Relationships in Triangles

Transcription

1 elationships in Triangles Lesson 5- Identify and use perpendicular bisectors, angle bisectors, medians, and altitudes of triangles. Lesson 5- pply properties of inequalities relating to the measures of angles and sides of triangles. Lesson 5- Use indirect proof with algebra and geometry. Lessons 5-4 and 5-5 pply the Triangle Inequality Theorem and and inequalities. Key Vocabulary perpendicular bisector (p. 8) median (p. 40) altitude (p. 4) indirect proof (p. 55) There are several relationships among the sides and angles of triangles. These relationships can be used to compare the length of a person s stride and the rate at which that person is walking or running. In Lesson 5-5, you will learn how to use the measure of the sides of a triangle to compare stride and rate. 4 hapter 5 elationships in Triangles Mike owell/getty Images

2 Triangle elations rerequisite kills To be successful in this chapter, you ll need to master these skills and be able to apply them in problem-solving situations. eview these skills before beginning hapter 5. For Lesson 5- Midpoint of a egment Find the coordinates of the midpoint of a segment with the given endpoints. (For review, see Lesson -.). (, 5), (4, 5). (, 5), (0, 0). E(9, 7), F( 0, ) For Lesson 5- Exterior ngle Theorem Find the measure of each numbered angle if. (For review, see Lesson 4-.) For Lesson 5- eductive easoning etermine whether a valid conclusion can be reached from the two true statements using the Law of etachment. If a valid conclusion is possible, state it. If a valid conclusion does not follow, write no conclusion. (For review, see Lesson -4.). () If the three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent. () and are congruent.. () The sum of the measures of the angles of a triangle is 80. () olygon JKL is a triangle. elationships in Triangles Make this Foldable to help you organize information about relations in triangles. egin with one sheet of notebook paper. Fold ut Fold lengthwise to the holes. ut along the lines to create five tabs. Label Label the edge. Then label the tabs using lesson numbers eading and Writing s you read and study each lesson, write notes and examples under the appropriate tab. hapter 5 elationships in Triangles 5

3 isectors, Medians, and ltitudes You can use the constructions for midpoint, perpendiculars, and angle bisectors to construct special segments in triangles. onstruction raw a triangle like. djust the compass to an opening greater than. lace the compass at vertex, and draw an arc above and below. onstruct the bisector of a side of a triangle. Using the same compass settings, place the compass at vertex. raw an arc above and below. Label the points of intersection of the arcs and. Use a straightedge to draw. Label the point where bisects as M. M M M by construction and M M by the eflexive roperty. because the arcs were drawn with the same compass setting. Thus, M M by. y T, M M. linear pair of congruent angles are right angles. o is not only a bisector of, but a perpendicular bisector.. onstruct the perpendicular bisectors for the other two sides.. What do you notice about the intersection of the perpendicular bisectors? onstruction raw intersecting arcs above and below. Label the points of intersection and. onstruct a median of a triangle. Use a straightedge to find the point where intersects. Label the midpoint M. M raw a line through and M. M is a median of. M 0 4 M. onstruct the medians of the other two sides. 4. What do you notice about the medians of a triangle? 6 Investigating lope-intercept Form 6 hapter 5 elationships in Triangles

4 review of Lesson 5- onstruction lace the compass at vertex and draw two arcs intersecting. Label the points where the arcs intersect the side X and Y. onstruct an altitude of a triangle. djust the compass to an opening greater than XY. lace the compass on X and draw an arc above. Using the same setting, place the compass on Y and draw an arc above. Label the intersection of the arcs H. Use a straightedge to draw H. Label the point where H intersects as. is an altitude of and is perpendicular to. H H X Y X Y X Y 5. onstruct the altitudes to the other two sides. (Hint: You may need to extend the lines containing the sides of your triangle.) 6. What observation can you make about the altitudes of your triangle? onstruction 4 lace the compass on vertex, and draw arcs through and. Label the points where the arcs intersect the sides as J and K. onstruct an angle bisector of a triangle. lace the compass on J, and draw an arc. Then place the compass on K and draw an arc intersecting the first arc. Label the intersection L. Use a straightedge to draw L. L is an angle bisector of. J J L J L K K K 7. onstruct the angle bisectors for the other two angles. 8. What do you notice about the angle bisectors? nalyze 9. epeat the four constructions for each type of triangle. a. obtuse scalene b. right scalene c. isosceles d. equilateral Make a onjecture 0. Where do the lines intersect for acute, obtuse, and right triangles?. Under what circumstances do the special lines of triangles coincide with each other? Investigating lope-intercept Form 7 Geometry ctivity isectors, Medians, and ltitudes 7

5 isectors, Medians, and ltitudes Identify and use perpendicular bisectors and angle bisectors in triangles. Identify and use medians and altitudes in triangles. Vocabulary perpendicular bisector concurrent lines point of concurrency circumcenter incenter median centroid altitude orthocenter can you balance a paper triangle on a pencil point? crobats and jugglers often balance objects while performing their acts. These skilled artists need to find the center of gravity for each object or body position in order to keep balanced. The center of gravity for any triangle can be found by drawing the medians of a triangle and locating the point where they intersect. EENIUL IETO N NGLE IETO The first construction you made in the Geometry ctivity on pages 6 and 7 was the perpendicular bisector of a side of a triangle. perpendicular bisector of a side of a triangle is a line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. erpendicular bisectors of segments have some special properties. These properties are described in the following theorems. tudy Tip eading Math Equidistant means the same distance. tudy Tip ommon Misconception Note that Theorem 5. states the point is on the perpendicular bisector. It does not say that any line containing that point is a perpendicular bisector. Theorems 5. ny point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. Example: If and bisects, then and. 5. ny point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. oints on erpendicular isectors Example: If, then lies on the perpendicular bisector of. If, then lies on the perpendicular bisector of. You will prove Theorems 5. and 5. in Exercises 0 and, respectively. 8 hapter 5 elationships in Triangles Michael. Yamashita/OI ecall that a locus is the set of all points that satisfy a given condition. perpendicular bisector can be described as the locus of points in a plane equidistant from the endpoints of a given segment. ince a triangle has three sides, there are three perpendicular bisectors in a triangle. The perpendicular bisectors of a triangle intersect at a common point. When three or more lines intersect at a common point, the lines are called concurrent lines, and their point of intersection is called the point of concurrency. The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter.

