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Warm-up Solve: 1) 2x + 1+4x +4x-11= 180 Compare greater than >, less than < or equal = 4+5___ 9 5+5__ 9 Find a number x. 6

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Solution: 1) 10x -10=180 10x=180+10 10x=190 x=190/10 x=19 4+5=9 5+5>9 6

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Essential Questions: 1-What does the sum of two sides of a triangle tell me about the third side? 2- In a triangle, what is the relationship of an angle of a triangle and its opposite side? 3-What is the relationship between the exterior angle of a triangle and either of its remote interior angles? 4- For two triangles with two pairs of congruent corresponding sides, what is the

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Essential Questions: Relationship between the included angle and the third side? 5- How do I use the Converse of the Pythagorean Theorem to determine if a triangle is a right triangle?

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Vocabulary 1. Triangle inequality 2. Side-angle inequality 3. 3. Side-side-side inequality 4. Exterior angle inequality

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Inequalities in One Triangle 6 3 2 6 3 3 4 3 6 Note that there is only one situation that you can have a triangle; when the sum of two sides of the triangle are greater than the third. They have to be able to reach!!

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Triangle Inequality Theorem AB + BC > AC A B C AB + AC > BC AC + BC > AB

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Triangle Inequality Theorem A B C Biggest Side Opposite Biggest Angle Medium Side Opposite Medium Angle Smallest Side Opposite Smallest Angle 3 5 m**
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Triangle Inequality Theorem Converse is true also Biggest Angle Opposite _____________ Medium Angle Opposite ______________ Smallest Angle Opposite _______________ B C A 65 30 Angle A > Angle B > Angle C So CB >AC > AB

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Example: List the measures of the sides of the triangle, in order of least to greatest. 10x - 10 = 180 Solving for x: Therefore, BC < AB < AC
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Using the Exterior Angle Inequality Example: Solve the inequality if AB + AC > BC x + 3 x + 2 A B C (x+3) + (x+ 2) > 3x - 2 3x - 22x + 5 > 3x - 2 x < 7

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Example: Determine if the following lengths are legs of triangles A)4, 9, 5 4 + 5 ? 9 9 > 9 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: B) 9, 5, 5 Since the sum is not greater than the third side, this is not a triangle 5 + 5 ? 9 10 > 9 Since the sum is greater than the third side, this is a triangle

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Example: a triangle has side lengths of 6 and 12; what are the possible lengths of the third side? 6 12 X = ? 12 + 6 = 18 12 – 6 = 6 Therefore: 6 < X < 18

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Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side

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Section 5.3 Inequalities in One Triangle. The definition of inequality and the properties of inequalities can be applied to the measures of angles.

Section 5.3 Inequalities in One Triangle. The definition of inequality and the properties of inequalities can be applied to the measures of angles.

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