Download Math 53 Winter Q09 2.1 The Parallel Postulate and Special Angles

Math 53 "Winter ’09"
2.1 "The Parallel Postulate and Special Angles"
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Objectives:
*
Develop more properties of parallel and perpendicular lines and the angles associated with them.
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Key Concepts:
Perpendicular Planes
Parallel Lines/Planes
Parallel Postulate
Transversal
Interior/Exterior/Corresponding/Alternate Interior/Alternate Exterior Angles
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Preliminaries:
De…nition:
"Perpendicular Lines"
kPerpendicular lines are two lines that meet to form congruent adjacent anglesk
Construction 6:
[Constructing the line that is perpendicular to a given line from a point not on the given line]
Given:
A line l and a point P not on l
!
Construct:
PQ ? l
The term perpendicular includes line-ray, line-plane, and plane-plane relationship.
Page: 1
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
2.1
We obtain the following Theorem:
Theorem 2.1.1:
kFrom a point not on a given line, there is exactly one line perpendicular to the given linek
Parallel Lines
De…nition:
"Parallel Lines"
kParallel lines are lines in the same plane that do not intersectk
Question:
If two parallel planes M and N are intersected by a third plane G.
How must the lines of intersection a and b be related?
Euclidean Geometry
The type of geometry found in this Elementary Geometry textbook is known as Euclidean geometry.
In this geometry, a plane is ‡at, two-dimensional surface in which the line segment joining any two
points of the plane lies entirely within the plane.
Postulate 10:
(Parallel Postulate)
Through a point not on a line, exactly one line is parallel to the given line
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Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
2.1
A transversal is a line that intersects two (or more) other lines at distinct points; all of the lines lie in the same plane.
t
1
2
3 4
r
s
5
7
6
8
Using the picture above, we obtain the following
Angles between r and s are interior angles
Angles outside r and s are exterior angles
Two angles that lie in the same relative positions (such as above and left) are called corresponding angles
Two interior angles that have di¤erent vertices and lie on opposite sides of the transversal are alternate interior.
Two exterior angles that have di¤erent vertices and lie on opposite sides of the transversal are alternate exterior.
Parallel Lines and Congruent Angles
Now we are going to consider two parallel lines cut by a transversal .
Postulate 11:
kIf two parallel lines are cut by a transversal, then the corresponding angles are congruentk
Example 1:
If l km and m 6 1 = 110 . Find:
1
3
5
7
6
8
2
4
a) m6 2
b) m6 5
c) m6 4
d)
m6 8
l
m
v
Page: 3
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
2.1
Theorem 2.1.2:
kIf two parallel lines are cut by a transversal, then the alternative interior angles are congruentk
Example 2: (Proof of Theorem 2.1.2)
Given:
Prove:
l km and transversal v
3=6 6
6
PROOF
Statements
Reasons
1
3
5
7
2
4
6
l
m
8
v
Theorem 2.1.3:
kIf two parallel lines are cut by a transversal, then the alternative exterior angles are congruentk
Parallel Lines and Supplementary Angles
Theorem 2.1.4:
kIf two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementaryk
Example 3: (Proof of Theorem 2.1.4)
!
!
!
Given:
T V W Y with transversal RS
Prove:
6
1 and
6
3 are supplementary
PROOF
Statements
Reasons
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Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
2.1
Theorem 2.1.5:
kIf two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementaryk
Example 4:
!
!
!
Given:
T V W Y with transversal RS
m 6 RU V = (x + 4) (x
m 6 W XS = x2
Find:
3)
3
x
Example 5: (Exercise 12)
Given:
AD BC , AB DC
m 6 A = 92
Find:
a)
m6 B
A
b)
m6 C
c)
m6
D
B
C
D
WARNING!
Do not assume properties from drawings that are not given.
For example, from the drawing above it seems that angles are right angles (90 ) but they are not!!!!
Page: 5
Bibiana Lopez
Elementary Geometry by Alexander and Koeberlein
2.1
Example 6: (Exercise 17)
Given:
m kn
and transversal k
m 6 3 = 6x + y
m 6 5 = 8x + 2y
m 6 6 = 4x + 7y
Find:
x; y
and
m6 7
Page: 6
Bibiana Lopez