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<x<18.

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Presentation transcript:

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<x<18

Solution: 1) 10x -10=180 10x= x=190 x=190/10 x=19 4+5=9 5+5>9 6<x<18 x=7, x=16 every number greater than 6 and less than 18.

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

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?

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

Inequalities in One Triangle 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!!

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

Triangle Inequality Theorem A B C Biggest Side Opposite Biggest Angle Medium Side Opposite Medium Angle Smallest Side Opposite Smallest Angle 3 5 m<B is greater than m<C

Triangle Inequality Theorem Converse is true also Biggest Angle Opposite _____________ Medium Angle Opposite ______________ Smallest Angle Opposite _______________ B C A Angle A > Angle B > Angle C So CB >AC > AB

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 <A = 2x + 1 <B = 4x <C = 4x -11 2x x + 4x - 11 =180 10x = 190 X = 19 Plugging back into our Angles: <A = 39 o ; <B = 76; <C = 65 Note: Picture is not to scale

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

Example: Determine if the following lengths are legs of triangles A)4, 9, ? 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 ? 9 10 > 9 Since the sum is greater than the third side, this is a triangle

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

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