6.5 Inequalities in Triangles and Indirect Proofs

6.5 Pear Deck

Review of Inequalities in Triangles The largest angle in a triangle is opposite the longest side. The shortest side of a triangle is opposite the smallest angle. The two shorter sides of triangle must add up to be longer than the third side.

Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angle of the first is smaller then the included angle of the second, then the third side of the first is shorter than the third side of the second.

Indirect Proof Make a temporary assumption that that desired conclusion is false. 2) Show that this assumption leads to a logical impossibility. 3) Therefore the original statement is true by contradiction.

Example of an Indirect Proof I want to prove that today is not Taco Tuesday. I first assume that today is Taco Tuesday. When I come home on Taco Tuesday, there is usually ground beef defrosting on the kitchen table. Since there was not ground beef defrosting on the kitchen table, my original assumption must be false. Therefore, today is not Taco Tuesday : (

Use an indirect proof to show that a triangle can only have at most one right angle.

Use indirect proof to show that in triangle ABC, if AB= 7 and BC = 8, then CA cannot equal 15.

Use indirect proof to show that if x and y are odd integers, then xy must be odd.