5 5 Indirect Proof triangle inequality exterior angle

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5 -5 Indirect Proof, triangle inequality, exterior angle inequality, Hinge Theorem and its converse

5 -5 Indirect Proof, triangle inequality, exterior angle inequality, Hinge Theorem and its converse Anna Chang T 2

Angle-Side Relationships in Triangles �The side that is opposite to the smallest angle will

Angle-Side Relationships in Triangles �The 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

Examples Smallest to largest <B, <A, <C

Examples Smallest to largest <B, <A, <C

Indirect Proof �At first, you assume that the statement is false and then show

Indirect Proof �At first, you assume that the statement is false and then show that this causes a contradiction with facts �Also called a proof by contradiction Writing 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

Examples

Q p R

Q p R

Triangle inequality �For the sum of the length of the two shorter sides must

Triangle inequality �For the sum of the length of the two shorter sides must always be longer than the third side (triangle)

Examples

Examples

Exterior angle inequality �supplementary to the adjacent interior angle and it is greater than

Exterior angle inequality �supplementary to the adjacent interior angle and it is greater than either of the non adjacent interior angles.

Examples

Examples

Hinge Theorem �If the two sides of two triangles are congruent but the third

Hinge Theorem �If 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 Theorem �If two sides of one triangle are congruent to two

Converse of Hinge Theorem �If 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

Examples

Section 5 3 Triangle Inequalities Exterior Angle InequalitySection 5 3 Triangle Inequalities Exterior Angle Inequality
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7 2 Exterior Angle Theorem 7 2 Exterior7 2 Exterior Angle Theorem 7 2 Exterior
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1 What is the Triangle Exterior Angle Theorem1 What is the Triangle Exterior Angle Theorem
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Proof of Work Proof of Stake Proof ofProof of Work Proof of Stake Proof of
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Indirect Proofs Steps for Writing an Indirect ProofIndirect Proofs Steps for Writing an Indirect Proof