6 Theorem 5. ircumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. Example: If J is the circumcenter of, then J J J. circumcenter J tudy Tip isectors In most triangles, a perpendicular bisector will not pass through the vertex opposite a given side. Likewise, an angle bisector usually will not bisect the side opposite an angle. tudy Tip Locus n angle bisector can be described as the locus of points in a plane equidistant from the sides of an angle. Theorem 5. roof Given:, m, and n are perpendicular bisectors of,, and, respectively. n J rove: J J J aragraph roof: m ince J lies on the perpendicular bisector of, it is equidistant from and. y the definition of equidistant, J J. The perpendicular bisector of also contains J. Thus, J J. y the Transitive roperty of Equality, J J. Thus, J J J. nother special line, segment, or ray in triangles is an angle bisector. Example Given: rove: roof: Use ngle isectors X bisects, X Y and X Z X Y X Z tatements easons. X bisects, X Y, and. Given X Z.. YX ZX. efinition of angle bisector. YX and ZX are right angles.. efinition of perpendicular 4. YX ZX 4. ight angles are congruent. 5. X X 5. eflexive roperty 6. YX ZX X Y X Z 7. T In Example, XY and XZ are lengths representing the distance from X to each side of. This is a proof of Theorem 5.4. Theorems 5.4 ny point on the angle bisector is equidistant from the sides of the angle. 5.5 ny point equidistant from the sides of an angle lies on the angle bisector. Y Z X oints on ngle isectors You will prove Theorem 5.5 in Exercise. Lesson 5- isectors, Medians, and ltitudes 9

7 s with perpendicular bisectors, there are three angle bisectors in any triangle. The angle bisectors of a triangle are concurrent, and their point of concurrency is called the incenter of a triangle. Theorem 5.6 Incenter Theorem The incenter of a triangle is equidistant from each side of the triangle. Example: If K is the incenter of, then K K K. incenter K tudy Tip Medians as isectors ecause the median contains the midpoint, it is also a bisector of the side of the triangle. You will prove Theorem 5.6 in Exercise. MEIN N LTITUE median is a segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex. Every triangle has three medians. The medians of a triangle also intersect at a common point. The point of concurrency for the medians of a triangle is called a centroid. The centroid is the point of balance for any triangle. Theorem 5.7 entroid Theorem The centroid of a triangle is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. Example: If L is the centroid of L = E, L = F, and L =. centroid L F E tudy Tip Eliminating Fractions You could also multiply the equation T by to eliminate the denominator. Example egment Measures LGE oints, T, and U are the midpoints of E, E F, and F, respectively. Find x, y, and z. Find x. T T 6 (x 5) x egment ddition ostulate ubstitution implify. T entroid Theorem 6 [x ] 6, T x 8 4x Multiply each side by and simplify. 6 4x ubtract from each side. 4 x ivide each side by 4. E y x 5 4z T U F 40 hapter 5 elationships in Triangles

8 Find y. E EU entroid Theorem y (y.9) E y, EU y.9 y y 5.8 y 5.8 Find z. F F Multiply each side by and simplify. ubtract y from each side. entroid Theorem 4.6 (4.6 4z) F 4.6, F z z Multiply each side by and simplify z ubtract 9. from each side z ivide each side by 8. n altitude of a triangle is a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side. Every triangle has three altitudes. The intersection point of the altitudes of a triangle is called the orthocenter. T M U orthocenter Finding the orthocenter can be used to help you construct your own nine-point circle. Visit online.com/webquest to continue work on your Webuest project. If the vertices of a triangle are located on a coordinate plane, you can use a system of equations to find the coordinates of the orthocenter. Example Use a ystem of Equations to Find a oint OOINTE GEOMETY The vertices of JKL are J(, ), K(, ), and L(, 0). Find the coordinates of the orthocenter of JKL. y J(, ) Find an equation of the altitude from J to K L. The slope of K L is, so the slope of the altitude is. (y y ) m(x x ) oint-slope form (y ) (x ) x, y, m = y x y x istributive roperty dd to each side. L(, 0) O x K(, ) Next, find an equation of the altitude from K to J L. The slope of J L is, so the slope of the altitude to J L is. (y y ) m(x x ) oint-slope form (y ) (x ) x, y, m y x 4 y x istributive roperty ubtract from each side. (continued on the next page) Lesson 5- isectors, Medians, and ltitudes 4

9 tudy Tip Graphing alculator Once you have two equations, you can graph the two lines and use the Intersect option on the alc menu to determine where the two lines meet. Then, solve a system of equations to find the point of intersection of the altitudes. Find x. eplace x with in one of the y x Equation of altitude from K equations to find the y-coordinate. x x ubstitution, y = x y x = 9x x Multiply each side by. y Multiply. x dd x to each side. x ivide each side by. The coordinates of the orthocenter of JKL are,. You can also use systems of equations to find the coordinates of the circumcenter and the centroid of a triangle graphed on a coordinate plane. pecial egments in Triangles Name Type oint of oncurrency perpendicular bisector line, segment, or ray circumcenter angle bisector line, segment, or ray incenter median segment centroid altitude segment orthocenter oncept heck. ompare and contrast a perpendicular bisector and a median of a triangle.. OEN ENE raw a triangle in which the circumcenter lies outside the triangle. Guided ractice. Find a counterexample to the statement n altitude and an angle bisector of a triangle are never the same segment. 4. OOINTE GEOMETY The vertices of are (, ), (, ), and (, 4). Find the coordinates of the circumcenter. 5. OOF Write a two-column proof. Given: X Y X Z Y M and Z N are medians. rove: Y M Z N M X N pplication 6. LGE Lines, m, and n are perpendicular bisectors of and meet at T. If T x, T y, and T 8, find x, y, and z. Z Y m x n y T 8 z hapter 5 elationships in Triangles

10 ractice and pply For Exercises ee Examples 0, 6, 6 7 9, 7 0 Extra ractice ee page 76. OOINTE GEOMETY The vertices of EF are (4, 0), E(, 4), and F(0, 6). Find the coordinates of the points of concurrency of EF. 7. centroid 8. orthocenter 9. circumcenter 0. OOF Write a paragraph proof of Theorem 5.. Given: is the perpendicular bisector of. E is a point on. rove: E E E OOF Write a two-column proof.. Given: UVW is isosceles with vertex angle UVW. Y V is the bisector of UVW. rove: Y V is a median. Y U W V. Given: G L is a median of EGH. J M is a median of IJK. EGH IJK rove: G L J M E L H G M I K J. LGE Find x and m if M is an altitude of MN, m x, and m 7x LGE If M is a median of MN, a 4, N a, and m M 7a, find the value of a. Is M also an altitude of MN? Explain. M Exercises and 4 N 5. LGE If W is a median and an angle bisector, y, H 7y 5, m HW x, m W x, and m HW 4x 6, find x and y. Is W also an altitude? Explain. 6. LGE If W is a perpendicular bisector, m WH 8q 7, m HW 0 q, 6r 4, and H r, find r, q, and m HW. H X W Exercises 5 and 6 tate whether each sentence is always, sometimes, or never true. 7. The three medians of a triangle intersect at a point in the interior of the triangle. 8. The three altitudes of a triangle intersect at a vertex of the triangle. 9. The three angle bisectors of a triangle intersect at a point in the exterior of the triangle. 0. The three perpendicular bisectors of a triangle intersect at a point in the exterior of the triangle. Lesson 5- isectors, Medians, and ltitudes 4

