are cut by a transversal, Ð1 and Ð2 are same side interior angles. If mÐ1 is three times mÐ2, can r be parallel to line s? Explain

Activity: Construction

Copying an Angle

Activity: Construction

Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.5: Using Properties of Parallel Lines

Example 1

Given: r || s and s || t

Prove: r || t

Example 2

The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe.

Describe your thinking as you prove that S1 and S13 are parallel.

How can you show that the first horizontal line on a piece of graph paper is parallel to the last horizontal line if the center vertical line is perpendicular to all horizontal lines?
Example 3

Fill in the boxes with mathematical statements, the first has been done

You are building a CD rack. You cut the sides, bottom, and top so that each corner is composed of two 45o angles. Prove that the top and bottom front edges of the CD rack are parallel.

Given: Prove:

Angle addition postulate Given Angle addition postulate Given

Substitution Property Substitution Property

Definition of a right angle Definition of a right angle

Definition of lines Definition of lines

How can you use the graph to determine if the slope of a line is positive or negative?

Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.6 Parallel Lines in the Coordinate Plane

Slope of Parallel Lines

The slope of a line is usually represented by the variable m. Slope is the change in the rise, or vertical change, over the change in the run, or horizontal change.

Example 1

Cog railway

A cog railway goes up the side of a Mount Washington, the tallest mountain in New England. At the steepest section, the train goes up about 4 feet for each 10 feet it goes forward. What is the slope of this section?

rise =______

slope = ------= ------

run =______

Example 2

The cog railway covers about 3.1 miles and gains about 3600 feet of altitude. What is the average slope of the track?

If you know the coordinates of two points on line p and two points on line q

how could you tell if p || q?

Example 3

Find the slope of a line that passes through the points (0,6) and (5,2).

x1= y1 =

x2= y2 =

slope = ------= ------

Example 4

Example 5

What information is contained in the equation of a line written in slope intercept form?

Write out the steps for graphing a line given an equation in slope intercept form.

Writing Equations of parallel Lines

In algebra, you learned that you can use the slope m of a non-vertical line to write the equation of the line in slope intercept form.

y = mx + b

Example 6

y = 2x + 5 y = -½x – 3

What is the slope?

What is the y-intercept?

Do you have enough information to graph the line?

Example 7

Write the equation of a line through the point (2,3) with a slope of 5.

Step 1: x =

y =

m =

Step 2: Substitute the values above into the equation y = mx + b. Solve for b

______= (______) (______) + b

y m x

Step 3: Rewrite the equation of the line in slope-intercept form, using m and b from

your solution to the equation above

y = ______x + ______

m b

What is the same in two parallel lines?

What is different?

What are some examples of slopes that you see around you. Do you think that any of them have the exact slope they do for a reason?

How important is having the right slope in carpentry, or architecture, or even skateboard parks and competitions?

Example 8

Line k1 has the equation ______.

Line k2 is parallel to line n1 passes through the point ______.

Line k1, what is the slope?

1.  What will the slope be for line k2?

2.  Write the equation of k2, using the information you know. Solve for b

3.  m = ______, b = ______

4.  Write an equation for the line k2 in slope-intercept form.

Example 9

Example 10

Activity: Investigating Slopes of Perpendicular Lines

You will need: an index card, a pencil and the graph below.

1.  Write about how the two lines are related. What do you see?

Geometry 1 Unit 3: Perpendicular and Parallel Lines

3.7 Perpendicular Lines in the Coordinate Plane

Example 1 j1 j2

1.  Label the point of intersection

And the x-intercept of each line.

2.  Find the slope of each line.

3.  Multiply the slopes.

Question: What do you notice?

4.  Look at the activity from the

start of class.

5.  Multiply the slopes of the lines.

Question: What do you notice?

What is true about the product of the slopes of perpendicular lines?

Example 2 Deciding Whether Lines Are Perpendicular

Decide whether and are perpendicular.

What is the product of the slopes of perpendicular lines? ______

Are these lines perpendicular? ______

1. Line r goes through (-2, 2) and (5, 8). Line s goes through (-8, 7) and (-2, 0).

Is r perpendicular to s?

2. An equation for line v is y = -3/8x + 5/8. An equation for line w is 8w + 3y = 10.

Is v perpendicular to w?
Example 3

Decide whether and are perpendicular.

What is the product of the slopes of perpendicular lines? ______

Are these lines perpendicular? ______

Example 4

Decide whether these lines are perpendicular.

line h: line j:

What is the product of the slopes of perpendicular lines? ______

Are these lines perpendicular? ______

Example 5

Decide whether these lines are perpendicular.

line r: line s:

What is the product of the slopes of perpendicular lines? ______

Are these lines perpendicular? ______

Question: How can you figure out if you have the right slope when you are finding the equation of a perpendicular line?

The slope of one of two perpendicular lines is a/b/. What is the slope of the other line?

Example 6

Line l1 has equation y = -2x +1. Find an equation for the line, l2 that passes through point (4, 0) and is perpendicular to l1.

What is the slope of l1? ______

What form is l1 written in? ______

What does the slope of l2 need to be if they are perpendicular? ______

With the point known (4, 0) , (it is in the original question), and the slope known for l2 , Can you find the y-intercept, b, of the perpendicular line?

x = ______

y = ______

m = ______

b = ______

What is the equation of the perpendicular line?