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Transcript
MM1G3b -Understand and use the triangle inequality, the side-angle inequality, and the exterior angle inequality. Important Triangle Facts A triangle has 6 parts: 3 sides 3 angles The sum of the angles in a triangle is always 180°. A triangle is named by its vertices. Name the triangle and its parts. ∆ ABC A AB C B BC <ACB <BAC <CBA ∆ RST R AC RS S T ST <RST <TRS <STR RT Classifying Triangles Classify triangles by the side lengths. Equilateral – all sides are equal Isosceles – at least two sides are Scalene – no sides are equal equal Classify triangles by angle measures. Right – has one 90⁰ angle Acute – all angles are less than 90⁰ Obtuse – has one angle greater than 90⁰ Equiangular – all angles are equal Do Parts I and II of the Triangle Notes Handout. Triangle Side Angle Inequalities Smaller angles are opposite shorter sides. Larger angles are opposite longer sides. Example. 1. In ∆ABC , list the sides in order from smallest to largest. 47⁰ 2. In ∆JKL , list the angles in order from smallest to largest. 98⁰ 35⁰ AB, BC, AC <JLK, <KJL, <JKL Handout straight edges and compasses Construct triangles: 3”, 4”, 5” 2”, 3”, 6” Triangle Inequality Theorem The sum of any two lengths of any two sides of a triangle is greater than the length of the third side. Could say that the sum of the two shorter sides must be greater than the longest side. If the 3rd side is equal to or less than the sum of the 2 other sides, then it can not form a triangle. Examples – Can these three sides form a triangle? A. 5, 8, 16 5 + 8 < 16 NO B. 6, 11, 14 6 + 11 > 14 YES C. 8, 13, 5 5 + 8 = 13 NO Triangle Inequality Theorem A. If two sides of a triangle are given, describe the possible lengths of the third side. 2 yd, 6 yd What possible values would work? Compare the sum of 2 shorter sides to longest side. If 3rd side is 1: 1 + 2 > 6 No If 3rd side is 6: 6 + 2 > 6 Yes If 3rd side is 2: 2 + 2 > 6 No If 3rd side is 7: 6 + 2 > 7 Yes If 3rd side is 3: 3 + 2 > 6 No If 3rd side is 8: 6 + 2 > 8 No If 3rd side is 4: 4 + 2 > 6 No If 3rd side is 9: 6 + 2 > 9 No If 3rd side is 5: 5 + 2 > 6 Yes If 3rd side is 10: 6 + 2 > 10 No So the third side has to be bigger than 4 and less than 8 or 4 < x < 8. Triangle Inequality Theorem A. If two sides of a triangle are given, describe the possible lengths of the third side. 2 yd, 6 yd So the third side has to be bigger than 4 and less than 8 or 4 < x < 8. In other words, 2+x>6 –2 –2 x > 4 or 2+6>x 8>x or x< 8 Therefore, the range is going to be x has to greater than the difference or less than the sum of the two given sides. Triangle Inequality Theorem B. If two sides of a triangle are given, describe the possible lengths of the third side. 4 in, 12 in C. 3 ft, 18 ft 4 + x > 12 and 4 + 12 > x 16 > x –4 –4 x > 8 and x < 16 8 < x < 16 3 + x > 18 and 3 + 18 > x –3 –3 21 > x x < 15 x < 21 15 < x < 21 Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent (remote) interior angles. The measure of an exterior angle is the sum of the remote interior angles. Example 6 1 2 What relationships do we know about the angles listed? 3 4 5 <5 > < 2 <5 > <3 <5 = <2 + <3 <4 > <1 <4 > <2 <4 = <1 + <2 <6 > <1 <6 > <3 <6 = <1 + <3 65° 53 3 (3x – 5)° What do you know about x? 3x – 5 > 53 3x – 5 > 65 3x – 5 = 53 + 65 3x – 5 = 118 + 5 = +5 3x = 123 x = 41 Classwork/Homework Textbook p287 (4-9,13-24 all)