11 . LGE Find x if is a. LGE Find x if is an median of. altitude of. (5x 4) (4x 6) x 7 x 5 0x 7 5x LGE For Exercises 6, use the following information. In, Z a, Z a 5, Y c, Y 4c, m Z 4b 7, m Z b 4, m Y 7b 6, and m X a 0. Z X. X is an altitude of. Find a. 4. If Z is an angle bisector, find m Z. 5. Find if Y is a median. Y 6. If Y is a perpendicular bisector of, find b. OOINTE GEOMETY For Exercises 7 0, use the following information. (, ), (, 6), and T(, 8) are the vertices of T, and X is a median. 7. What are the coordinates of X? 8. Find X. 9. etermine the slope of X. 0. Is X an altitude of T? Explain. Orienteering The International Orienteering Federation World up consists of a series of nine races held throughout the world, in which the runners compete for points based on their completion times. ource: OOF Write a two-column proof for each theorem.. Theorem 5. Given: rove: and are on the perpendicular bisector of.. Theorem 5.5. Theorem OIENTEEING Orienteering is a competitive sport, originating in weden, that tests the skills of map reading and cross-country running. ompetitors race through an unknown area to find various checkpoints using only a compass and topographical map. On an amateur course, clues were given to locate the first flag. The flag is as far from the Grand Tower as it is from the park entrance. If you run from tearns oad to the flag or from mesbury oad to the flag, you would run the same distance. escribe how to find the first flag. E tearns oad Entrance Grand Tower mesbury oad 44 hapter 5 elationships in Triangles Getty Images

12 TTITI For Exercises 5 8, use the following information. The mean of a set of data is an average value of the data. uppose has vertices (6, 8), (, 4), and ( 6, ). 5. Find the mean of the x-coordinates of the vertices. 6. Find the mean of the y-coordinates of the vertices. 7. Graph and its medians. 8. Make a conjecture about the centroid and the means of the coordinates of the vertices. 9. ITIL THINKING raw any XYZ with median X N and altitude X O. ecall that the area of a triangle is one-half the product of the measures of the base and the altitude. What conclusion can you make about the relationship between the areas of XYN and XZN? FT ractice tandardized Test ractice 40. WITING IN MTH nswer the question that was posed at the beginning of the lesson. How can you balance a paper triangle on a pencil point? Include the following in your answer: which special point is the center of gravity, and a construction showing how to find this point. F 4. In FGH, which type of segment is F J? angle bisector perpendicular bisector G median altitude J 4. LGE If xy 0 and x 0.y, then y H?. x Maintain Your kills Mixed eview osition and label each triangle on the coordinate plane. (Lesson 4-7) 4. equilateral with base n units long 44. isosceles EF with congruent sides a units long and base a units long 45. right GHI with hypotenuse G I, HI is three times GH, and GH is x units long Getting eady for the Next Lesson For Exercises 46 49, refer to the figure at the right. (Lesson 4-6) 46. If 9 0, name two congruent segments. 47. If N L L, name two congruent angles. 48. If L T L, name two congruent angles. 49. If 4, name two congruent segments. N L 50. INTEIO EIGN tacey is installing a curtain rod on the wall above the window. To ensure that the rod is parallel to the ceiling, she measures and marks 6 inches below the ceiling in several places. If she installs the rod at these markings centered over the window, how does she know the curtain rod will be parallel to the ceiling? (Lesson -6) I KILL eplace each with or to make each sentence true M T Lesson 5- isectors, Medians, and ltitudes 45

13 Math Words and Everyday Words everal of the words and terms used in mathematics are also used in everyday language. The everyday meaning can help you to better understand the mathematical meaning and help you remember each meaning. This table shows some words used in this chapter with the everyday meanings and the mathematical meanings. Word Everyday Meaning Geometric Meaning median a paved or planted strip a segment of a dividing a highway into triangle that lanes according to connects the direction of travel vertex to the midpoint of the opposite side Y X Z altitude the vertical elevation of a segment from a an object above a vertex of a triangle surface that is perpendicular to the line containing the opposite side T bisector something that divides a segment that into two usually divides an angle equal parts or a side into two parts of equal measure ource: Merriam-Webster ollegiate ictionary Notice that the geometric meaning is more specific, but related to the everyday meaning. For example, the everyday definition of altitude is elevation, or height. In geometry, an altitude is a segment of a triangle perpendicular to the base through the vertex. The length of an altitude is the height of the triangle. eading to Learn. How does the mathematical meaning of median relate to the everyday meaning?. EEH Use a dictionary or other sources to find alternate definitions of vertex.. EEH Median has other meanings in mathematics. Use the Internet or other sources to find alternate definitions of this term. 4. EEH Use a dictionary or other sources to investigate definitions of segment. 46 Investigating lope-intercept Form 46 hapter 5 elationships in Triangles

14 Inequalities and Triangles unshine tate tandards M...4. ecognize and apply properties of inequalities to the measures of angles of a triangle. ecognize and apply properties of inequalities to the relationships between angles and sides of a triangle. can you tell which corner is bigger? am is delivering two potted trees to be used on a patio. The instructions say for the trees to be placed in the two largest corners of the patio. ll am has is a diagram of the triangular patio that shows the measurements 45 feet, 48 feet, and 5 feet. am can find the largest corner because the measures of the angles of a triangle are related to the measures of the sides opposite them. 5 ft 45 ft 48 ft NGLE INEULITIE In algebra, you learned about the inequality relationship between two real numbers. This relationship is often used in proofs. efinition of Inequality For any real numbers a and b, a b if and only if there is a positive number c such that a b c. Example: If 6 4, 6 4 and 6. The properties of inequalities you studied in algebra can be applied to the measures of angles and segments. omparison roperty roperties of Inequalities for eal Numbers For all numbers a, b, and c a b, a b, or a b Transitive roperty. If a b and b c, then a c.. If a b and b c, then a c. ddition and. If a b, then a c b c and a c b c. ubtraction roperties. If a b, then a c b c and a c b c. Multiplication and. If c 0 and a b, then ac bc and a c b c. ivision roperties. If c 0 and a b, then ac bc and a c b c.. If c 0 and a b, then ac bc and a c b c. 4. If c 0 and a b, then ac bc and a c b c. Lesson 5- Inequalities and Triangles 47

15 Example ompare ngle Measures etermine which angle has the greatest measure. Explore ompare the measure of to the measures of and. lan Use properties and theorems of real numbers to compare the angle measures. olve ompare m to m. y the Exterior ngle Theorem, m m m. ince angle measures are positive numbers and from the definition of inequality, m m. ompare m to m. gain, by the Exterior ngle Theorem, m m m. The definition of inequality states that if m m m, then m m. Examine m is greater than m and m. Therefore, has the greatest measure. The results from Example suggest that the measure of an exterior angle is always greater than either of the measures of the remote interior angles. This relationship can be stated as a theorem. tudy Tip ymbols for ngles and Inequalities The symbol for angle ( ) looks similar to the symbol for less than ( ), especially when handwritten. e careful to write the symbols correctly in situations where both are used. Theorem 5.8 Exterior ngle Inequality Theorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. Example: m 4 m m 4 m 4 The proof of Theorem 5.8 is in Lesson 5-. Example Exterior ngles Use the Exterior ngle Inequality Theorem to list all of the angles that satisfy the stated condition. a. all angles whose measures are less than m 8 y the Exterior ngle Inequality Theorem, m 8 m 4, m 8 m 6, m 8 m, and m 8 m 6 m 7. Thus, the measures of 4, 6,, and 7 are all less than m 8. b. all angles whose measures are greater than m y the Exterior ngle Inequality Theorem, m 8 m and m 4 m. Thus, the measures of 4 and 8 are greater than m hapter 5 elationships in Triangles NGLE-IE ELTIONHI ecall that if two sides of a triangle are congruent, then the angles opposite those sides are congruent. In the following Geometry ctivity, you will investigate the relationship between sides and angles when they are not congruent.

