The Triangle Inequality LESSON 5–5 The Triangle Inequality

Five-Minute Check (over Lesson 5–4) TEKS Then/Now Theorem 5.11: Triangle Inequality Theorem Example 1: Identify Possible Triangles Given Side Lengths Example 2: Find Possible Side Lengths Example 3: Real-World Example: Proof Using Triangle Inequality Theorem Lesson Menu

A. ABC is a right triangle. B. A = D 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 ___  5-Minute Check 1

A. RS is a perpendicular bisector. B. RS is not an angle bisector. 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. mR = 90° ___ 5-Minute Check 2

A. X is a not a right angle. B. mX < 90° 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. mX < 90° C. mX > 90° D. mX = 90° 5-Minute Check 3

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 5-Minute Check 4

A. ΔMNO is a right triangle. B. ΔMNO is an isosceles triangle. 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 5-Minute Check 5

Which statement is a contradiction to the statement that AB  CD? ___ A. AB = CD B. AB > CD C. CD  AB D. AB ≤ CD ___ 5-Minute Check 6

G.5(D) Verify the Triangle Inequality theorem using Targeted TEKS G.5(D) Verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems. G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(F) TEKS

Use the Triangle Inequality Theorem to identify possible triangles. 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. Then/Now

Concept

Identify Possible Triangles Given Side Lengths A. Is it possible to form a triangle with side lengths of 6 , 6 , and 14 ? If not, explain why not. __ 1 2 Check each inequality.  X Answer: Example 1

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 7.2 + 5.1 > 6.8 6.8 + 5.1 > 7.2 14 > 5.1 12.3 > 6.8  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 Example 1

A. yes B. no Example 1

B. Is it possible to form a triangle given the side lengths 4. 8, 12 B. Is it possible to form a triangle given the side lengths 4.8, 12.2, and 15.1? A. yes B. no Example 1

In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR? A 7 B 9 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 Example 2

You need to determine which value is not valid. 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 Example 2

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. Example 2

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 Example 2

In ΔXYZ, XY = 6, and YZ = 9. Which measure cannot be XZ? Example 2

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. Example 3

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. Example 3

A. Jacinda is correct, HC + CG > HG. 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. Example 3

The Triangle Inequality LESSON 5–5 The Triangle Inequality