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Five-Minute Check (over Lesson 5–4) CCSS Then/Now Theorem 5.11: Triangle Inequality Theorem Example 1: Identify Possible Triangles Given Side Lengths Example 2: Standardized Test Example: Find Possible Side Lengths Example 3: Real-World Example: Proof Using Triangle Inequality Theorem Over Lesson 5–4 State the assumption you would make to start an indirect proof of the statement. ΔABC ΔDEF A. ABC is a right triangle. B. A = D ___ ___ C. AB = DE D. ΔABC / ΔDEF Over Lesson 5–4 State the assumption you would make to start an indirect proof of the statement. ___ RS is an angle bisector. ___ A. RS is a perpendicular bisector. ___ B. RS is not an angle bisector. ___ C. R is the midpoint of ST. D. mR = 90° Over Lesson 5–4 State the assumption you would make to start an indirect proof of the statement. X is a right angle. A. X is a not a right angle. B. mX < 90° C. mX > 90° D. mX = 90° Over Lesson 5–4 State the assumption you would make to start an indirect proof of the statement. If 4x – 3 ≤ 9, then x ≤ 3. A. 4x – 3 ≤ 9 B. x > 3 C. x > 1 D. 4x ≤ 6 Over Lesson 5–4 State the assumption you would make to start an indirect proof of the statement. ΔMNO is an equilateral triangle. A. ΔMNO is a right triangle. B. ΔMNO is an isosceles triangle. C. ΔMNO is not an equilateral triangle. D. MN = NO = MO Over Lesson 5–4 Which is a contradiction to the statement ___statement ___ that AB CD? A. AB = CD B. AB > CD ___ ___ C. CD AB D. AB ≤ CD Content Standards G.CO.10 Prove theorems about triangles. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively. You recognized and applied properties of inequalities to the relationships between the angles and sides of a triangle. • Use the Triangle Inequality Theorem to identify possible triangles. • Prove triangle relationships using the Triangle Inequality Theorem. Identify Possible Triangles Given Side Lengths A. Is it possible to form a triangle with side lengths 1 , 6 __ 1 , and 14 __ 1 ? If not, explain why not. of 6 __ 2 2 2 Check each inequality. Answer: X Identify Possible Triangles Given Side Lengths B. Is it possible to form a triangle with side lengths of 6.8, 7.2, 5.1? If not, explain why not. Check each inequality. 6.8 + 7.2 > 5.1 14 > 5.1 7.2 + 5.1 > 6.8 12.3 > 6.8 6.8 + 5.1 > 7.2 11.9 > 7.2 Since the sum of all pairs of side lengths are greater than the third side length, sides with lengths 6.8, 7.2, and 5.1 will form a triangle. Answer: yes A. yes B. no B. Is it possible to form a triangle given the side lengths 4.8, 12.2, and 15.1? A. yes B. no Find Possible Side Lengths In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 C 11 D 13 Find Possible Side Lengths Read the Test Item You need to determine which value is not valid. Solve the Test Item Solve each inequality to determine the range of values for PR. or n < 12.4 Find Possible Side Lengths Notice that n > –2 is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both inequalities is n > 2 and n < 12.4, which can be written as 2 < n < 12.4. Find Possible Side Lengths Examine the answer choices. The only value that does not satisfy the compound inequality is 13 since 13 is greater than 12.4. Thus, the answer is choice D. Answer: D In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot be XZ? A. 4 B. 9 C. 12 D. 16 Proof Using Triangle Inequality Theorem TRAVEL The towns of Jefferson, Kingston, and Newbury are shown in the map below. Prove that driving first from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury. Proof Using Triangle Inequality Theorem Abbreviating the vertices as J, K, and N: JK represents the distance from Jefferson to Kingstown; KN represents the distance from Kingston to Newbury; and JN the distance from Jefferson to Newbury. Answer: By the Triangle Inequality Theorem, JK + KN > JN. Therefore, driving from Jefferson to Kingston and then Kingston to Newbury is a greater distance than driving from Jefferson to Newbury. Jacinda is trying to run errands around town. She thinks it is a longer trip to drive to the cleaners and then to the grocery store, than to the grocery store alone. Determine whether Jacinda is right or wrong. A. Jacinda is correct, HC + CG > HG. B. Jacinda is not correct, HC + CG < HG.