16 Inequalities for ides and ngles of Triangles Model raw an acute scalene triangle, and label the vertices,, and. Measure each side of the triangle. ecord the measures in a table. ide Measure Measure each angle of the triangle. ecord each measure in a table. ngle Measure nalyze. escribe the measure of the angle opposite the longest side in terms of the other angles.. escribe the measure of the angle opposite the shortest side in terms of the other angles.. epeat the activity using other triangles. Make a onjecture 4. What can you conclude about the relationship between the measures of sides and angles of a triangle? The Geometry ctivity suggests the following theorem. Theorem 5.9 If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. tudy Tip Theorem 5.9 The longest side in a triangle is opposite the largest angle in that triangle. roof Given: rove: Theorem 5.9 N m m N (continued on the next page) Lesson 5- Inequalities and Triangles 49

17 roof: tatements.,, N.. m m 4. m m 5. m m m 6. m m 7. m m 8. m m easons. Given. Isosceles Triangle Theorem. efinition of congruent angles 4. Exterior ngle Inequality Theorem 5. ngle ddition ostulate 6. efinition of inequality 7. ubstitution roperty of Equality 8. Transitive roperty of Inequality Example ide-ngle elationships etermine the relationship between the measures of the given angles. a., The side opposite is longer than the side opposite 5, so m m. b., m m m m m m m m m m The converse of Theorem 5.9 is also true. Theorem 5.0 If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. Treehouses The strength of the tree is the most important concern when building a treehouse. It is important to look for a tree that has branches thick and strong. ource: Example 4 You will prove Theorem 5.0 in Lesson 5-, Exercise 6. ngle-ide elationships TEEHOUE Mr. Jackson is constructing the framework for part of a treehouse for his daughter. He plans to install braces at the ends of a certain floor support as shown. Which supports should he attach to and? Theorem 5.9 states that if one angle of a triangle has a greater measure, then the side opposite that angle is longer than the side opposite the other angle. Therefore, Mr. Jackson should attach the longer brace at the end marked and the shorter brace at the end marked hapter 5 elationships in Triangles Tony Freeman/hotoEdit

18 oncept heck. tate whether the following statement is always, sometimes, or never true. In JKL with right angle J, if m J is twice m K, then the side opposite J is twice the length of the side opposite K.. OEN ENE raw. List the angle measures and side lengths of your triangle from greatest to least.. FIN THE EO Hector and Grace each labeled. Hector Grace Who is correct? Explain. Guided ractice etermine which angle has the greatest measure. 4.,, 4 5.,, 5 6.,,, 4, Use the Exterior ngle Inequality Theorem to list all angles that satisfy the stated condition. 7. all angles whose measures are less than m all angles whose measures are greater than m 6 9. all angles whose measures are less than m etermine the relationship between the measures of the given angles. 0. WXY, XYW. XZY, XYZ. WYX, XWY X W Y 7 Z etermine the relationship between the lengths of the given sides.. E, E 4. E, 5., E E pplication 6. ELL uring a baseball game, the batter hits the ball to the third baseman and begins to run toward first base. t the same time, the runner on first base runs toward second base. If the third baseman wants to throw the ball to the nearest base, to which base should he throw? Explain Lesson 5- Inequalities and Triangles 5

19 ractice and pply For Exercises ee Examples 4 Extra ractice ee page 76. etermine which angle has the greatest measure. 7.,, 4 8., 4, 6 9., 5, 7 0.,, 6. 5, 7, 8., 6, 8 Use the Exterior ngle Inequality Theorem to list all angles that satisfy the stated condition.. all angles whose measures are less than m 5 4. all angles whose measures are greater than m 6 5. all angles whose measures are greater than m Use the Exterior ngle Inequality Theorem to list all angles that satisfy the stated condition. 6. all angles whose measures are less than m 7. all angles whose measures are greater than m 9 8. all angles whose measures are less than m etermine the relationship between the measures of J the given angles KJ, JK 0. MJY, JYM 0. MJ, MJ. KJ, JK K M 9. MYJ, JMY 4. JY, JY 0 Y OOF Write a two-column proof. 5. Given: J M J L 6. Given: J L K L rove: m m rove: m m L K J M etermine the relationship between the lengths of the given sides. 7. Z Y, Y 8., Z 9. Z, 40. Z Y, Z 4. T Y, Z Y 4. T Y, Z T Z T Y OOINTE GEOMETY Triangle KLM has vertices K(, ), L(, 5), and M(, 7). List the angles in order from the least to the greatest measure. 5 hapter 5 elationships in Triangles 44. If > > in and M, N, and O are the medians of the triangle, list M, N, and O in order from least to greatest.

20 45. TVEL plane travels from es Moines to hoenix, on to tlanta, and then completes the trip directly back to es Moines as shown in the diagram. Write the lengths of the legs of the trip in order from greatest to least. es Moines (8x 4) (x 7) (5x ) hoenix tlanta Travel One sixth of adult mericans have never flown in a commercial aircraft. ource: U.. ureau of Transportation tatistics LGE Find the value of n. List the sides of in order from shortest to longest for the given angle measures. 46. m 9n 9, m 9 5n, m 0n 47. m n 9, m 6 n, m 6n 48. m 9n 4, m 4n 6, m 68 n 49. m n 0, m n 7, 4n m 4n 6, m 67 n, n OO The wedge at the right is used as a door stopper. The values of x and y are in inches. Write an inequality relating x and y. Then solve the inequality for y in terms of x. x 75 (y ) 6 5. OOF Write a paragraph proof for the following statement. If a triangle is not isosceles, then the measure of the median to any side of the triangle is greater than the measure of the altitude to that side. 5. ITIL THINKING Write and solve an inequality for x. x 5 (y ) (y 8) (4y ) 4x 7 FT ractice tandardized Test ractice 54. WITING IN MTH nswer the question that was posed at the beginning of the lesson.. How can you tell which corner is bigger? Include the following in your answer: the name of the theorem or postulate that lets you determine the comparison of the angle measures, and which angles in the diagram are the largest. 55. In the figure at the right, what is the value of p in terms of m and n? m n 80 m n 80 m n (m n) n p m 56. LGE If x x, then x? Lesson 5- Inequalities and Triangles 5 Jeff Greenberg/hotoEdit

