http://www.ck12.org Chapter 5. Relationships with Triangles
Example A (Algebra Example)
Ifx=2, then 3x− 56 =10. Prove this statement is true by contradiction.
Remember that in an indirect proof the first thing you do is assume the conclusion of the statement isfalse.In this
case, we will assume theoppositeof "Ifx=2, then 3x− 56 =10":
Ifx=2, then 3x− 5 =10.
Take this statement as true and solve forx.
3 x− 5 = 10
3 x= 15
x= 5Butx= 5 contradictsthe given statement thatx=2. Hence, ourassumption is incorrectand 3x− 56 =10 istrue.
Example B (Geometry Example)
If 4 ABCis isosceles, then the measure of the base angles cannot be 92◦. Prove this indirectly.
Remember, to start assume theoppositeof the conclusion.
The measure of the base anglesare 92◦.
If the base angles are 92◦, then they add up to 184◦. Thiscontradictsthe Triangle Sum Theorem that says the three
angle measures of all triangles add up to 180◦. Therefore, the base angles cannot be 92◦.
Example C (Geometry Example)
If^6 Aand^6 Bare complementary then^6 A≤ 90 ◦. Prove this by contradiction.
Assume theoppositeof the conclusion.
(^6) A> 90 ◦.
Consider first that the measure of^6 Bcannot be negative. So if^6 A> 90 ◦this contradicts the definition of comple-
mentary, which says that two angles are complementary if they add up to 90◦. Therefore,^6 A≤ 90 ◦.
Watch this video for help with the Examples above.
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/52533CK-12 Foundation: Chapter5IndirectProofB
Vocabulary
AnIndirect Proof or Proof by Contradictionis a method of proof where the conclusion from a hypothesis is
assumed to be false (or opposite of what it states) and then a contradiction is reached from the given or deduced
statements.