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
Parallel LinesSkew Lines
Parallel Planes
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
Postulate / What does it say? / What does it LOOK likeParallel Postulate
Perpendicular Postulate
Constructing Perpendicular Lines
Name of Figure / Describe It / Sketch ItTransversal
Corresponding Angles
Alternate Exterior Angles
Alternate Interior Angles
Consecutive Interior Angles
(same side interior)
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 ______
Two-Column ProofParagraph Proof
Flow Proof
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.
Theorem / Explanation / SketchCongruent Linear Pair Theorem
Adjacent Complementary Angle Theorem
Perpendicular Lines Intersection Theorem
Example 1-Method 1
Given: AB = CD
Prove: AC = BD
Statements / Reasons1. / 1.
2. / 2.
3. / 3.
4. / 4.
Example 1-Method 2
Given: AB = CD
Prove: AC = BD
Two-Column Proof
Statements / Reasons1. 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
Statements / Reasons1. / 1.
2. / 2.
3. / 3.
4. / 4.
5. / 5.
6. / 6.
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
Postulate/TheoremDescriptive Name / Description / Sketch
Corresponding Angles Postulate
Alternate Interior Angles Theorem
Consecutive Interior Angles Theorem
Alternate Exterior Angles Theorem
Perpendicular Transversal Theorem
Example 1
Given: p || q
Prove: mÐ1 + mÐ2 = 180°
Statements / Reasons1. / 1.
2. / 2.
3. / 3.
4. / 4.
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
Postulate or TheoremDescriptive Name / Description / Sketch
Corresponding Angles Converse Postulate
Alternate Interior Angle Converse Theorem
Consecutive Interior Angles Converse Theorem
Alternate Exterior Angles Converse Theorem
Example 1
Given: m p, m q
Prove: p || q
Statements / Reasons1. / 1.
2. / 2.
3. / 3.
4. / 4.
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
Lines Parallel to a Third Line TheoremLines Perpendicular to a Third Line Theorem
Example 1
Given: r || s and s || t
Prove: r || t
Statements / Reasons1. / 1.
2. / 2.
3. / 3.
4. / 4.
5. / 5.
6. / 6.
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 = ------= ------
Slopes of Parallel Lines PostulateExample 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
Slopes of Perpendicular Lines PostulateExample 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
AD
C
B
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
B
C
D
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? ______
Slope of a line / Slope of the perpendicular line / Product of the slopes7
4
-1
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?