21 Maintain Your kills Mixed eview LGE For Exercises 57 59, use the following information. (Lesson 5-) Two vertices of are (, 8) and (9, ). is a median with at (, ). 57. What are the coordinates of? 58. Is an altitude of? Explain. 59. The graph of point E is at (6, 6). E F intersects at F. If F is at 0, 7, is E F a perpendicular bisector of? Explain. TTITI For Exercises 60 and 6, refer to the graph at the right. (Lesson 4-7) 60. Find the coordinates of if the x-coordinate of is the mean of the x-coordinates of the vertices of and the y-coordinate is the mean of the y-coordinates of the vertices of. 6. rove that is the intersection of the medians of. O y (b, c) (0, 0) (a, 0) x Getting eady for the Next Lesson ractice uiz LGE Use. (Lesson 5-). Find x if is a median of.. Find y if is an altitude of. Name the corresponding congruent angles and sides for each pair of congruent triangles. (Lesson 4-) 6. TUV XYZ 6. G W 64. F GH 65. Find the value of x so that the line containing points at (x, ) and ( 4, 5) is perpendicular to the line containing points at (4, 8) and (, ). (Lesson -) I KILL etermine whether each equation is true or false if a, b 5, and c 6. (To review evaluating expressions, see page 76.) 66. ab c(b a) a c > a b tate whether each statement is always, sometimes, or never true. (Lesson 5-). The medians of a triangle intersect at one of the vertices of the triangle. 4. The angle bisectors of a triangle intersect at a point in the interior of the triangle. 5. The altitudes of a triangle intersect at a point in the exterior of the triangle. 6. The perpendicular bisectors of a triangle intersect at a point on the triangle. 4x 9 (y 6) 7x 6 Lessons 5- and 5-7. escribe a triangle in which the angle bisectors all intersect in a point outside the triangle. If no triangle exists, write no triangle. (Lesson 5-) T 8. List the sides of TU in order from longest to shortest. (Lesson 5-) LGE In, m x 0, m x 7, and m 4x 5. (Lesson 5-) 9. etermine the measure of each angle. 0. List the sides in order from shortest to longest. 4 7 uestion 8 9 U 54 hapter 5 elationships in Triangles

22 Indirect roof unshine tate tandards M...4. Vocabulary indirect reasoning indirect proof proof by contradiction Use indirect proof with algebra. Use indirect proof with geometry. is indirect proof used in literature? In The dventure of the lanched oldier, herlock Holmes describes his detective technique, stating, That process starts upon the supposition that when you have eliminated all which is impossible, then whatever remains,... must be the truth. The method herlock Holmes uses is an example of indirect reasoning. tudy Tip Truth Value of a tatement ecall that a statement must be either true or false. To review truth values, see Lesson -. INIET OOF WITH LGE The proofs you have written so far use direct reasoning, in which you start with a true hypothesis and prove that the conclusion is true. When using indirect reasoning, you assume that the conclusion is false and then show that this assumption leads to a contradiction of the hypothesis, or some other accepted fact, such as a definition, postulate, theorem, or corollary. ince all other steps in the proof are logically correct, the assumption has been proven false, so the original conclusion must be true. proof of this type is called an indirect proof or a proof by contradiction. The following steps summarize the process of an indirect proof.. ssume that the conclusion is false. teps for Writing an Indirect roof. how that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem, or corollary.. oint out that because the false conclusion leads to an incorrect statement, the original conclusion must be true. Example tating onclusions tate the assumption you would make to start an indirect proof of each statement. a. MN MN b. is an isosceles triangle. is not an isosceles triangle. c. x 4 If x 4 is false, then x 4 or x 4. In other words, x 4. d. If 9 is a factor of n, then is a factor of n. The conclusion of the conditional statement is is a factor of n. The negation of the conclusion is is not a factor of n. Lesson 5- Indirect roof 55 Joshua Ets-Hokin/hotoisc

23 Indirect proofs can be used to prove algebraic concepts. Example Given: x 7 rove: x 5 lgebraic roof Indirect roof: tep ssume that x 5. That is, assume that x 5 or x 5. tep tep Make a table with several possibilities for x given that x 5 or x 5. This is a contradiction because when x 5 or x 5, x 7. In both cases, the assumption leads to the contradiction of a known fact. Therefore, the assumption that x 5 must be false, which means that x 5 must be true. x x Indirect reasoning and proof can be used in everyday situations. Example Use Indirect roof HOING Lawanda bought two skirts for just over $60, before tax. few weeks later, her friend Tiffany asked her how much each skirt cost. Lawanda could not remember the individual prices. Use indirect reasoning to show that at least one of the skirts cost more than $0. Given: The two skirts cost more than $60. rove: t least one of the skirts cost more than $0. That is, if x y 60, then either x 0 or y 0. hopping The West Edmonton Mall in Edmonton, lberta, anada, is the world s largest entertainment and shopping center, with an area of 5. million square feet. The mall houses an amusement park, water park, ice rink, and aquarium, along with over 800 stores and services. ource: Indirect roof: tep ssume that neither skirt costs more than $0. That is, x 0 and y 0. tep tep If x 0 and y 0, then x y 60. This is a contradiction because we know that the two skirts cost more than $60. The assumption leads to the contradiction of a known fact. Therefore, the assumption that x 0 and y 0 must be false. Thus, at least one of the skirts had to have cost more than $0. INIET OOF WITH GEOMETY prove statements in geometry. Example 4 Given: rove: m Geometry roof Indirect roof: tep ssume that. Indirect reasoning can be used to n m 56 hapter 5 elationships in Triangles James Marshall/OI

24 tep tep and are corresponding angles. If two lines are cut by a transversal so that corresponding angles are congruent, the lines are parallel. This means that m. However, this contradicts the given statement. ince the assumption leads to a contradiction, the assumption must be false. Therefore,. Indirect proofs can also be used to prove theorems. roof Given: rove: Exterior ngle Inequality Theorem is an exterior angle of. m m and m m 4 4 Indirect roof: tep tep Make the assumption that m m or m m 4. In other words, m m or m m 4. You only need to show that the assumption m m leads to a contradiction as the argument for m m 4 follows the same reasoning. m m means that either m m or m m. ase : m m m m m 4 Exterior ngle Theorem m m m 4 ubstitution 0 m 4 ubtract m from each side. This contradicts the fact that the measure of an angle is greater than 0, so m m. ase : m m y the Exterior ngle Theorem, m m m 4. ince angle measures are positive, the definition of inequality implies m m and m m 4. This contradicts the assumption. tep In both cases, the assumption leads to the contradiction of a theorem or definition. Therefore, the assumption that m m and m m 4 must be true. oncept heck. Explain how contradicting a known fact means that an assumption is false.. ompare and contrast indirect proof and direct proof. ee margin.. OEN ENE tate a conjecture. Then write an indirect proof to prove your conjecture. Lesson 5- Indirect roof 57

