5-5 The Triangle Inequality
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1 Is it possible to form a triangle with the given side lengths? If not, explain why not cm, 7 cm, 10 cm Yes; > 10, > 7, and > m, 14 m, 10 m Yes; > 10, > 14, and > 6. PROOF Write a two-column proof. 5. Given: Prove: Think backwards when considering this proof. Notice that what you are trying to prove is an inequality statement. However, it isn't exactly related to, except for instead of side being used, it is Since it is given that, you can easily use this in a substitution step. Given: Prove: YZ + ZW > XW Statements (Reasons) 1. (Given) 2. XW = YW (Def. of segments) 3. YZ + ZW > YW ( Inequal. Thm.) 4. YZ + ZW > XW (Substitution Property.) esolutions Manual - Powered by Cognero Page 1
2 Is it possible to form a triangle with the given side lengths? If not, explain why not mm, 21 mm, 16 mm Yes; > 16, > 21, and > in., 4.2 in., 7.9 in. No; The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. 11. Yes; Find the range for the measure of the third side of a triangle given the measures of two m, 11 m Let n represent the length of the third side. If n is the largest side, then n must be less than Therefore, n < 16. If n is not the largest side, then 11 is the largest and 11 must be less than 5 + n. Therefore, 6 < n. Combining these two inequalities, we get 6 < n < in., 9.2 in. Let n represent the length of the third side. If n is the largest side, then n must be less than Therefore, n < 13. If n is not the largest side, then 9.2 is the largest and 9.2 must be less than n. Therefore, 5.4 < n. Combining these two inequalities, we get 5.4 < n < 13. esolutions Manual - Powered by Cognero Page 2
3 17. Let n represent the length of the third side. If n is the largest side, then n must be less than. Therefore, n <. If n is not the largest side, then is the largest and must be less than. Therefore,. Combining these two inequalities, we get. PROOF Write a two-column proof. 19. Given: Prove: KJ +KL> LM Think backwards when considering this proof. Notice that what you are trying to prove is an inequality statement. However, it isn't exactly related to, except for instead of side being used, it is Since it is given that, you can easily use this in a substitution step Proof: Statements (Reasons) 1. (Given) 2. JL = LM (Def. of segments) 3. KJ + KL > JL ( Inequality Thm.) 4. KJ + KL > LM (Substitution Property) esolutions Manual - Powered by Cognero Page 3
4 CCSS SENSE-MAKING Determine the possible values of x. 21. Set up and solve each of the three triangle inequalities. Notice that is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both inequalities is and, which can be written as. esolutions Manual - Powered by Cognero Page 4
5 PROOF Write a two-column proof. 23. PROOF Write a two-column proof. Given: ΔABC Prove: AC + BC > AB (Triangle Inequality Theorem) (Hint: Draw auxiliary segment, so that C is between B and D and.) Proof: Statements (Reasons) 1. Construct so that C is between B and D and. (Ruler Postulate) 2. CD = AC (Definition of congruence) 3. (Isosceles Triangle Theorem) 4. m CAD = m ADC (Definition. of congruence angles ) 5. m BAC + m CAD = m BAD ( Addition Postulate) 6. m BAC + m ADC = m BAD (Substitution) 7. m ADC < m BAD (Definition of inequality) 8. AB < BD (Angle Side Relationships in Triangles) 9. BD = BC + CD (Segment Addition Postulate) 10. AB < BC + CD (Substitution) 11. AB < BC + AC (Substitution (Steps 2, 10)) Find the range of possible measures of x if each set of expressions represents measures of the sides of a triangle. 25. x, 4, 6 If x is the largest side, then x must be less than Therefore, x < 4+6 or x <10. If x is not the largest side, then 6 is the largest and 6 must be less than 4 + x. Therefore, 4 + x >6 or x > 2. Combining these two inequalities, we get 2 < x < 10. esolutions Manual - Powered by Cognero Page 5
6 27. x + 1, 5, 7 If x + 1 is the largest side, then x + 1 must be less than Therefore, x + 1 < or x + 1 < 12 or x <11. If x + 1 is not the largest side, then 7 is the largest and 7 must be less than 5 + (x + 1). Therefore, 5 + x + 1 > 7 or 6 + x > 7 or x > 1. Combining these two inequalities, we get 1 < x < x + 2, x + 4, x + 6 Set up and solve each of the three triangle inequalities. Notice that and are always true for any whole number measure for x. So, the only required inequality is x > 0. esolutions Manual - Powered by Cognero Page 6
7 CCSS REASONING Determine whether the given coordinates are the vertices of a triangle. Explain. 39. F( 4, 3), G(3, 3), H(4, 6) We can graphically show that given coordinates form a triangle by graphing them, as shown below. We can algebraically prove that the given coordinates form a triangle by proving that the length of the longest side is greater than the sum of the two shorter Use the distance formula. has endpoints F( 4,3) and G(3, 3). has endpoints G(3, 3) and H(4, 6). has endpoints H(4, 6) and F( 4, 3). Here, FG + GH > FH, FG + FH > GH, and GH + FH > FG. Thus, the coordinated form a triangle. esolutions Manual - Powered by Cognero Page 7
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