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Transcript
5-5 INEQUALITIES IN TRIANGLES (p. 273-279) If a generous geometry teacher gives two different students packages of oatmeal from a box containing exactly five packages, then he unfortunately can not distribute them equally if he gives them all away. However, the amount that he gives to one student will always be less than the original amount. This is an edible example of the Comparison Property of Inequality. Property Comparison Property of Inequality If a = b + c and c > 0, then a > b. This property allows you to prove a corollary to the Exterior Angle Theorem for triangles. Recall that this theorem says that the measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. Example: Use the following diagram to answer this question. According to the Exterior Angle theorem for triangles, m ACD _______ _______ (fill in the blanks). A B D C 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. A B D C According to this corollary, blanks) m ACD ______ and m ACD ______ (fill in the Example: Discuss how you can use the exterior angle theorem for triangles and the comparison property of inequality to prove this corollary. Example: In the following diagram, AB XY. 1. Why is 5 2 ? 2. Why is m 4 m 2 ? 3. Why is m 4 m 5 ? B 5 Y 2 4 A 1 3 X C If two sides of a triangle are congruent, then the angles opposite those sides are congruent. We learned this as Theorem 4-3 (Isosceles Triangle Theorem). What was the short-cut way of remembering this theorem? What do you think you can conclude if two sides of a triangle are not congruent? Theorem 5-10 It two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. B A If BC > AB, then C m A m C. Example: In DOG, DO = 11, DG = 7, and OG = 13. List the angles from largest to smallest. _______ > _______ > _______ (fill in the blanks) If two angles of a triangle are congruent, then the sides opposite those angles are congruent. We learned this as Theorem 4-4 (Converse of Isosceles Triangle Theorem). What was the short-cut way of remembering this theorem? What do you think you can conclude if two angles of a triangle are not congruent? Theorem 5-11 If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. X Z Y If m Y m X, then XZ YZ. There is no need to cover the indirect proof of Theorem 5-11. Example: In CAT, m C 84, m A 32, and m T 64. List the sides from shortest to longest. _______ < _______ < _______ (fill in the blanks) Not every set of three segments can form a triangle. Example: Consider three segments with lengths of 6 cm, 8 cm, and 18 cm. Make a sketch to show how these three segments can not “bridge the gap” to form a triangle. Theorem 5-12 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. R T A RA + AT > RT RA + RT > AT AT + RT > RA Since the shortest distance between two points is a segment, the sum of the lengths of two different noncollinear segments (that start and end at these two points) must be greater. Example: Can a triangle have sides with the given lengths? Explain. 1. 3 cm, 3 cm, 6 cm 2. 8 ft, 19 ft, 12 ft Do 4 on p. 276. Example: In ABC, AB 7 and BC 18. Describe the possible lengths of AC. Let x equal the length of AC and solve these inequalities. 1. x + 7 > 18 2. x + 18 > 7 (Note: You can really ignore this second inequality because the first inequality actually includes this condition) 3. 7 + 18 > x (Note: This means the same as x 7 18) Alternative method for the above example: Find the possible lengths of AC by using simple subtraction and addition. To do this, find the difference of the two known lengths and the sum of the two known lengths and insert these numbers in the following inequality from left to right. _______ < x < _______ Do 5 on p. 276. Homework p. 276-279: 1,6,8,11,15,18,20,25,27,30a-c,31,32,36,42-44,51,55,59 31. The shortcut across the grass is shorter than the sum of the lengths of the other two sides of the triangle (the two cement walkways). 42. Solve 6a 12 114 a = 17, so m P 46, m Q 68, and m QRP 66. PR is the longest length.