25 Guided ractice Write the assumption you would make to start an indirect proof of each statement. 4. If 5x 5, then x Two lines that are cut by a transversal so that alternate interior angles are congruent are parallel. 6. If the alternate interior angles formed by two lines and a transversal are congruent, the lines are parallel. OOF Write an indirect proof. 7. Given: a 0 8. Given: n is odd. rove: a 0 rove: n is odd. 9. Given: 0. Given: m n rove: There can be no more than rove: Lines m and n intersect at one obtuse angle in. exactly one point.. OOF Use an indirect proof to show that the hypotenuse of a right triangle is the longest side. pplication. IYLING The Tour de France bicycle race takes place over several weeks in various stages throughout France. uring two stages of the 00 Tour de France, riders raced for just over 70 miles. rove that at least one of the stages was longer than 5 miles. ractice and pply For Exercises 8 9, 0,,, 4, 5 ee Examples, 4 Extra ractice ee page 76. Write the assumption you would make to start an indirect proof of each statement.. T 4. If x, then x 4. a 5. If a rational number is any number that can be expressed as, where b a and b are integers, and b 0, 6 is a rational number. 6. median of an isosceles triangle is also an altitude. 7. oints,, and are collinear. 8. The angle bisector of the vertex angle of an isosceles triangle is also an altitude of the triangle. OOF Write an indirect proof. 9. Given: a 0 0. Given: n is even. rove: a is negative. rove: n is divisible by 4.. Given:. Given: m m rove: m rove: Z is not a median of. t m Z 58 hapter 5 elationships in Triangles

26 OOF Write an indirect proof.. If a 0, b 0, and a b, then b a. 4. If two sides of a triangle are not congruent, then the angles opposite those sides are not congruent. 5. Given: and are equilateral. is not equilateral. rove: is not equilateral. 6. Theorem 5.0 Given: m m rove: 7. TVEL amon drove 75 miles from eattle, Washington, to ortland, Oregon. It took him three hours to complete the trip. rove that his average driving speed was less than 60 miles per hour. EUTION For Exercises 8 0, refer to the graphic at the right. 8. rove the following statement. The majority of college-bound seniors stated that they received college information from a guidance counselor. 9. If 500 seniors were polled for this survey, verify that 5 said they received college information from a friend. 0. id more seniors receive college information from their parents or from teachers and friends? Explain. U TOY napshots ources of college information eople whom college-bound high school seniors say they receive college information from: arents High school teacher Friend ollege student Other relative 9% % 5% Guidance counselor 8% % ource: tamats ommunications Teens Talk, % y indy Hall and Marcy E. Mullins, U TOY. LW uring the opening arguments of a trial, a defense attorney stated, My client is innocent. The police report states that the crime was committed on November 6 at approximately 0:5.M. in an iego. I can prove that my client was on vacation in hicago with his family at this time. verdict of not guilty is the only possible verdict. Explain whether this is an example of indirect reasoning.. EEH Use the Internet or other resource to write an indirect proof for the following statement. In the tlantic Ocean, the percent of tropical storms that developed into hurricanes over the past five years varies from year to year. Lesson 5- Indirect roof 59

27 . ITIL THINKING ecall that a rational number is any number that can be expressed in the form b a, where a and b are integers with no common factors and b 0, or as a terminating or repeating decimal. Use indirect reasoning to prove that is not a rational number. FT ractice tandardized Test ractice 4. WITING IN MTH nswer the question that was posed at the beginning of the lesson. How is indirect proof used in literature? Include the following in your answer: an explanation of how herlock Holmes used indirect proof, and an example of indirect proof used every day. 5. Which statement about the value of x is not true? x 60 x 40 x x 60 x OILITY bag contains 6 blue marbles, 8 red marbles, and white marbles. If three marbles are removed at random and no marble is returned to the bag after removal, what is the probability that all three marbles will be red? Maintain Your kills Mixed eview For Exercises 7 and 8, refer to the figure at the right. (Lesson 5-) 7. Which angle in MO has the greatest measure? 8. Name the angle with the least measure in LMN. O L N 7 M 5 OOF Write a two-column proof. (Lesson 5-) 9. If an angle bisector of a triangle is also an altitude of the triangle, then the triangle is isosceles. 40. The median to the base of an isosceles triangle bisects the vertex angle. 4. orresponding angle bisectors of congruent triangles are congruent. 4. TONOMY The ig ipper is a part of the larger constellation Ursa Major. Three of the brighter stars in the constellation form. If m 4 and m 09, find m. (Lesson 4-) Ursa Major Write an equation in point-slope form of the line having the given slope that contains the given point. (Lesson -4) 4. m, (4, ) 44. m, (, ) 45. m, ( 4, 9) Getting eady for the Next Lesson EEUIITE KILL etermine whether each inequality is true or false. (To review the meaning of inequalities, see pages 79 and 740.) hapter 5 elationships in Triangles

28 The Triangle Inequality unshine tate tandards M...4. pply the Triangle Inequality Theorem. etermine the shortest distance between a point and a line. can you use the Triangle Inequality Theorem when traveling? huck Noland travels between hicago, Indianapolis, and olumbus as part of his job. Mr. Noland lives in hicago and needs to get to olumbus as quickly as possible. hould he take a flight that goes from hicago to olumbus, or a flight that goes from hicago to Indianapolis, then to olumbus? hicago eoria nn rbor pringfied Indianapolis Fort Wayne olumbus THE TINGLE INEULITY In the example above, if you chose to fly directly from hicago to olumbus, you probably reasoned that a straight route is shorter. This is an example of the Triangle Inequality Theorem. Theorem 5. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Examples: You will prove Theorem 5. in Exercise 40. The Triangle Inequality Theorem can be used to determine whether three segments can form a triangle. tudy Tip Inequality If the sum of the smallest number and the middle number is greater than the largest number, then each combination of inequalities are true. Example Identify ides of a Triangle etermine whether the given measures can be the lengths of the sides of a triangle. a., 4, 5 heck each inequality. 4? 5 5? 4 4 5? ll of the inequalities are true, so, 4, and 5 can be the lengths of the sides of a triangle. b. 6, 8, 4 6 8? 4 ecause the sum of two measures equals the measure of the 4 4 third side, the sides cannot form a triangle. Lesson 5-4 The Triangle Inequality 6

