7.3 Triangle Inequalities
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1 Name lass Date 7.3 Triangle Inequalities Essential Question: How can you use inequalities to describe the relationships among side lengths and angle measures in a triangle? Resource Locker Explore Exploring Triangle Inequalities triangle can have sides of different lengths, but are there limits to the lengths of any of the sides? onsider a where you know two side lengths, = 4 inches and = 2 _ inches. On a separate piece of paper, draw so that it is 4 inches long. To _ determine all possible locations for with = 2 inches, set your compass to 2 inches. Draw a circle with center at. hoose and label a final vertex point so it is located on the circle. Using a straightedge, draw the segments to form a triangle. Houghton Mifflin Harcourt Publishing ompany D re there any places on the circle where point cannot lie? Explain. Measure and record the lengths of the three sides of your triangle. Module Lesson 3
2 E The figures below show two other examples of that could have been formed. What are the values that _ approaches when point approaches _? Reflect 1. Use the side lengths from your table to make the following comparisons. What do you notice? +? +? +? 2. Measure the angles of some triangles with a protractor. Where is the smallest angle in relation to the shortest side? Where is the largest angle in relation to the longest side? 3. Discussion How does your answer to the previous question relate to isosceles triangles or equilateral triangles? Explain 1 Using the Triangle Inequality Theorem The Explore shows that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. This can be summarized in the following theorem. Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. + > + > + > Houghton Mifflin Harcourt Publishing ompany To be able to form a triangle, each of the three inequalities must be true. So, given three side lengths, you can test to determine if they can be used as segments to form a triangle. To show that three lengths cannot be the side lengths of a triangle, you only need to show that one of the three triangle inequalities is false. Module Lesson 3
3 Example 1 Use the Triangle Inequality Theorem to tell whether a triangle can have sides with the given lengths. Explain. 4, 8, 10??? > > > 4 12 > > 8 18 > 4 onclusion: The sum of each pair of side lengths is greater than the third length. So, a triangle can have side lengths of 4, 8, and 10. 7, 9, 18??? + > 18 + > > > 18 > 9 > onclusion: Reflect 4. an an isosceles triangle have these side lengths? Explain. 5, 5, How do you know that the Triangle Inequality Theorem applies to all equilateral triangles? Houghton Mifflin Harcourt Publishing ompany Your Turn Determine if a triangle can be formed with the given side lengths. Explain your reasoning units, 4 units, 17 units cm, 8 cm, 30 cm Module Lesson 3
4 Explain 2 Finding Possible Side Lengths in a Triangle From the Explore, you have seen that if given two side lengths for a triangle, there are an infinite number of side lengths available for the third side. ut the third side is also restricted to values determined by the Triangle Inequality Theorem. b a Example 2 Find the range of values for x using the Triangle Inequality Theorem. Find possible values for the length of the third side using the Triangle Inequality Theorem. 12 x x + 10 > 12 x + 12 > > x x > 2 x > > x 2 < x < Ignore the inequality with a negative value, since a triangle cannot have a negative side length. ombine the other two inequalities to find the possible values for x. 15 x 15 + > + > + > > > > < x < Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
5 Reflect 8. Discussion Suppose you know that the length of the base of an isosceles triangle is 10, but you do not know the lengths of its legs. How could you use the Triangle Inequality Theorem to find the range of possible lengths for each leg? Explain. Your Turn Find the range of values for x using the Triangle Inequality Theorem x 9 x 18 Explain 3 Ordering a Triangle s ngle Measures Given Its Side Lengths From the Explore Step D, you can see that changing the length of _ also changes the measure of in a predictable way. Houghton Mifflin Harcourt Publishing ompany s side gets shorter, m approaches 0 s side gets longer, m approaches 180 Side-ngle Relationships in Triangles If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. > m > m Module Lesson 3
6 Example 3 For each triangle, order its angle measures from least to greatest Longest side length: Greatest angle measure: m Shortest side length: Least angle measure: m Order of angle measures from least to greatest: m, m, m Longest side length: Greatest angle measure: Shortest side length: Least angle measure: Order of angle measures from least to greatest: Your Turn For each triangle, order its angle measures from least to greatest Explain 4 Ordering a Triangle s Side Lengths Given Its ngle Measures From the Explore Step D, you can see that changing the the measure of also changes length of _ in a predictable way. Houghton Mifflin Harcourt Publishing ompany s m approaches 0, side gets shorter s m approaches 180, side gets longer ngle-side Relationships in Triangles If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. Module Lesson 3
