Download Geometry Fall 2012 Lesson 030 _Proving lines parallel

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Lesson Plan #30
Class: Geometry
HW 30:
Date: Wednesday November 7th, 2012
Topic: Proving lines are parallel?
Aim: How do we prove lines are parallel?
Objectives:
1) Students will be able to prove lines parallel.
Note:
From a previous lesson recall the theorem The measure of an exterior angle of a triangle is greater than
the measure of either nonadjacent interior angle.
Do Now: Examine this proof by contradiction or indirect proof.
Sample Test Question:
1) Which type of proof begins by assuming the opposite
of what you want to prove?
A) Two column
B) Indirect
C) Paragraph
D) Flow
2) If two lines are each parallel to a third line, then
A) They are parallel to each other.
B) They are perpendicular to each other.
C) They are a pair of skew lines.
D) Their relationship can’t be determined.
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Assignment #1:
Go to
http://www.mathopenref.com/transversal.html
Drag one of the two lines cut by the transversal.
What is true of the two lines cut by the
transversal
when the alternate interior angles are congruent?
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Let’s try to prove that “if two coplanar lines are cut by a transversal so that alternate interior angles formed are congruent, then
the two lines are parallel.”
Given:

and
are cut
by transversal
at
points E and F
respectively.

Prove:

1.
2.
3.
Statements
and
are cut by transversal
at points E and F respectively
Let
not be parallel to
and
intersect at some point
P, forming
4.
5.
6.
7.
But
Reasons
1.
Given
2.
3.
Assumption
If coplanar lines are not parallel, then they
are intersecting.
4. The measure of an exterior angle of a
triangle is greater than the measure of either
non adjacent interior angle
5.Given
6. Congruent angles are equal in measure
7. Contradiction in steps 4 and 6, therefore the
assumption in step 2 is false.
Theorem: If two coplanar lines are cut by a transversal, so that alternate interior angles
formed are congruent, then the two lines are parallel.
Let’s examine the two interior angles on the same side of the transversal at http://www.mathopenref.com/transversal.html
What can you tell about those two angles?
Let’s see if we can prove that if two lines are cut by a transversal so that interior angles on the same side of the transversal are
supplementary, then the lines are parallel. Finish the proof
Given:
intersects
and
.
is the supplement of
Prove:
1.
2.
Statements
intersects
and
.
and
form a linear pair
3.
4.
is the supplement of
5.
6
7.
is the supplement of
Reasons
1.Given
2. A linear pair of angles are two adjacent angles
whose sum is a straight angle (1)
3. A straight angle is an angle whose degree
measure is 180o. (1)
4. Supplementary angles are two angles the sum
of whose degree measures is 180o. (2)
5. Given
6.
7.
Theorem: If two coplanar lines are cut by a transversal so that the interior angles on the
same side of the transversal are supplementary, then the lines are parallel.
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Other ways to prove lines are parallel (presented without proof)
Theorem: If two coplanar lines are cut by a transversal, so that corresponding angles are
congruent, then the two lines are parallel
Theorem: If two lines are perpendicular to the same line, then they are parallel.
Summary of ways to prove lines parallel
1) A pair of alternate interior angles congruent
2) A pair of corresponding angles congruent
3) A pair of interior angles on the same side of the transversal are supplementary
4) Both lines are perpendicular to the same line
5) Both lines are parallel to the same line.
6) If two lines are cut by a transversal forming a pair of alternate exterior angles congruent, then the two lines are parallel
Assignment:
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