Transcript 3.5 Proving Lines Parallel

3.5 Proving Lines Parallel
Objectives

Recognize angle conditions that occur with
parallel lines

Prove that two lines are parallel based on
given angle relationships
Postulate 3.4
Converse of the Corresponding Angles Postulate
If two lines in a plane are cut by a
transversal so that corresponding angles are
congruent, then the lines are parallel.
Abbreviation: If corr. s are , then lines are ║.
Postulate 3.5
Parallel Postulate
If given a line and a point not on the line,
then there exists exactly one line through
the point that is parallel to the given line.
Theorem 3.5
Converse of the Alternate Exterior Angles Theorem
If two lines in a plane are cut by a transversal
so that a pair of alternate exterior angles is
congruent, then the two lines are parallel.
Abbreviation: If alt ext. s are , then lines are ║.
Theorem 3.6
Converse of the Consecutive Interior Angles Theorem
If two lines in a plane are cut by a transversal
so that a pair of consecutive interior angles is
supplementary, then the lines are parallel.
Abbreviation: If cons. int. s are supp., then lines
are ║.
Proof of the Converse of the Consecutive
Interior Angles Theorem
Given: 4 and 5 are supplementary
Prove: g ║ h
g
6
5
4
h
Paragraph Proof of the Converse of the
Consecutive Interior Angles Theorem
You are given that 4 and 5 are
supplementary. By the Supplement
Theorem, 5 and 6 are also
supplementary because they form a linear
pair. If 2 s are supplementary to the
same , then 4  6. Therefore, by the
Converse of the Corresponding s Angles
Postulate, g and h are parallel.
Theorem 3.7
Converse of the Alternate Interior Angles Theorem
If two lines in a plane are cut by a transversal
so that a pair of alternate interior angles is
congruent, then the lines are parallel.
Abbreviation: If alt. int. s are , then lines are ║.
Proof of the Converse of the Alternate
Interior Angles Theorem
Given: 1  2
Prove: m ║ n
3
m
2
1
n
Two - Column Proof of the Converse of
the Alternate Interior Angles Theorem
Statements:
1.
2.
3.
4.
1  2
2  3
1  3
m║n
Reasons:
1.
2.
3.
4.
Given
Vertical Angles are 
Transitive prop.
If corres. s are ,
then lines are ║
Theorem 3.8
In a plane, if two lines are perpendicular to
the same line, then they are parallel.
Abbreviation: If 2 lines are ┴ to the same line,
then the lines are ║.
Example 1:
Determine which lines,
if any, are parallel.
consecutive
interior angles are supplementary. So,
consecutive
interior angles are not supplementary. So, c is not parallel
to a or b.
Answer:
Your Turn:
Determine which lines, if any, are parallel.
Answer:
Example 2:
ALGEBRA Find x and mZYN so that
Explore From the figure, you know that
and
You also know that
are alternate exterior angles.
Example 2:
Plan For line PQ to be parallel to MN, the alternate exterior
angles must be congruent.
Substitute the given angle measures into this equation
and solve for x. Once you know the value of x, use
substitution to find
Solve
Alternate exterior angles
Substitution
Subtract 7x from each side.
Add 25 to each side.
Divide each side by 4.
Example 2:
Original equation
Simplify.
Examine Verify the angle measure by using the value of x to
find
Since
Answer:
Your Turn:
ALGEBRA Find x and mGBA so that
Answer:
Example 3:
Given:
Prove:
Example 3:
Proof:
Statements
;
1.
2.
3.
4.  7 + 6 = 180
Reasons
1. Given
2. Consecutive Interior Thm.
3. Def. of congruent s
4. Def. Suppl. s
5.  4 + 6 = 180
6.  4 and 6 are suppl
5. Substitution
6. Def. Suppl. s
7.
7. If cons. int.
then lines are .
s are suppl.,
Your Turn:
Given: a || b
1  12
Prove: x || y
Your Turn:
Proof:
Statements
1. a || b; 1  12
2. 1  13
3. 13  15
4. 12  15
5. x || y
Reasons
1. Given
2. Corres. s Postulate
3. Vertical s are 
4. Substitution
5. If corres. s are , then lines are ||
Example 4:
Answer:
Your Turn:
Answer: Since the slopes are not equal, r is not parallel to s.
Assignment

Geometry:
Pg. 155 #13 – 31, 34, 35

Pre-AP Geometry:
Pg. 155 #13 – 31, 34 - 39