Download Geometry 2 - Proving Parallel Lines Transversals_1

Math 20-2
Geometry: Lesson #2
Proving Properties of Parallel Lines & Transversals
Objective: By the end of this lesson, you should be able to:
- Prove, using deductive reasoning, properties of angles formed by transversals and
parallel lines.
- Identify and correct errors in a given proof of a property that involves angles.
Last class, we made conjectures about the relationships between angles formed by parallel lines
and a transversal, using inductive reasoning. Remember: Inductive reasoning does not prove that
something is true. This lesson, we will be using deductive reasoning to prove the angle
properties.
Vocabulary:

Supplementary angles –
The type of proof that we will use most often in this unit is called the Two-Column Proof. A
two-column proof looks like this:
Statement
Reason
-
-
True mathematical
(geometric) facts
Reason that you know
each statement is true
The “Reasons” in these proofs can be the reasons we gave in the Warm-Up:
-
opposite angles (equal)
-
alternate angles (interior or exterior) (equal)
-
corresponding angles (equal)
-
interior angles (supplementary)
Another common “Reason” is the Transitive Property:
To prove a statement, you may use other properties of the angles formed by parallel lines and a
transversal, but you may not use the property that you are trying to prove. This would be using
circular reasoning, which is not a valid method of proof.
Math 20-2
Geometry: Lesson #2
e.g. 1) Prove that opposite angles are always equal.
Statement
Reason
e.g. 2) Prove that when a transversal intersects parallel lines, the alternate interior angles are
equal.
Statement
Reason
The converse of the statements about angles formed by parallel lines and a transversal are also
true. If:
 alternate angles (interior or exterior) are equal, OR
 corresponding angles are equal, OR
 the interior angles on the same side of the transversal are supplementary
then the lines are parallel.
This means that in a proof, you can use any one of these statements as the reason for stating that
two lines are parallel.
Math 20-2
Geometry: Lesson #2
e.g. 3) In the diagram below, CBD  BDE . Prove that ABE  BED .
C
Statement
Reason
D
B
A
E
e.g. 4) Identify and correct the error in the proof that Z  VWX .
Given: VWX  VXW
WX YZ
V
W
Y
Assignment:
X
Z
p. 78-82 #1-3, 8, 12, 16, 19
Statement
Reason
VWX  VXW
Given
VXW  Z
Alternate interior angles
Z  VWX
Transitive property