Transcript 3-3 Proving Lines Parallel


Students will be able to
◦ Determine whether two lines are parallel
◦ Write flow proofs
◦ Define and apply the converse of the theorems from
the previous section

You can use certain angle pairs to determine
if two lines are parallel




What is the corresponding angles theorem?
If a transversal intersects two parallel lines,
then corresponding angles are congruent
What is the converse of the corresponding
angles theorem?
If two lines and a transversal form congruent
corresponding angles, then the lines are
parallel

Which lines are parallel if <6 ≅ <7?
◦ m || l

Which lines are parallel if <4 ≅ <6
◦ a || b

If two lines and a transversal form alternate
interior angles that are congruent, then the
two lines are parallel

If two lines and a transversal form same side
interior angles that are supplementary, then
the two lines are parallel.

If two lines and a transversal form alternate
exterior angles that are congruent, then the
two lines are parallel.




If corresponding angles are congruent, then
the lines are parallel
If alternate interior lines are congruent, then
the lines are parallel
If alternate exterior lines are congruent, then
the lines are parallel
If same side interior angles are
supplementary, then the lines are parallel



In order to use the theorems relating to
parallel lines, you must first prove the lines
are parallel if it is not given/stated in the
problem.
Even if lines appear to be parallel, you cannot
assume they are parallel
Always assume diagrams are NOT drawn to
scale, unless otherwise stated



Third way to write a proof
In a flow proof, arrows show the flow, or the
logical connections, between statements.
Reasons are written below the statements


Given: <4 ≅ <6
Prove: l || m
<4 ≅ <6
Given
<2 ≅ <6
<2 ≅ <4
Vert. <s are ≅
Trans. Prop
of ≅
L || m
Converse of
Corresponding
Angles Thm.
*You cannot use the Corresponding Angles Thm to say
<2 ≅ <6 because we do not know if the lines are
parallel




Given: m<5 = 40, m<2 = 140
Prove: a || b
Start with what you know
◦ The given statement
◦ What you can conclude
from your picture.
What you need to know
◦ Which theorem you can use to show a||b

Given: m<5 = 40, m<2 = 140

Prove: a || b
<5 = 40
Given
<5 and <2 are
Supp. <s
Def. of Supp. <s
<2 = 140
Given
<5 and <2 are
Same side Interior
Angles
Def. of Same
Side Interior <s
a || b
Converse of
Same Side
Int. <s Thm

You now have four ways to prove if two lines
are parallel



What is the value of x for which a || b?
Work backwards. What must be true of the
given angles for a and b to be parallel?
How are the angles related?
◦ Same side interior
◦ Therefore, they must add to be 180



What is the value of x for which a || b?
Work backwards. What must be true of the
given angles for a and b to be parallel?
How are the angles related?
◦ Corresponding Angles
◦ Therefore, the angles are congruent



Pg. 160 – 162
# 7 – 16, 21 – 24, 28, 32
16 Problems