Download Geometry Fall 2015 Lesson 032 _Properties of Parallel Lines

1
Lesson Plan #32
Class: Geometry
Date: Wednesday December 2nd, 2015
Topic: Properties of parallel lines (continued)
Aim: How do we use the properties of parallel lines?
Objectives:
Students will be able to use the properties of parallel lines.
HW #32:
Page 80 #’s 1-10 , classroom exercises
Recall the previously stated postulate “Through a
given point not on a given line, there exists one and
only one line parallel to the given line.”
Do Now:
Note:
R
B
1) Construct a line through point R that is parallel to AB . Justify!
2)
A
3)
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Assignment #1:
1) Write the converse of the statement “If I live in New York City, then I live in New York State”
2) If a statement is true, is the converse of that statement always true?
3
Write the converse of the statement “If two lines are cut by a transversal so that alternate interior angles are congruent, then
the two lines are parallel”
4) Is the converse of the statement in (3) true or false?
2
Let’s see if we can prove the converse of the theorem “If two planar lines are cut by a transversal so that alternate interior angles
are congruent, then the two lines are parallel”, namely “If two parallel lines are cut by a transversal, then the
alternate interior angles formed are congruent.” Let’s try a proof by contradiction using a narrative proof.
Given:
Prove:
2
1
4
m
3
Let’s try a proof by contradiction:
5
6
7
8
n
Plan: Assume
Then draw a line through the vertex of angle 4 so it that it forms an angle
congruent to
. This would lead to congruent alternate interior angles, meaning
this new line is parallel to line n. But we already had one line (m) parallel to the
given line (n) through a point not on the line.
But this contradicts our previously stated postulate which states through a point not
on a given line, there is one and only one line parallel to the given line.
2
1
4
m
3
5
6
7
8
Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles
formed are congruent.
Theorems that are converses of other previously stated theorems (presented without proof)



If two parallel lines are cut by a transversal, then the corresponding angles are congruent
If two parallel lines are cut by a transversal, then the two interior angles on the same side of the
transversal are supplementary
If a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
Assignment #2:
Assignment #3:
n
3
Assignment #4: In your groups, complete the sample test questions. Do question 8 first, then 20, then come back to #1
4
5
6
Sample Test Questions:
1)
2)
3)
4)
5)
6)
7
Statement and reason version of earlier proof.
Given:
Prove:
Let’s try a proof by contradiction:
Reasons
Statements
1. m || n
1. Given
2.  4 not congruent to  6
2. Assumption
3. Draw line k through point x creating
 9  6
3. A line may be drawn through a point
4. k || n
5.  4  6
4. If 2 coplanar lines are cut by a transversal
forming congruent alternate interior
angles, then the lines are parallel (3)
5. Contradiction in statements 1 and 4.
Through a point (x) not on a line (n), there can be
one and only one line parallel to the given line (n).