Download Parallel Lines, Part 2

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2:1:32:Parallel Lines, Part 2
TITLE OF LESSON
Geometry Unit 1 Lesson 32– Parallel Lines, Part 2
Prove it! What’s on the outside? What’s on the inside? Of Geometry
TIME ESTIMATE FOR THIS LESSON
One class period
ALIGNMENT WITH STANDARDS
California – Geometry
7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties
of quadrilaterals, and the properties of circles.
MATERIALS
None
LESSON OBJECTIVES
To introduce
• Proofs related to parallel lines
FOCUS AND MOTIVATE STUDENTS
1) Homework Check – Stamp/initial complete homework assignments. Pass back graded work and have
students place in the appropriate sections of their binders.
2) Agenda – Have students copy the agenda.
3) Review – (7 minutes) Go over yesterday’s homework. Have students describe in their own words what the
parallel postulate means. Next, have students present the 5 situations they came up with in which reality
offers us a good approximation of parallel lines. Discuss why they chose the examples they chose and
whether or not they work. Collect homework.
ACTIVITIES – INDIVIDUAL AND GROUP
1.
Theorems – (2 minutes) Ask students to recall that a theorem can be written as IF (some statement), THEN
(some other statement). For instance we can make statements such as (Write these on the board.):
IF I am at least 16, THEN I am old enough to vote.
IF this month is July, THEN next month is August.
IF the sun is up, THEN it is daytime. (Unless you live way up north or way down south.)
IF I have 15 dollars, THEN I can buy a CD.
Ask students to try to come up with some other statements of this sort. How about some statements from
geometry?
IF two parallel lines are cut by a transversal THEN alternate interior angles are equal.
IF two parallel lines are cut by a transversal THEN alternate exterior angles are equal.
IF two parallel lines are cut by a transversal THEN the consecutive interior angles are supplementary.
IF two parallel lines are cut by a transversal THEN corresponding angles are equal.
2.
Converse Theorems – (7 minutes) When we reverse the statements within the theorems, we get what we call the
converse of the theorem. For instance if we had said:
IF today is Tuesday THEN yesterday was Monday.
the converse would be:
1
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2:1:32:Parallel Lines, Part 2
IF yesterday was Monday THEN today is Tuesday.
What would the converse be of:
IF tomorrow is Friday THEN the day after tomorrow is Saturday.
(IF the day after tomorrow is Saturday THEN today is Friday.)
Working in the notes section of their binders, ask students to create the converse of the statements in the # 1
above. Do not include the geometric statements. You’ll do those in the next section.
3.
4.
Statement/Converse – (5 minutes) Now, let’s look at the four statements from yesterday and have students state
their converses. As they get them, have one student record each of these on the board, both the statement and
the converse.
a.
Statement
If two parallel lines are cut by a transversal then alternate interior angles are equal.
Converse
(If alternate interior angles are equal then the lines cut by a transversal are parallel.)
b.
Statement
If two parallel lines are cut by a transversal then alternate exterior angles are equal.
Converse
(If alternate exterior angles are equal then the lines cut by a transversal are parallel.)
c.
Statement
If two parallel lines are cut by a transversal then the consecutive interior angles are supplementary.
Converse
(If consecutive interior angles are supplementary then the lines cut by a transversal are parallel.)
d.
Statement
If two parallel lines are cut by a transversal then corresponding angles are equal.
Converse
(If corresponding angles are equal then the lines cut by a transversal are parallel.)
Demonstration – (5 minutes) Have students form the life-size ‘transversal with two parallel lines’ from class
yesterday. You can have them tie string to desks, etc, or just have six students hold the three pieces of string.
Let the students direct the formation of these lines as much as possible. Start by simply suggesting that this is
what you’d like them to do without giving any directions. Only step in if you have to.
For each of the statements in #3 above, ask two students to demonstrate by standing in the appropriate angles
formed by the string. Have students demonstrate all sets of angles for each. Then, ask for volunteers to form the
converse of each of these statements.
5.
Visual Aid – (2 Minutes) Draw the figure from yesterday on the board, or have a student do it. Have students
point out in the drawing which angles are covered in each of the statements in #3 above.
A
1
2
B
4
5
3
6
C
8
7
Figure 32.1 Parallel Lines
Cut by a Transversal
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6.
2:1:32:Parallel Lines, Part 2
Discussion – (20 minutes) Write the following on the board:
If two parallel lines are cut by a transversal and m∠6 is 2x+20 and m∠2 is x+30 what is x?
Ask if anyone can answer this question. They’re going to have to go back to their algebra, but they should pretty
easily be able to set up the equation. What do we know? (We know that angles 2 and 6 are corresponding angles
and are, therefore, equal. Thus, we know that 2x + 20 = x + 30.) What do we have to do to find out the answer?
(We have to know how to solve the equation for x.)
To do this, we have to get all x’s on one side of the equation, so we’re left with only numbers on the other side.
As they should recall, that means we have to subtract x from each side of the equation, leaving us with x + 20 =
30. Then, we have to subtract 20 from each side of the equation, leaving us with x = 10.
∠6
= ∠2
2x + 20 = x + 30
-x
-x
x + 20 = 30
- 20 = - 20
x = 30
(Since we have to do the same thing to both sides to keep the sides equal.)
(Ditto.)
So we know that x = 10. And that’s all we know for now. But, we can use that! Using this information,
continue with the following questions. Try to elicit the answers for each. If no one answers, give some hints. If
all else fails explain the answer.
What is m∠2? (x + 30 = 10 = 30 = 40)
What is m∠6? (2x + 20 = 20 + 20 = 40) (Or, because they’re corresponding angles, and therefore equal,
m∠6 equals m∠2 and therefore m∠6 equals 40 also.)
If the m∠4 is 2x + 20 and the m∠5 is x + 40 what is x?
(Since ∠4 and ∠5 are consecutive interior angles, they sum to 180°. Therefore 2x + 20 + x + 40 = 180.
This implies that 3x + 60 = 180. If we subtract 60 from each side we get 3x = 120. If we divide both sides
by 3 we get x = 40)
What is the measure of angle 4?
(2 * 40 + 20 = 100)
What is the measure of angle 5?
(40 + 40 = 80)
7.
Homework – (2 minutes) Have students copy down their homework.
HOMEWORK
1) If two parallel lines are cut by a transversal and m∠2 is 3x + 5 and m∠8 is x + 10 what is x?
What is the size of angle 2?
What is the size of angle 8?
2) If m∠1 is 3x + 50 and m∠7 is x + 90 what is x?
What is the measure of angle 1?
What is the measure of angle 7?
3) Create outline for final project: proof scramble. Due Lesson 35.
GROUP ROLES
None
DOCUMENTATION FOR PORTFOLIO
3
© 2001 ESubjects Inc. All rights reserved.
Prove It
How do we create truth?
2:1:32:Parallel Lines, Part 2
Homework and notes during discussion and group work should be added to binders.
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© 2001 ESubjects Inc. All rights reserved.