1. 6.5 Indirect proof inequalities in one triangle

2. What we will learn Write indirect proofs
List sides and angles of a triangle in order by size
Use Triangle Inequality Thm to find possible side lengths of triangles

3. Ex. 1 writing indirect proof
Given: ∆𝐴𝐵𝐶
Prove: ∆𝐴𝐵𝐶 can have at most one right angle
Step 1:
Assume triangle ABC can have two right angles.
Step 2:
∠𝐶 = 180, so ∠𝐶=0. A triangle cannot have an angle of 0. This contradicts the given of ABC is a triangle.
Step 3:
So the assumption is false, which proves triangle ABC can have at most one right angle
Steps
1. write prove statement with assume and write opposite of what proving
2. find a contradiction
3. write assumption is false, and rewrite prove statement

4. Your practice Given: Triangle QRS and 𝑚∠𝑄+𝑚∠𝑅=90 Prove: 𝑚∠𝑆=90 Step 1:
Assume 𝑚∠𝑆≠90
Step 2:
All angles of a triangle add up to 180. So Q + R + S = S = S = 90 degrees. This contradicts the assumption.
Step 3:
Assumption is false, which means 𝑚∠𝑆=90.

5. Exs. 3 and 4 Ordering sides and angles
List angles in order of smallest to largest
Relative to side lengths
Look at opposite sides
∠𝐿, ∠𝐽, 𝑎𝑛𝑑 ∠𝐾
List sides in order of smallest to largest
Relative to angle measures
Look at opposite angle
𝐷𝐹 , 𝐸𝐹 , 𝑎𝑛𝑑 𝐷𝐸

6. Ex. 5 finding possible side lengths
A triangle has one side of length 14 and another side of length 9. Describe the possible lengths of the third side.
𝑥+9> 𝑥<9+14
−9 − 𝑥<23
𝑥>5
5<𝑥<23
Steps
1. set up two inequalities
Length of any two sides has to be greater than third side
One with shortest side and one with addition of both sides
Third side must be less than addition of two other sides
2. solve each inequality
3. write as single inequality
Put x in middle, small number on left and large number on right, and put in appropriate inequality symbol

7. Your practice
A triangle has one side of length 12 and another side of length 20. Describe the possible lengths of the third side.
𝑥+12>20
−12−12
𝑥>8
𝑥<12+20
𝑥<32
8<𝑥<32