Accelerated Math I Unit 2 Concept: Triangular Inequalities The Hinge Theorem

Accelerated Math I Unit 2 Concept: Triangular Inequalities The Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second

Accelerated Math I Unit 2 Concept: Triangular Inequalities The Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second

Accelerated Math I Unit 2 Concept: Triangular Inequalities The Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second

Accelerated Math I Unit 2 Concept: Triangular Inequalities

The Hinge Theorem Converse If two sides of a triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second

Accelerated Math I Unit 2 Concept: Triangular Inequalities The Hinge Theorem Converse If two sides of a triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second

Accelerated Math I Unit 2 Concept: Triangular Inequalities The Hinge Theorem Converse If two sides of a triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second

Example of Hinge Accelerated Math I Unit 2 Concept: Triangular Inequalities

Solution Accelerated Math I Unit 2 Concept: Triangular Inequalities

Example Accelerated Math I Unit 2 Concept: Triangular Inequalities

Solution Accelerated Math I Unit 2 Concept: Triangular Inequalities

Example Accelerated Math I Unit 2 Concept: Triangular Inequalities

Solution Accelerated Math I Unit 2 Concept: Triangular Inequalities

Indirect Proof Start by making an assumption that the conclusion is false. By showing that this assumption leads to a logical impossibility, you prove the original statement true by contradiction Accelerated Math I Unit 2 Concept: Triangular Inequalities

Steps in an Indirect Proof: Assume that the opposite of what you are trying to prove is true. From this assumption, see what conclusions can be drawn. These conclusions must be based upon the assumption and the use of valid statements. Search for a conclusion that you know is false because it contradicts given or known information. Oftentimes you will be contradicting a piece of GIVEN information. Accelerated Math I Unit 2 Concept: Triangular Inequalities

Since your assumption leads to a false conclusion, the assumption must be false. If the assumption (which is the opposite of what you are trying to prove) is false, then you will know that what you are trying to prove must be true. Accelerated Math I Unit 2 Concept: Triangular Inequalities

Recognize When Indirect Proof is needed Accelerated Math I Unit 2 Concept: Triangular Inequalities Generally, the word "not" or the presence of a "not symbol" (such as the not equal sign ) in a problem indicates a need for an Indirect Proof.

Proof by Contradiction is also known as reductio ad absurdum which from Latin means reduced to an absurdity Accelerated Math I Unit 2 Concept: Triangular Inequalities

Example Accelerated Math I Unit 2 Concept: Triangular Inequalities