29 When you know the lengths of two sides of a triangle, you can determine the range of possible lengths for the third side. FT ractice tandardized Test ractice Example Multiple-hoice Test Item etermine ossible ide Length In XYZ, XY 8, and XZ 4. Which measure cannot be YZ? X Y Z Test-Taking Tip Testing hoices If you are short on time, you can test each choice to find the correct answer and eliminate any remaining choices. ead the Test Item You need to determine which value is not valid. olve the Test Item olve each inequality to determine the range of values for YZ. Let YZ n. XY XZ YZ XY YZ XZ YZ XZ XY 8 4 n 8 n 4 n 4 8 n or n n 6 n 6 Graph the inequalities on the same number line Graph n. Graph n 6. Graph n 6. Find the intersection. The range of values that fit all three inequalities is 6 n. Examine the answer choices. The only value that does not satisfy the compound inequality is 6 since 6 6. Thus, the answer is choice. ITNE ETWEEN OINT N LINE ecall that the distance between point and line is measured along a perpendicular segment from the point to the line. It was accepted without proof that was the shortest segment from to. The theorems involving the relationships between the angles and sides of a triangle can now be used to prove that a perpendicular segment is the shortest distance between a point and a line. Theorem 5. The perpendicular segment from a point to a line is the shortest segment from the point to the line. Example: is the shortest segment from to. shortest distance 6 hapter 5 elationships in Triangles

30 tudy Tip hortest istance to a Line If a line is horizontal, the shortest distance from a point to that line will be along a vertical line. Likewise, the shortest distance from a point to a vertical line lies along a horizontal line. Example Given: rove: roof: rove Theorem 5. is any nonperpendicular segment from to. > tatements easons.. Given. and are right angles.. lines form right angles... ll right angles are congruent. 4. m m 4. efinition of congruent angles 5. m m 5. Exterior ngle Inequality Theorem 6. m m 6. ubstitution roperty If an angle of a triangle is greater than a second angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. orollary 5. follows directly from Theorem 5.. orollary 5. The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Example: is the shortest segment from to lane M. shortest distance M You will prove orollary 5. in Exercise. oncept heck. Explain why the distance between two nonhorizontal parallel lines on a coordinate plane cannot be found using the distance between their y-intercepts.. FIN THE EO Jameson and noki drew EFG with FG and EF 5. They each chose a possible measure for GE Jameson G noki G 0 8 F 5 E F 5 E Who is correct? Explain.. OEN ENE Find three numbers that can be the lengths of the sides of a triangle and three numbers that cannot be the lengths of the sides of a triangle. Justify your reasoning with a drawing. Lesson 5-4 The Triangle Inequality 6

31 Guided ractice etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain. 4. 5, 4, 5. 5, 5, , 0.8, , 0., 5. Find the range for the measure of the third side of a triangle given the measures of two sides and 9. 4 and 0. and 4. 5 and 8 FT ractice tandardized Test ractice. OOF Write a proof for orollary 5.. Given: plane M rove: is the shortest segment from to plane M.. n isosceles triangle has a base 0 units long. If the congruent side lengths have whole number measures, what is the shortest possible length of the sides? ractice and pply For Exercises ee Examples Extra ractice ee page 764. etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain. 4.,, 5., 6, 6. 8, 8, 5 7., 6, ,, 9. 9,, , 7, 9. 7, 0, , 7.,.5. 4, 0.9, ,, , 0., 0.5 Find the range for the measure of the third side of a triangle given the measures of two sides and 7. 7 and and 5 9. and 8 0. and 47. and 6. 0 and and and and and and OOF Write a two-column proof. 8. Given: 9. Given: H E E G rove: rove: HE FG EF H E G F 40. Given: rove: (Triangle Inequality Theorem) (Hint: raw auxiliary segment, so that is between and and.) 64 hapter 5 elationships in Triangles

32 LGE etermine whether the given coordinates are the vertices of a triangle. Explain. 4. (5, 8), (, 4), (, ) 4. L( 4, 9), M(, 0), N( 5, 7) 4. X(0, 8), Y(6, ), Z(8, 5) 44. (, 4), (, 0), T(5, ) FT For Exercises 45 and 46, use the following information. arlota has several strips of trim she wishes to use as a triangular border for a section of a decorative quilt she is going to make. The strips measure centimeters, 4 centimeters, 5 centimeters, 6 centimeters, and centimeters. 45. How many different triangles could arlota make with the strips? 46. How many different triangles could arlota make that have a perimeter that is divisible by? 47. HITOY The early Egyptians used to make triangles by using a rope with knots tied at equal intervals. Each vertex of the triangle had to occur at a knot. How many different triangles can be formed using the rope below? History ncient Egyptians used pieces of flattened, dried papyrus reed as paper. urviving examples include the hind apyrus and the Moscow apyrus, from which we have attained most of our knowledge about Egyptian mathematics. ource: OILITY For Exercises 48 and 49, use the following information. One side of a triangle is feet long. Let m represent the measure of the second side and n represent the measure of the third side. uppose m and n are whole numbers and that 4 m 7 and n List the measures of the sides of the triangles that are possible. 49. What is the probability that a randomly chosen triangle that satisfies the given conditions will be isosceles? 50. ITIL THINKING tate and prove a theorem that compares the measures of each side of a triangle with the differences of the measures of the other two sides. FT ractice tandardized Test ractice 5. WITING IN MTH nswer the question that was posed at the beginning of the lesson. How can you use the Triangle Inequality when traveling? Include the following in your answer: an example of a situation in which you might want to use the greater measures, and an explanation as to why it is not always possible to apply the Triangle Inequality when traveling. 5. If two sides of a triangle measure and 7, which of the following cannot be the perimeter of the triangle? LGE How many points of intersection exist if the equations (x 5) (y 5) 4 and y x are graphed on the same coordinate plane? none one two three Lesson 5-4 The Triangle Inequality 65 ritish Museum, London/rt esource, NY

33 Maintain Your kills Mixed eview 54. OOF Write an indirect proof. (Lesson 5-) Given: is a point not on line. rove: is the only line through perpendicular to. LGE List the sides of in order from longest to shortest if the angles of have the given measures. (Lesson 5-) 55. m 7x 8, m 8x 0, m 7x m x 44, m 68 x, m x 6 etermine whether JKL given the coordinates of the vertices. Explain. (Lesson 4-4) 57. J(0, 5), K(0, 0), L(, 0), (4, 8), (4, ), (6, ) 58. J(6, 4), K(, 6), L( 9, 5), (0, 7), (5, ), (5, 8) 59. J( 6, ), K(, 5), L(, ), (, ), (5, 4), (0, 0) Getting eady for the Next Lesson EEUIITE KILL olve each inequality. (To review solving inequalities, see pages 79 and 740.) 60. x x 4 x x 7 80 ractice uiz Lessons 5- and 5-4 Write the assumption you would make to start an indirect proof of each statement. (Lesson 5-). The number 7 is divisible by.. m m Write an indirect proof. (Lesson 5-). If 7x 56, then x Given: M O O N, M N 5. Given: m m rove: MO NO rove: is not an altitude of. M O N etermine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain. (Lesson 5-4) 6. 7, 4, , 5, ,, , 0, 6 0. If the measures of two sides of a triangle are 57 and, what is the range of possible measures of the third side? (Lesson 5-4) 66 hapter 5 elationships in Triangles

