CK-12 Geometry Concepts

5.8. Indirect Proof in Algebra and Geometry http://www.ck12.org

Guided Practice

1. Ifnis an integer andn^2 is odd, thennis odd. Prove this is true indirectly.


2. Prove the SSS Inequality Theorem is true by contradiction. (The SSS Inequality Theorem says: “If two sides of
   a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third
   side of the second triangle, then the included angle of the first triangle’s two congruent sides is greater in measure
   than the included angle of the second triangle’s two congruent sides.”)


3. Ifx=3, then 4x+ 16 =17. Prove this statement is true by contradiction.

Answers:

1. First, assume theoppositeof “nis odd.”

niseven.

Now, squarenand see what happens.

Ifnis even, thenn= 2 a, whereais any integer.

n^2 = ( 2 a)^2 = 4 a^2

This means thatn^2 is a multiple of 4. No odd number can be divided evenly by an even number, so thiscontradicts
our assumptionthatnis even. Therefore,nmust be odd ifn^2 is odd.

2. First, assume the opposite of the conclusion.

The included angle of the first triangle islessthanorequalto the included angle of the second triangle.

If the included angles areequal then the two triangles would be congruent by SAS and the third sides would be
congruent by CPCTC. This contradicts the hypothesis of the original statement “the third side of the first triangle is
longer than the third side of the second.” Therefore, the included angle of the first triangle must be larger than the
included angle of the second.

3. 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=3, then 4x+ 16 =17":

Ifx=3, then 4x+ 1 = 17

Take this statement as true and solve forx.

4 x+ 1 = 17

4 x= 16

x= 4

x= 4 contradictsthe given statement thatx=3. Hence, ourassumption is incorrectand 4x+ 16 =17 istrue.

Practice

Prove the following statements true indirectly.

1. Ifnis an integer andn^2 is even, thennis even.


2. Ifm^6 A 6 =m^6 Bin 4 ABC, then 4 ABCis not equilateral.


3. Ifx>3, thenx^2 >9.


4. The base angles of an isosceles triangle are congruent.