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Transcript

Triangle Inequalities Part 2 Exterior Angle Inequality Theorem • If an angle is an exterior angle of a triangle then its measure is greater than the measure of either of its corresponding remote interior angles. 2 m4 > m1 1 3 4 m4 > m2 Example Which angles are less than m3? D m3 > m 4 6 m3 > m 6 C 5 3 Why is m 2 > m 6? 4 2 Since m 2 = m 3 becuase of the converse of the isosceles trainge theorem, then m 2 must also be greater than m 6 A 1 B Triangle Sides • If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. B C is the largest angle B is the middle angle 5 9 A is the smallest angle C A 7 Examples • List the Angles from largest to smallest Largest P Largest C Medium M Medium B Smallest N Smallest A Triangle Angles • If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. E DFis the largest side 77° DEis the middle side 65° D 38° F EFis the smallest side Examples • List the sides of triangle from largest to smallest Largest QS Largest CB Medium QR Medium CA Smallest RS Smallest AB Triangle Inequality Theorem • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. – Always check using the two smallest sides, they must be larger than the third. If this is true the numbers will represent a triangle. Example • Do these numbers represent a triangle? 1.) 9, 7, 12 Yes 2.) 5, 5, 10 No 3.) 1, 4, 6 No 4.) 6, 6, 2 Yes Finding Range of Third Side • If you are given two sides of a traingle you can determine the range that the third side must fall in. – To find the smallest possible side length you subtract the larger side from the smaller side. The value you get can not be a side, however everything larger will work – To find the largest your third side could be you add your two given sides, although this value will not work everything less than it will. Example • If you have two sides of a triangle 4 in and 7 in what is the range for the possible third side, n. 3in < n < 11in • If you have two sides of a triangle 8 in and 12 in what is the range for the possible third side, n. 4in < n < 20in Hinge Theorem • If two sides of one triangle are congruent two two sides of another triangle, and the included angles are not congruent, then the longest side is opposite the larger included angle. If m B > m X Then AC > DF D A F E C B Example What is the relationship between AC and DE? C 7 B E 123 ° 4 4 A 78 ° D 7 AC > DE F Converse of the 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 opposite the longer third side. If AC > DF Then m B > m X D A F E C B Example If AC > DE , what is the relationship between B and F? C 7 B E 4 4 A F D mB > m F 7