Due MON 12/9
5.1 Indirect Proof
p. 213 # 6-8,11-15
5.2 Proving That Lines are Parallel
p. 219 # 10,12,15,19, 22-27
When disproving all options except for the one you want, you are doing an
indirect proof. Since all of the other possibilities are incorrect, you are left
with one correct option.
Usually an indirect proof is used when proving things are not true.
The steps are:
A. List all possibilities for the conclusion.
B. Assume that all possibilities you do not want to prove are true and
use it as a given. (Assume that the negation of the desired conclusion
is correct.)
C. Write a chain of reasoning until you reach an impossibility. (A
contradiction of:
(a) the given information
(b) a known fact (an already proved theorem, a definition, a
postulate, etc.)
D. State the remaining possibility as the desired conclusion.
B
Given : DB is an altitude of DABC;
DB is not a median of DABC
Pr ove : AB @/ CB
A
D
C
A. List all possibilities for the conclusion.
Either AB @/ CB or AB @ CB
B. Assume that all possibilities you do not want to prove are true and
use it as a given. (Assume that the negation of the desired conclusion
is correct.)
Assume AB @ CB
B
Given : DB is an altitude of DABC;
DB is not a median of DABC
Pr ove : AB @/ CB
A
D
C
C. Write a chain of reasoning until you reach an impossibility. (A contradiction of:
(a) the given information
(b) a known fact (an already proved theorem, a definition, a postulate, etc.)
Statements
1. AB @ CB
Reasons
1. Assumed True
2. Given
2. DB is an alt. of DABC
3. In an isos. D the alt. is the median
3. DB is a median of DABC
4. DB is not a median of DABC 4. Given
D. State the remaining possibility as the desired conclusion.
5. AB @/ CB
5. Steps 3 and 4 contradict each other,
\ the assumption in step 1 is false.
D
Given : DA^AB; DA^AC; ÐB @/ ÐC
Pr ove : AB @/ AC
m
B
What assumption should we
Begin with?
AB @ AC
A
C
Adjacent
Exterior
Angle
Interior Angle
Interior Angles
Remote
Interior Angle
Remote
Remote
Interior Angle
Adjacent
Interior Angle
Interior Angle
Exterior Angle
Exterior Angle
Adjacent
Interior Angle
Note: for the remainder of this presentation,
the angle symbol ( Ð ) will appear as .
This is the Babylonio-Sumerian symbol for angle.
No, not really.
just a formatting bug I can’t figure out…
A
D
B
C
M Ð A + mÐ B + mÐACB = 180
ACB and ACD form a straight 
180 = mACB + mACD
mA + mB + mACB = mACB + mACD
mA + mB = mACD
mACD > mA and mACD > mB
The measure of an exterior angle is greater than the
measure of each remoter interior angle.
The measure of an exterior angle equals the sum of the
measures of the remoter interior angle.
t
4
6
m
n
If 2 lines are cut by a transversal such that 2 alternate interior angles
are congruent, then the lines are parallel.
Alt. int. s  || lines
If 4  6 then m || n.
t
4
6
m
n
Given : Ð4 @ Ð6
Pr ove : m || n
/
Either m || n or m || n
/
Assume m || n is true
t
4 is an ext. .
6 is a remote int.
4 > 6
4
This contradicts the given that 4 = 6
6
n
m
 M || n
t
1
7
m
n
If 2 lines are cut by a transversal such that 2 alternate exterior angles
are congruent, then the lines are parallel.
Alt. ext. s  || lines
If 1  7 then m || n.
t
1
3
5
7
m
n
Given : Ð1@ Ð7
Pr ove : m || n
1  7
3  1
7  5
3  5
m || n
Alt. int. s  || lines
t
2
6
m
n
If 2 lines are cut by a transversal such that 2 corresponding angles are
congruent, then the lines are parallel.
Corr. s  || lines
If 2  6 then m || n.
t
2
4
6
m
n
Given : Ð2 @ Ð6
Pr ove : m || n
2  6
4  2
4  6
m || n
Alt. int. s  || lines
t
4
5
m
n
If 2 lines are cut by a transversal such that 2 same side interior angles
are supplementary, then the lines are parallel.
Same side int. s supp.  || lines
If 4  5 then m || n.
t
4 3
5
m
n
Given : Ð4 is supp. Ð5
Pr ove : m || n
4 is supp. 5
4 and 3 form a straight 
4 and 3 are supp.
5  3
m || n
Alt. int. s  || lines
t
1
8
m
n
If 2 lines are cut by a transversal such that 2 same side exterior angles
are supplementary, then the lines are parallel.
Same side ext. s supp.  || lines
If 1  8 then m || n.
t
1
4
m
n
8
Given : Ð1 is supp. Ð8
Pr ove : m || n
1 is supp. 8
1 and 4 form a straight 
1 and 4 are supp.
4  8
m || n
corr. s  || lines
a
b
c
If 2 coplanar lines are perpendicular to a third line, then they are parallel.
Given: a  c and b  c
Prove: a || b
corr. s  || lines