5-5Indirect Proof, triangle

inequality, exterior angle inequality, Hinge Theorem

and its converseAnna Chang

T2

Angle-Side Relationships in TrianglesThe side that is opposite to the smallest

angle will be always the shortest side and the side that is opposite to the largest angle will be the longest side

ExamplesSmallest to largest<B, <A, <C

Indirect ProofAt first, you assume that the statement is

false and then show that this causes a contradiction with facts

Also called a proof by contradictionWriting an Indirect Proof

1. Identify the conjecture to be proven

2. Assume the opposite of the conclusion is true

3. Use direct reasoning to show that the asssumption leads to a contradiction

4. Conclude that since the assumption is false, the original conjecture must be true

Examples

p

Q

R

Triangle inequalityFor the sum of the length of the two shorter

sides must always be longer than the third side (triangle)

Examples

Exterior angle inequalitysupplementary to the adjacent interior angle

and it is greater than either of the non adjacent interior angles.

Examples

Hinge TheoremIf the two sides of two triangles are

congruent but the third side is not congruent then the triangle with the longer side will have a larger included angle.

Converse of Hinge TheoremIf two sides of one triangle are congruent to

two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side.

Examples