Download Proving Lines Parallel

Geometry (Holt 3-3)
K.Santos
Converse of the Corresponding
Angles Postulate (3-3-1)
If two lines and a transversal form corresponding angles
that are congruent, then the two lines are parallel.
Given: <1 ≅<2
Then: a || b
a
1
2
b
Parallel Postulate (3-3-2)
Through any point P not on line m, there is exactly one
line parallel to line m
P
m
Converse of the Alternate Interior
Angles Theorem (3-3-3)
If two lines and a transversal form alternate interior
angles that are congruent, then the two lines are parallel.
a
b
Given: <1 ≅<2
Then: a || b
1
2
Converse of the Alternate Exterior
Angles Theorem(3-3-4)
If two lines and a transversal form alternate exterior
angles that are congruent, then the two lines are parallel.
a
1
Given: <1 ≅<2
Then: a || b
2
b
Converse of the Same-Side Interior
Angles Theorem
If two lines and a transversal form same-side interior
angles that are supplementary, then the two lines are
parallel.
Given: <1 𝑎𝑛𝑑<2 are supplementary
Then: a || b
a
1
b
2
Converse of the Same-Side Exterior
Angles Theorem (3-3-5)
If two lines and a transversal form same-side exterior
angles that are supplementary, then the two lines are
parallel.
Given: <1 𝑎𝑛𝑑<2 are supplementary
Then: a || b
1
a
b
2
Summary
To prove two lines are parallel show one pair of…
corresponding angles
are congruent
alternate interior angles
are congruent
alternate exterior angles
are congruent
same-side interior angles
are supplementary
same-side exterior angles
are supplementary
Example
Use the theorems and given information to show that r||s.
r
s
3
5 6
7 8
1 2
4
<4 ≅<8
<4 & <8 are congruent corresponding angles, so the lines are parallel by
the converse of the corresponding angles postulate
<3 and <5 are supplementary
<3 and <5 same side interior angles that are supplementary, so the lines
are parallel by the converse of the same-side interior angles theorem
Example
Find the value of x for which m || n.
Alternate interior angles would be
congruent if lines were parallel.
m
14 + 3x
n
14 + 3x = 5x – 66
3x = 5x -80
-2x = -80
x = 40
5x -66
Example
Show lines m and n are parallel.
m
m<3 = (4x – 80)° and m<7 =(3x – 50)°
and x = 30.
n
3
7
m<3 = 4(30) - 80= 40°
m<7 = 3(30) – 50 = 40°
This makes sense since these are corresponding angles and
they are congruent. So the lines are parallel by the converse
of the corresponding angles theorem.
Proof of Alternate Interior Angles
Theorem
Given: a || b
Prove: <1 ≅<3
Statements
1. a || b
2. <1 ≅<4
3. <3 ≅<4
4. <1≅ < 3
4
a
b
3
1
Reasons
1. Given
2. If lines are parallel then
corresponding angles are
congruent
3. Vertical angles are congruent
4. Transitive Property (2, 3)