1. Lesson 5 – 4 Indirect Proof
Geometry
Lesson 5 – 4
Indirect Proof
Objective:
Write indirect algebraic proofs.
Write indirect geometric proofs.

2. Indirect reasoning Indirect reasoning
Reasoning where you assume a conclusion was false and then showing that this assumption led to a contradiction.
Indirect proof (proof by contradiction)
Assume that what you are trying to prove is false and then by showing this assumption to be logically impossible, you prove your assumption false and the original conclusion true.

3. Indirect Proof Step 1 Step 2 Step 3
Identify the conclusion you are asked to prove. Make the assumption that this conclusion is false by assuming that the opposite is true.
Step 2
Use logical reasoning to show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem, or corollary.
Step 3
Point out that since the assumption leads to a contradiction, the original conclusion, what you were asked to prove, must be true.

4. State the Assumption for starting an Indirect Proof
If 6 is a facto of n, then 2 is a factor of n.
is an obtuse angle.
2 is not a factor of n
Don’t just
Put “not obtuse”
is an acute or right angle.

5. State the Assumption for starting an Indirect Proof
x > 5
Points J, K, and L are collinear
Triangle XYZ is an equilateral triangle.
x < 5
Points J, K, and L are noncollinear
Triangle XYZ is a scalene or isosceles triangle.

6. Write an Indirect Proof
Write an indirect proof to show that if
–3x + 4 > 16, then x < -4.
Given: -3x + 4 > 16
Prove: x < -4
Step 1: negation of conclusion
Assume x > -4 or x = -4.
Step 2: Show your assumption is wrong. x
-3x+4 -4 -3 -2 -1 16 13 10 7
Step 3: Point out the wrong assumption and state original is true
When x > -4 or x = -4 –3x + 4 < 16
This is a contradiction, therefore x < -14 must be true.

7. Write an indirect proof: If –c is positive, then c is negative
Given: -c is positive
Prove: c is negative
Assume c is positive. c -c 5 4 3 -5 -4 -3
When c is positive –c is negative.
This is a contradiction so therefore c is negative.

8. Write an indirect proof to show that if x + 2 is an even integer, then x is an even integer.
Given: x + 2 is an even integer
Prove: x is an even integer
Assume that x is an odd integer. An even integer can be represented by 2k so let x be an odd integer x= 2k + 1
x + 2 = (2k + 1) + 2
= 2k
= 2(k + 1) + 1
x + 2=2(k + 1) + 1 is of the form of an odd number.
This is a contradiction and therefore x is an even
integer.

9. If an angle is an exterior angle of a triangle, prove that its measure is greater than the measures of either of its corresponding remote interior angles.
Step 1: Draw a picture and identify
given,prove, and assume.
Given:
Prove:
Assume
Cont…

10. Step 2: Show that your assumption leads to a
contradiction.
Case 1:
Ext. Angle Thm.
This is a contradiction that an angle must be greater than 0.
Case 2:
This is a contradiction to the Exterior Angle Thm.
Step 3: conclusion
Both cases lead to contradictions, therefore
is true.

11. Homework
Pg – 8 all, 12 – 20 E, 46 – 58 E