OMMON ORE Learning Standard HSG-O..0 6.5 Indirect Proof and Inequalities in One riangle Essential Question How are the sides related to the angles of a triangle? How are any two sides of a triangle related to the third side? omparing ngle Measures and Side Lengths Work with a partner. Use dynamic geometry software. raw any scalene. a. Find the side lengths and angle measures of the triangle. 5 4 0 0 4 5 6 7 Sample Points (, ) (5, ) (7, 4) Segments =? =? =? ngles m =? m =? m =? ENING O PREISION o be proficient in math, you need to express numerical answers with a degree of precision appropriate for the content. b. Order the side lengths. Order the angle measures. What do you observe? c. rag the vertices of to form new triangles. Record the side lengths and angle measures in a table. Write a conjecture about your findings. Relationship of the Side Lengths of a riangle Work with a partner. Use dynamic geometry software. raw any. a. Find the side lengths of the triangle. b. ompare each side length with the sum of the other two side lengths. 4 0 0 4 5 6 Sample Points (0, ) (, ) (5, ) Segments =? =? =? c. rag the vertices of to form new triangles and repeat parts (a) and (b). Organize your results in a table. Write a conjecture about your findings. ommunicate Your nswer. How are the sides related to the angles of a triangle? How are any two sides of a triangle related to the third side? 4. Is it possible for a triangle to have side lengths of, 4, and 0? Explain. Section 6.5 Indirect Proof and Inequalities in One riangle 5
6.5 Lesson What You Will Learn ore Vocabulary indirect proof, p. 6 Previous proof inequality Write indirect proofs. List sides and angles of a triangle in order by size. Use the riangle Inequality heorem to find possible side lengths of triangles. Writing an Indirect Proof Suppose a student looks around the cafeteria, concludes that hamburgers are not being served, and explains as follows. t fi rst, I assumed that we are having hamburgers because today is uesday, and uesday is usually hamburger day. here is always ketchup on the table when we have hamburgers, so I looked for the ketchup, but I didn t see any. So, my assumption that we are having hamburgers must be false. he student uses indirect reasoning. In an indirect proof, you start by making the temporary assumption that the desired conclusion is false. y then showing that this assumption leads to a logical impossibility, you prove the original statement true by contradiction. ore oncept How to Write an Indirect Proof (Proof by ontradiction) Step Identify the statement you want to prove. ssume temporarily that this statement is false by assuming that its opposite is true. Step Step Reason logically until you reach a contradiction. Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false. Writing an Indirect Proof Write an indirect proof that in a given triangle, there can be at most one right angle. Given Prove can have at most one right angle. REING You have reached a contradiction when you have two statements that cannot both be true at the same time. Step ssume temporarily that has two right angles. hen assume and are right angles. Step y the definition of right angle, m = m = 90. y the riangle Sum heorem (heorem 5.), m + m + m = 80. Using the Substitution Property of Equality, 90 + 90 + m = 80. So, m = 0 by the Subtraction Property of Equality. triangle cannot have an angle measure of 0. So, this contradicts the given information. Step So, the assumption that has two right angles must be false, which proves that can have at most one right angle. Monitoring Progress Help in English and Spanish at igideasmath.com. Write an indirect proof that a scalene triangle cannot have two congruent angles. 6 hapter 6 Relationships Within riangles
Relating Sides and ngles of a riangle Relating Side Length and ngle Measure raw an obtuse scalene triangle. Find the largest angle and longest side and mark them in red. Find the smallest angle and shortest side and mark them in blue. What do you notice? longest side smallest angle largest angle shortest side OMMON ERROR e careful not to confuse the symbol meaning angle with the symbol < meaning is less than. Notice that the bottom edge of the angle symbol is horizontal. he longest side and largest angle are opposite each other. he shortest side and smallest angle are opposite each other. he relationships in Example are true for all triangles, as stated in the two theorems below. hese relationships can help you decide whether a particular arrangement of side lengths and angle measures in a triangle may be possible. heorems heorem 6.9 riangle Longer Side heorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. 8 5 Proof Ex. 4, p. 4 >, so m > m. heorem 6.0 riangle Larger ngle heorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Proof p. 7 50 0 m > m, so >. OMMON ERROR e sure to consider all cases when assuming the opposite is true. riangle Larger ngle heorem Given m > m Prove > Indirect Proof Step ssume temporarily that. hen it follows that either < or =. Step If <, then m < m by the riangle Longer Side heorem. If =, then m = m by the ase ngles heorem (hm. 5.6). Step oth conclusions contradict the given statement that m > m. So, the temporary assumption that cannot be true. his proves that >. Section 6.5 Indirect Proof and Inequalities in One riangle 7
