Anna Chang T2. Angle-Side Relationships in Triangles The side that is opposite to the smallest angle...
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Transcript of Anna Chang T2. Angle-Side Relationships in Triangles The side that is opposite to the smallest angle...
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