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Geometry unit 5.6

This document contains explanations and examples regarding triangle theorems and properties including: the exterior angle theorem, isosceles triangle theorem, relationship between side length and angle size, triangle inequality theorem, and determining possible third side lengths given two sides of a triangle. Examples are provided to demonstrate applying these triangle concepts to determine largest angles, longest/shortest sides, and possible side lengths.

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 Corollary to the Triangle Exterior Angle Theorem: The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. 2 3 1 m<1 > m<2 and m<1 > m<3
 m<2 = m<1 by the Isosceles Triangle Theorem. Explain why m<2 > m>3. 1 2 4 3 m<1 > m<3 + m<4 because <1 is the exterior angle, so m<1 > m<3. By substitution property, m< 2 > m<3, since m<2 = m<1.
 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. X Y 11 12 14 Z <Y is the largest angle.
 A landscape architect is designing a triangular deck. She wants to place benches in the two larger corners. Which corners have the larger angles? 27ft 21ft A 18ft B C <B and <A are the larger angles, <C is the smallest.
 If two sides of a triangle are not congruent, then the longer side lies opposite the larger angle. X Y Z 48 98 34 XZ is the longest side.
 Which side is the shortest? 66 TV is the shortest side. 52 62 U T V 40 60 X Y Z 80 YZ is the shortest side
 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 + b > c b + c > a c + a > b
 Can a triangle have sides with the given lengths?  3ft, 7ft, 8ft Yes, 3 + 7 = 10 and 10 > 8  3cm, 6cm, 10cm No, 3 + 6 = 9 and 9 is not greater than 10
 Can a triangle have sides with the given lengths?  2m, 7m, 9m No, 2 + 7 = 9, and 9 is not greater than 9  4yd, 6yd, 9yd Yes, 4 + 6 = 10 and 10 is greater than 9
 A triangle has side lengths of 8cm and 10cm. Describe the possible lengths of the third side. To answer this kind of question, add the numbers together and Subtract the small number from the larger number. 8 + 10 = 18 10 – 8 = 2 The value of the third side must be greater Than 2 and less than 18. (x > 2 and x < 18) 2cm < x < 18cm
 A triangle has side lengths of 3in and 12in. Describe the possible lengths of the third side. To answer this kind of question, add the numbers together and Subtract the small number from the larger number. 3 + 12 = 15 12 – 3 = 9 The value of the third side must be greater Than 9 and less than 15. (x > 9 and x < 15) 9in < x < 15in
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