34 Inequalities Involving Two Triangles unshine tate tandards M...4. pply the Inequality. pply the Inequality. does a backhoe work? Many objects, like a backhoe, have two fixed arms connected by a joint or hinge. This allows the angle between the arms to increase and decrease. s the angle changes, the distance between the endpoints of the arms changes as well. INEULITY The relationship of the arms and the angle between them illustrates the following theorem. tudy Tip Inequality The Inequality Theorem is also called the Hinge Theorem. Theorem 5. Inequality/Hinge Theorem If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle. Example: Given,, if m m, then. roof Inequality Theorem Given: and EF F, E F m F m E rove: E F roof: We are given that F and E F. We also know that m F m. raw auxiliary ray FZ such that m FZ m and that Z F. This leads to two cases. ase : If Z lies on E, then FZ by. Thus, Z by T and the definition of congruent segments. y the egment ddition ostulate, E EZ Z. lso, E Z by the definition of inequality. Therefore, E by the ubstitution roperty. F E Z Lesson 5-5 Inequalities Involving Two Triangles 67 Jeremy Walker/Getty Images

35 ase : If Z does not lie on E, then let the intersection of F Z and E be point T. Now draw another auxiliary segment F V such that V is on E and EFV VFZ. E F V T Z ince F Z and E F, we have F Z E F by the Transitive roperty. lso V F is congruent to itself by the eflexive roperty. Thus, EFV ZFV by. y T, E V Z V or EV ZV. lso, FZ by. o, Z by T or Z. In VZ, V ZV Z by the Triangle Inequality Theorem. y substitution, V EV Z. ince E V EV by the egment ddition ostulate, E Z. Using substitution, E or E. Example Use Inequality in a roof Write a two-column proof. Given: Y Z X Z Z is the midpoint of. m ZY m ZX Y X rove: X Y roof: tatements easons. Y Z X Z. Given Z is the midpoint of. m ZY m ZX Y X. Z Z. efinition of midpoint. Y X. Inequality 4. Y X 4. efinition of congruent segments 5. Y Y X X 5. ddition roperty 6. Y Y 6. egment ddition ostulate X X ubstitution roperty INEULITY The converse of the Inequality Theorem is the Inequality Theorem. Z Theorem 5.4 Inequality If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. Example: Given,, if, then m m. You will prove Theorem 5.4 in Exercise hapter 5 elationships in Triangles

36 Example rove Triangle elationships Given: rove: Flow roof: m O m O O Given O O Given lt. Int. s Th. O O T O O eflexive roperty ( ) Given m O m O Inequality m O m O ubstitution ( ) O O O O Vert. s are m O m O m O m O ef. of s You can use algebra to relate the measures of the angles and sides of two triangles. Example elationships etween Two Triangles Write an inequality using the information in the figure. a. ompare m and m. In and,,, and. The Inequality allows us to conclude that m m. (5x 4) b. Find the range of values containing x. y the Inequality, m m, or m m. m m Inequality 5x 4 46 ubstitution 5x 60 dd 4 to each side. x ivide each side by 5. lso, recall that the measure of any angle is always greater than 0. 5x 4 0 5x 4 dd 4 to each side. x 4 or.8 ivide each side by 5. 5 The two inequalities can be written as the compound inequality.8 x. Lesson 5-5 Inequalities Involving Two Triangles 69

37 Inequalities involving triangles can be used to describe real-world situations Health hysical therapists help their patients regain range of motion after an illness or injury. They also teach patients exercises to help prevent injury. ource: Example 4 Use Triangle Inequalities HELTH ange of motion describes the amount that a limb can be moved from a straight position. To determine the range of motion of a person s forearm, determine the distance from his or her wrist to the shoulder when the elbow is bent as far as possible. uppose Jessica can bend her left arm so her wrist is 5 inches from her shoulder and her right arm so her right wrist is inches from her shoulder. Which of Jessica s arms has the greater range of motion? Explain. in. 5 in. The distance between the wrist and shoulder is smaller on her right arm. ssuming that both her arms have the same measurements, the inequality tells us that the angle formed at the elbow is smaller on the right arm. This means that the right arm has a greater range of motion. oncept heck Guided ractice. OEN ENE escribe a real-world object that illustrates either or inequality.. ompare and contrast the Inequality Theorem to the ostulate for triangle congruence. Write an inequality relating the given pair of angles or segment measures.., 4. m, m Write an inequality to describe the possible values of x x x 7 6 (7x 4) OOF Write a two-column proof. 7. Given: 8. Given: T U U rove: U V rove: T UV T T U V 70 hapter 5 elationships in Triangles ob aemmrich/the Image Works

38 pplication 9. TOOL lever is used to multiply the force applied to an object. One example of a lever is a pair of pliers. Use the or Inequality to explain how to use a pair of pliers. ractice and pply For Exercises , 6 ee Examples 4 Extra ractice ee page 764. Write an inequality relating the given pair of angles or segment measures. 0., F. m, m F. m F, m F Write an inequality relating the given pair of angles or segment measures.., 4. O, O 5. m O, m O F O Write an inequality to describe the possible values of x x x x 8 T x 8 x x 8 8. In the figure, M M,, m 5x 0 and m 8x 00. Write an inequality to describe the possible values of x. M 9. In the figure, m V 5 5x, m VT 0x 0, T, and TV TV. Write an inequality to describe the possible values of x. V T OOF Write a two-column proof. 0. Given:. Given: rove: rove: m m 4 Lesson 5-5 Inequalities Involving Two Triangles 7 quared tudios/hotoisc

39 OOF Write a two-column proof.. Given:. Given: E F m m rove: m m is the midpoint of. E F rove: 4 F E More bout OOF Use an indirect proof to prove the Inequality Theorem (Theorem 5.4). Given: U W T W V T UV rove: m m W U T W V 5. OO Open a door slightly. With the door open, measure the angle made by the door and the door frame. Measure the distance from the end of the door to the door frame. Open the door wider, and measure again. How do the measures compare? E Landscape rchitect Landscape architects design the settings of buildings and parklands by arranging both the location of the buildings and the placement of plant life so the site is functional, beautiful, and environmentally friendly. Online esearch For information about a career as a landscape architect, visit: com/careers 6. LNING When landscapers plant new trees, they usually brace the tree using a stake tied to the trunk of the tree. Use the or Inequality to explain why this is an effective method for supporting a newly planted tree. 7. ITIL THINKING The Inequality states that the base of an isosceles triangle gets longer as the measure of the vertex angle increases. escribe the effect of changing the measure of the vertex angle on the measure of the altitude. IOLOGY For Exercises 8 0, use the following information. The velocity of a person walking or running can be estimated using the formula v 0.7 8s.67 h., 7 where v is the velocity of the person in meters per second, s is the length of the stride in meters, and h is the height of the hip in meters. 8. Find the velocities of two people that each have a hip height of 0.85 meters and whose strides are.0 meter and. meters. 9. opy and complete the table at the right tride (m) Velocity (m/s) for a person whose hip height is 0.5. meters iscuss how the stride length is related to either the Inequality of the 0.75 Inequality hapter 5 elationships in Triangles achel Epstein/hotoEdit