Ordering ngle Measures of a riangle You are constructing a stage prop that shows a large triangular mountain. he bottom edge of the mountain is about feet long, the left slope is about 4 feet long, and the right slope is about 6 feet long. List the angles of KL in order from smallest to largest. K raw the triangle that represents the mountain. Label the side lengths. he sides from shortest to longest are K, KL, and L. he angles opposite these sides are L,, and K, respectively. So, by the riangle Longer Side heorem, the angles from smallest to largest are L,, and K. L K 4 ft 6 ft ft L Ordering Side Lengths of a riangle List the sides of EF in order from shortest to longest. First, find m F using the riangle Sum heorem (heorem 5.). m + m E + m F = 80 5 + 47 + m F = 80 m F = 8 F he angles from smallest to largest are E,, and F. he sides opposite these angles are F, EF, and E, respectively. So, by the riangle Larger ngle heorem, the sides from shortest to longest are F, EF, and E. Monitoring Progress Help in English and Spanish at igideasmath.com. List the angles of PQR in order from smallest to largest. 5 47 6.8 E Q. List the sides of RS in order from shortest to longest. S P 5.9 R 6. R 9 0 8 hapter 6 Relationships Within riangles
Using the riangle Inequality heorem Not every group of three segments can be used to form a triangle. he lengths of the segments must fit a certain relationship. For example, three attempted triangle constructions using segments with given lengths are shown below. Only the first group of segments forms a triangle. 4 5 5 5 When you start with the longest side and attach the other two sides at its endpoints, you can see that the other two sides are not long enough to form a triangle in the second and third figures. his leads to the riangle Inequality heorem. heorem heorem 6. riangle Inequality heorem he sum of the lengths of any two sides of a triangle is greater than the length of the third side. + > + > + > Proof Ex. 47, p. 4 Finding Possible Side Lengths triangle has one side of length 4 and another side of length 9. escribe the possible lengths of the third side. REING You can combine the two inequalities, x > 5 and x <, to write the compound inequality 5 < x <. his can be read as x is between 5 and. Let x represent the length of the third side. raw diagrams to help visualize the small and large values of x. hen use the riangle Inequality heorem to write and solve inequalities. Small values of x Large values of x 4 x x 9 4 9 x + 9 > 4 9 + 4 > x x > 5 > x, or x < he length of the third side must be greater than 5 and less than. Monitoring Progress Help in English and Spanish at igideasmath.com 4. triangle has one side of length inches and another side of length 0 inches. escribe the possible lengths of the third side. ecide whether it is possible to construct a triangle with the given side lengths. Explain your reasoning. 5. 4 ft, 9 ft, 0 ft 6. 8 m, 9 m, 8 m 7. 5 cm, 7 cm, cm Section 6.5 Indirect Proof and Inequalities in One riangle 9
6.5 Exercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. VOULRY Why is an indirect proof also called a proof by contradiction?. WRIING How can you tell which side of a triangle is the longest from the angle measures of the triangle? How can you tell which side is the shortest? Monitoring Progress and Modeling with Mathematics In Exercises 6, write the first step in an indirect proof of the statement. (See Example.). If WV + VU inches and VU = 5 inches, then WV 7 inches. 4. If x and y are odd integers, then xy is odd. 5. In, if m = 00, then is not a right angle. 6. In KL, if M is the midpoint of KL, then M is a median. In Exercises 7 and 8, determine which two statements contradict each other. Explain your reasoning. 7. LMN is a right triangle. L N LMN is equilateral. 8. oth X and Y have measures greater than 0. oth X and Y have measures less than 0. m X + m Y = 6 In Exercises 9 and 0, use a ruler and protractor to draw the given type of triangle. Mark the largest angle and longest side in red and the smallest angle and shortest side in blue. What do you notice? (See Example.) 9. acute scalene 0. right scalene In Exercises and, list the angles of the given triangle from smallest to largest. (See Example.). R 0 S. 6 9 8 5 K L In Exercises 6, list the sides of the given triangle from shortest to longest. (See Example 4.). 67 6 5 4. 5. 6. M N 7 9 P X Y 6 In Exercises 7 0, describe the possible lengths of the third side of the triangle given the lengths of the other two sides. (See Example 5.) 7. 5 inches, inches 8. feet, 8 feet 9. feet, 40 inches 0. 5 meters, 5 meters In Exercises 4, is it possible to construct a triangle with the given side lengths? If not, explain why not.. 6, 7,., 6, 9. 8, 7, 46 4. 5, 0, 5 5. ERROR NLYSIS escribe and correct the error in writing the first step of an indirect proof. Step F Show that is obtuse. ssume temporarily that is acute. G Z 40 hapter 6 Relationships Within riangles