7 Example 4 For each triangle, order the side lengths from least to greatest Greatest angle measure: m Longest side length: Least angle measure: m Shortest side length: Order of side lengths from least to greatest:,, Greatest angle measure: Longest side length: Least angle measure: Shortest side length: Order of side lengths from least to great: Your Turn For each triangle, order the side lengths from least to greatest Houghton Mifflin Harcourt Publishing ompany Elaborate 15. When two sides of a triangle are congruent, what can you conclude about the angles opposite those sides? 16. What can you conclude about the side opposite the obtuse angle in an obtuse triangle? 17. Essential Question heck-in Suppose you are given three values that could represent the side lengths of a triangle. How can you use one inequality to determine if the triangle exists? Module Lesson 3
8 Evaluate: Homework and Practice Use a compass and straightedge to decide whether each set of lengths can form a triangle cm, 9 cm, 18 cm 2. 2 in., 4 in., 5 in. Online Homework Hints and Help Extra Practice 3. 1 in., 2 in., 10 in cm, 10 cm, 11 cm Determine whether a triangle can be formed with the given side lengths ft, 3 ft, 15 ft in., 4 in., 15 in in., 12 in., and 18 in m, 59 m, and 89 m Find the range of possible values for x using the Triangle Inequality Theorem x x triangle with side lengths 22.3, 27.6, and x 12. nalyze Relationships Suppose a triangle has side lengths,, and x, where = 2. Find the possible range for x in terms of. Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
9 For each triangle, write to order the angle measures from least to greatest D E F 15. nalyze Relationships Suppose a triangle has side lengths PQ, QR, and PR, where PR = 2PQ = 3QR. Write the angle measures in order from least to greatest. For each triangle, write the side lengths in order from least to greatest E D 33 F 18. In JKL, m J = 53, m K = 68, and m L = In PQR, m P = 102 and m Q = 25. Houghton Mifflin Harcourt Publishing ompany Image redits: arlos Davila/Photographer's hoice RF/Getty Images 20. Represent Real-World Problems Rhonda is traveling from New York ity to Paris and is trying to decide whether to fly via Frankfurt or to get a more expensive direct flight. Given that it is 3,857 miles from to Frankfurt and another 278 miles from Frankfurt to Paris, what is the range of possible values for the direct distance from New York ity to Paris? Module Lesson 3
10 21. Represent Real-World Problems large ship is sailing between three small islands. To do so, the ship must sail between two pairs of islands, avoiding sailing between a third pair. The safest route is to avoid the closest pair of islands. Which is the safest route for the ship? X 58 N Y 73 Z 22. Represent Real-World Problems hole on a golf course is a dogleg, meaning that it bends in the middle. golfer will usually start by driving for the bend in the dogleg (from to ), and then using a second shot to get the ball to the green (from to ). Sandy believes she may be able to drive the ball far enough to reach the green in one shot, avoiding the bend (from direct to ). Sandy knows she can accurately drive a distance of 250 yd. Should she attempt to drive for the green on her first shot? Explain. 102 yd 135 yd 23. Represent Real-World Problems Three cell phone towers form a triangle, PQR. The measure of Q is 10 less than the measure of P. The measure of R is 5 greater than the measure of Q. Which two towers are closest together? 24. lgebra In PQR, PQ = 3x + 1, QR = 2x 2, and PR = x + 7. Determine the range of possible values of x. Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
11 25. In any triangle, suppose you know the lengths of _ and _, and suppose that >. If x is the length of the third side, _, use the Triangle Inequality Theorem to prove that < x < +. That is, x must be between the difference and the sum of the other two side lengths. Explain why this result makes sense in terms of the constructions shown in the figure. 26. Given the information in the diagram, prove that m DE < m. 9 4 D 7 9 E n isosceles triangle has legs with length 11 units. Which of the following could be the perimeter of the triangle? hoose all that apply. Explain your reasoning. a. 22 units b. 24 units c. 34 units Houghton Mifflin Harcourt Publishing ompany d. 43 units e. 44 units H.O.T. Focus on Higher Order Thinking 28. ommunicate Mathematical Ideas Given the information in the diagram, prove that PQ < PS. Q R P S Module Lesson 3
12 29. Justify Reasoning _ In obtuse, m < m. The auxiliary line segment D perpendicular to (extended beyond ) creates right triangles D and D. Describe how you could use the Pythagorean Theorem to prove that <. D 30. Make a onjecture _ In acute DEF, m D < m E. The auxiliary line segment FG creates EFG, where EF = FG. What would you need to prove about the points D, G, and E to prove that DGF is obtuse, and therefore that EF < DF? Explain. F D G E Lesson Performance Task s captain of your orienteering team, it s your job to map out the shortest distance from point to point H on the map. Justify each of your decisions. G 48 E 94 H 53 F D 58 I Houghton Mifflin Harcourt Publishing ompany Module Lesson 3
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