Inequalities in One Triangle

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Presentation transcript:

Inequalities in One Triangle Lesson 5-3 Inequalities in One Triangle Lesson 5-3: Inequalities in One Triangle

Content Standards G-CO.10 Prove theorems about triangles. Mathematical Practices 1 Make sense of problems and persevere in solving them 2 Reason abstractly and quantitatively. 6 Attend to precision.

You found the relationship between the angle measures of a triangle You found the relationship between the angle measures of a triangle. (Lesson 4–2) Recognize and apply properties of inequalities to the measures of the angles of a triangle. Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.

Lesson 5-3: Inequalities in One Triangle Triangle Inequality The smallest side is across from the smallest angle. The largest angle is across from the largest side. 54 ° 37 89 B C A BC = 3.2 cm AB = 4.3 cm AC = 5.3 cm Lesson 5-3: Inequalities in One Triangle

Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. C A B 4 cm 6 cm 5 cm Lesson 5-3: Inequalities in One Triangle

Triangle Inequality – examples… For the triangle, list the sides in order from shortest to longest measure. 8x-10 7x+6 7x+8 C A B (7x + 8) ° + (7x + 6 ) ° + (8x – 10 ) ° = 180° 22 x + 4 = 180 ° 22x = 176 X = 8 m<C = 7x + 8 = 64 ° m<A = 7x + 6 = 62 ° m<B = 8x – 10 = 54 ° 54 ° 62 ° 64 ° Lesson 5-3: Inequalities in One Triangle

Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of wither of its corresponding remote interior angles. Lesson 5-3: Inequalities in One Triangle

Lesson 5-3: Inequalities in One Triangle Converse Theorem & Corollaries Converse: If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle. Corollary 1: The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 2: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Lesson 5-3: Inequalities in One Triangle

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a + b > c a + c > b b + c > a Example: Determine if it is possible to draw a triangle with side measures 12, 11, and 17. 12 + 11 > 17  Yes 11 + 17 > 12  Yes 12 + 17 > 11  Yes Therefore a triangle can be drawn. Lesson 5-3: Inequalities in One Triangle