6. ERROR NLYSIS escribe and correct the error in labeling the side lengths,, and on the triangle. 60 0 5. MOELING WIH MHEMIS You can estimate the width of the river from point to the tree at point by measuring the angle to the tree at several locations along the riverbank. he diagram shows the results for locations and. 7. RESONING You are a lawyer representing a client who has been accused of a crime. he crime took place in Los ngeles, alifornia. Security footage shows your client in New York at the time of the crime. Explain how to use indirect reasoning to prove your client is innocent. 8. RESONING Your class has fewer than 0 students. he teacher divides your class into two groups. he first group has 5 students. Use indirect reasoning to show that the second group must have fewer than 5 students. 40 50 50 yd 5 yd a. Using and, determine the possible widths of the river. Explain your reasoning. b. What could you do if you wanted a closer estimate? 6. MOELING WIH MHEMIS You travel from Fort Peck Lake to Glacier National Park and from Glacier National Park to Granite Peak. 9. PROLEM SOLVING Which statement about UV is false? U UV > U UV + V > U UV < V UV is isosceles. 84 48 V Glacier National Park 489 km 565 km MONN x km Fort Peck Lake Granite Peak 0. PROLEM SOLVING In RS, which is a possible side length for S? Select all that apply. 7 8 9 0. PROOF Write an indirect proof that an odd number is not divisible by 4.. PROOF Write an indirect proof of the statement In QRS, if m Q + m R = 90, then m S = 90.. WRIING Explain why the hypotenuse of a right triangle must always be longer than either leg. 4. RIIL HINKING Is it possible to decide if three side lengths form a triangle without checking all three inequalities shown in the riangle Inequality heorem (heorem 6.)? Explain your reasoning. S 8 65 R 56 a. Write two inequalities to represent the possible distances from Granite Peak back to Fort Peck Lake. b. How is your answer to part (a) affected if you know that m < m and m < m? 7. RESONING In the figure, XY bisects WYZ. List all six angles of XYZ and WXY in order from smallest to largest. Explain your reasoning. X 4. 5.6 5 W 6.4 Z Y.7 8. MHEMIL ONNEIONS In EF, m = (x + 5), m E = (x 4), and m F = 6. List the side lengths and angle measures of the triangle in order from least to greatest. Section 6.5 Indirect Proof and Inequalities in One riangle 4
9. NLYZING RELIONSHIPS nother triangle inequality relationship is given by the Exterior ngle Inequality heorem. It states: he measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles. Explain how you know that m > m and m > m in with exterior angle. MHEMIL ONNEIONS In Exercises 40 and 4, describe the possible values of x. 40. x + K 5x 9 x + 0 L 4. 6x x + U x 4. HOW O YOU SEE I? Your house is on the corner of Hill Street and Eighth Street. he library is on the corner of View Street and Seventh Street. What is the shortest route to get from your house to the library? Explain your reasoning. V 44. USING SRUURE he length of the base of an isosceles triangle is. escribe the possible lengths for each leg. Explain your reasoning. 45. MKING N RGUMEN Your classmate claims to have drawn a triangle with one side length of inches and a perimeter of feet. Is this possible? Explain your reasoning. 46. HOUGH PROVOKING ut two pieces of string that are each 4 centimeters long. onstruct an isosceles triangle out of one string and a scalene triangle out of the other. Measure and record the side lengths. hen classify each triangle by its angles. 47. PROVING HEOREM Prove the riangle Inequality heorem (heorem 6.). Given Prove + >, + >, and + > 48. ENING O PREISION he perimeter of HGF must be between what two integers? Explain your reasoning. Eighth St. Hill St. Washington ve. Seventh St. G 5 4 F H View St. 4. PROVING HEOREM Use the diagram to prove the riangle Longer Side heorem (heorem 6.9). Given >, = Prove m > m 49. PROOF Write an indirect proof that a perpendicular segment is the shortest segment from a point to a plane. P Given P plane M Prove P is the shortest segment from P to plane M. M Maintaining Mathematical Proficiency Name the included angle between the pair of sides given. (Section 5.) 50. E and E 5. and 5. and 5. E and E Reviewing what you learned in previous grades and lessons E 4 hapter 6 Relationships Within riangles