Inequalities Within a Triangle
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1 7 3 Inequalities ithin a Triangle hat You ll earn You ll learn to identify the relationships between the sides and angles of a triangle. hy It s Important urveying Triangle relationships are important in undersea surveying. ee xample. Florists often use triangles as guides in their flower arrangements. There are special relationships between the side measures and angle measures of each triangle. You will discover these relationships in the following activity. uppose in triangle, the inequality > holds true. Is there a similar relationship between the angles and, which are across from those sides? raphing alculator Tutorial ee pp tep Use the Triangle tool on F to draw and label. tep elect easure from the F5 menu. Then use the istance & ength tool and the ngle tool on F6 to display the measures of the sides and angles of. Try These. efer to the triangle drawn using the steps above. a. hat is the measure of the largest angle in your triangle? b. hat is the measure of the side opposite the largest angle? c. hat is the measure of the smallest angle in your triangle? d. hat is the measure of the side opposite the smallest angle?. rag vertex to a different location. a. hat are the lengths of the longest and shortest sides of the new triangle? b. hat can you conclude about the measures of the angles of a triangle and the measures of the sides opposite these angles? 3. Use the erpendicular isector tool on F3 to draw the perpendicular bisector of side. rag vertex very close to the perpendicular bisector. hat do you observe about the measures of the sides and angles? 90 hapter 7 Triangle Inequalities
2 The observations you made in the previous activity suggest the following theorem. Theorem 7 6 ords: odel: If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides are unequal in the same order. 6 6 ymbols: m m m The converse of Theorem 7 6 is also true. Theorem 7 7 ords: odel: If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles are unequal in the same order. 45 K ymbols: m m m K K K xample In, list the angles in order from least to greatest measure. in. 4 in. 8 in. First, write the segment measures in order from least to greatest. Then, use Theorem 7 6 to write the measures of the angles opposite those m m m sides in the same order. The angles in order from least to greatest measure are,, and. Your Turn a. In T, list the sides in order from least to greatest measure T esson 7 3 Inequalities ithin a Triangle 9
3 eal orld xample urveying ink cientists are developing automated robots for underwater surveying. These undersea vehicles will be guided along by sonar and cameras. If NQ represents the intended course for an undersea vehicle, which segment of the trip will be the longest? tart Q 48 4 N First, write the angle measures in order from least to greatest. m N m Q m Then, use Theorem 7 7 to write the measures of the sides opposite those Q N QN angles in the same order. o, Q N, the first segment of the course, will be the longest. Your Turn Undersea obot Vehicle, Oberon b. If represents a course for an undersea vehicle, which turn will be the sharpest that is, which angle has the least measure? 43 m 45 m 5 m xample illustrates an argument for the following theorem. Theorem 7 8 ords: odel: In a right triangle, the hypotenuse is the side with the greatest measure. 3 5 ymbols: Y YX Y X X 4 Y heck for Understanding ommunicating athematics. Name the angle opposite Z H in HZ.. hoose the correct value for x in HZ without using the ythagorean Theorem: 4, 6, or 0. xplain how you made your choice. cm x cm Z 6 cm xercises H 9 hapter 7 Triangle Inequalities
4 3. Identify the shortest segment from point to line. rite a conjecture in your journal about the shortest segment from a point to a line uided ractice xample 4. ist the angles in order from 5. ist the sides in order from least to greatest measure. least to greatest measure. 5.5 km 3.5 km Q xample 6 km 6. Identify the angle with the 7. Identify the side with the greatest measure. greatest measure yd 4 yd yd 8 30 N 8. riving The road sign indicates that a steep hill is ahead. a. Use a ruler to measure the sides of T to the nearest millimeter. Then list the sides in order from least to greatest measure. b. ist the angles in order from least to greatest measure. xample T 8% ombard treet, an Francisco xercises ractice ist the angles in order from least to greatest measure I. X 9 ft 3.0 m Y 5 cm.0 m 9 cm 8 ft. m H ft cm F esson 7 3 Inequalities ithin a Triangle 93 Z
5 Homework Help For xercises 9, 5 7,, 3 4, 8 0,, 4 ee xamples xtra ractice ee page 739. ist the sides in order from least to greatest measure.. T 3. N Identify the angle with the greatest measure. 5. K.3 km km km V 38 9 in. 98 Q 44 6 in. T mi 6 mi 5 7 mi 5 4 in. Z Identify the side with the greatest measure. 8. I K H. In, m 30, m 45, and m 05. hich side of has the greatest measure?. In QF, Q QF F. hich angle of QF has the greatest measure? pplications and roblem olving 3. rchaeology gyptian carpenters used a tool called an adze to smooth and shape wooden objects. oes, the angle the copper blade makes with the handle, have a measure less than or greater than the measure of, the angle the copper blade makes with the work surface? xplain. F 75 in. 8 in. 4 in. 4. aps Two roads meet at an angle of 50 at point. third road from to makes an angle of 45 with the road from to. hich intersection, or, is closer to? xplain hapter 7 Triangle Inequalities
6 5. ritical Thinking In an obtuse triangle, why is the longest side opposite the obtuse angle? ixed eview 6. The measures of two interior angles of a triangle are 7 and 68. hat is the measure of the exterior angle opposite these angles? (esson 7 ) 7. lgebra If m 48 and m x 0, what values of x make m m? (esson 7 ) omplete each congruence statement. (esson 5 4) 8. Q 9. X K F Y K? YX? tandardized Test ractice 30. hort esponse ketch at least three different quilt patterns that could be made using transformations of the basic square shown at the right. Identify each transformation. (esson 5 3) Quiz essons 7 through 7 3 > eplace each with,, or to make a true sentence. (esson 7 ). T. m T m TN T 5 N Find the measure of each angle. (esson 7 ) N 5. eography erth, arwin, and ydney are three cities in ustralia. hich two of the cities are the farthest apart? (esson 7 3) 6 hark ay erth 54 eographe ay reat andy esert estern lateau 0 arwin 8 Northern Territory UTI outh ustralia reat ustralian ight O risbane 45 ydney elbourne esson 7 3 Inequalities ithin a Triangle 95
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