1. Sketch an obtuse scalene triangle. Label its interior angles 1, 2, and 3. Then draw its exterior angles. Shade the exterior angles.

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1 UI I Vocabular heck oncept heck. ketch an obtuse scalene triangle. abel its interior angles, 2, and 3. hen draw its eterior angles. hade the eterior angles. In the figure, Q Æ Æ and Æ fi Q Æ. omplete the sentence. 2. Q Æ is the? of the right triangle Q. 3. In Q, Q Æ is the side opposite angle?. kill heck Æ 4. Q is the? of the isosceles triangle Q. 5. he legs of are? and?. In ercises 6 8, classif the triangle b its angles and b its sides he measure of one interior angle of a triangle is 25. he other interior angles are congruent. ind the measures of the other interior angles. I IIO U H tra ractice to help ou master skills is on p HI I In ercises 0 5, match the triangle description with the most specific name. 0. ide lengths: 2 cm, 3 cm, 4 cm. ide lengths: 3 cm, 2 cm, 3 cm 2. ide lengths: 4 cm, 4 cm, 4 cm 3. ngle measures: 60, 60, ngle measures: 30, 60, ngle measures: 20, 45, 5. uilateral. calene. Obtuse. uiangular. Isosceles. ight IYI I lassif the triangle b its angles and b its sides U H HOWOK H ample : s. 0 26, ample 2: s. 27, 28, 45 ample 3: s ample 4: s U 2. J 85 V K 98 hapter 4 ongruent riangles

2 OI OI omplete the statement using alwas, sometimes, or never. 22. n isosceles triangle is? an euilateral triangle. 23. n obtuse triangle is? an isosceles triangle. 24. n interior angle of a triangle and one of its adjacent eterior angles are? supplementar. 25. he acute angles of a right triangle are? complementar. 26. triangle? has a right angle and an obtuse angle. IIYI O I efer to the triangles in ercises Identif the legs and the hpotenuse of an right triangles. 28. Identif the legs and the base of an isosceles triangles. Which isosceles triangle has a base that is also the hpotenuse of a right triangle? UI Use the graph. he segment Æ is a leg of an isosceles right triangle. (?,?) 29. ind the coordinates of point. op the graph and sketch. 30. ind the coordinates of a point that forms a Æ different isosceles right triangle with leg. Include a sketch with our answer. (2, 2) (5, 2) U H II U ind the measure of the numbered angles. HOWOK H Visit our Web site for help with s I UI he variable epressions represent the angle measures of a triangle. ind the measure of each angle. hen classif the triangle b its angles. 34. m = 35. m = 36. m W = ( º 5) m = 2 m = 7 m Y = (2 º 65) m = (2 + 5) m = m Z = 90 XIO ind the measure of the eterior angle shown (2 8) 3 38 (7 ) (0 9) 40. HOOY Use geometr software to demonstrate the riangle um heorem or the terior ngle heorem. escribe our procedure. (2 2) 4. riangles and ngles 99

3 4. UI In Q, the measure of is 36. he measure of Q is five times the measure of. ind m Q and m. 42. UI he measure of an eterior angle of a triangle is 20. he interior angles that are not adjacent to this eterior angle are congruent. ind the measures of the interior angles of the triangle. 43. II K You want to make a wooden billiard rack. he rack will be an euilateral triangle whose side length is 33.5 centimeters. You have a strip of wood that is 00 centimeters long. o ou need more wood? plain. 44. O H You are bending a wire to make a coat hanger. he length of the wire is 88 centimeters, and 20 centimeters are needed to make the hook portion of the hanger. he triangular portion of the hanger is an isosceles triangle. he length of one leg of this triangle is 3 5 the length of the base. ketch the hanger. ive the dimensions of the triangular portion. HYOOY hdrologist studies how water circulates in the atmosphere, on the ground, and under the ground. hdrologist might use a wing deflector to minimize the effects of erosion on the bank of a stream. IK I I OU O WI O In ercises 45 and 46, use the information about wing deflectors. wing deflector is a structure built with rocks to redirect the flow of water in a stream and increase the rate of the water s flow. Its shape is a right triangle. 45. Identif the legs and the hpotenuse of the right triangle formed b the wing deflector. 46. It is generall recommended that the upstream angle should range from 30 to 45. ive a range of angle measures for the downstream angle. 47. VOI OO ill in the missing steps in the two-column proof of the terior ngle heorem. IV is an eterior angle of. OV m = m + m tatements. is an eterior angle of.. iven 2. and are a linear pair. 2. efinition of eterior angle 3. m + m = 80 3.? 4.? easons 5. m + m = 5.? m + m + m 6. m = m + m 6.? upstream angle downstream angle 4. riangle um heorem 48. WO-OU OO Write a two-column proof of the orollar to the riangle um heorem on page hapter 4 ongruent riangles

4 est reparation hallenge X H UI HOI he lengths of the two legs of an isosceles triangle are represented b the epressions (2 º 5) and ( + 7). he perimeter of the triangle is 50 cm. ind the length of the base of the triangle. cm 9 cm 2 cm 26 cm 32 cm 50. UI HOI Which of the terms below can be used to describe a triangle with two 45 interior angles? cute ight calene Obtuse uilateral 5. IV OO here is often more than one wa to prove a theorem. In the diagram, is constructed parallel to Q. his construction is the first step of a proof of the riangle um heorem. Use the diagram to prove the riangle um heorem. IV Q OV m + m 2 + m 3 = IX VIW VUI Use the figure to determine whether the statement is true or false. (eview.5 for 4.2) m + m = Æ Æ 56. Æ bisects VOI OO Is it possible to prove that lines p and are parallel? If so, state the postulate or theorem ou would use. (eview 3.4) 57. p 58. p 59. p WII QUIO Write an euation of the line that passes through the given point and has the given slope. (eview 3.6) 60..(0, º2), m = 0 6. (4, 7), m = 62. (º3, º5), m = º 63. (9, º), m = (º, º), m = (º2, º3), m = º (5, 2), m = (8, 3), m = º (º6, º4), m = º 3 4. riangles and ngles 20

5 In this lesson, ou have learned to prove that two triangles are congruent b the definition of congruence that is, b showing that all pairs of corresponding angles and corresponding sides are congruent. In upcoming lessons, ou will learn more efficient was of proving that triangles are congruent. he properties below will be useful in such proofs. HO HO 4.4 roperties of ongruent riangles XIV OY O OU I ver triangle is congruent to itself. YI OY O OU I If, then. IIV OY O OU I If and JK, then JK. J K UI I Vocabular heck oncept heck kill heck. op the congruent triangles shown at the right. hen label the vertices of our triangles so that JK. Identif all pairs of congruent corresponding angles and corresponding sides. O YI Use the information and the diagram below. On an eam, a student sas that because the corresponding angles of the triangles are congruent. 2. How does the student know that the corresponding angles are congruent? 3. Is? plain our answer. Use the diagram at the right, where Q. 4. What is the measure of? What is the measure of? 6. What is the measure of? 7. What is the measure of? Æ? 8. Which side is congruent to Q 9. Which side is congruent to Æ? ongruence and riangles 205

6 I IIO U H tra ractice to help ou master skills is on p II OU I In the diagram, UV. omplete the statement. 0.? Æ. V? U 59 8 cm 2. VU? 55 V 3. =? 4. m = m? =? 5. Which of the statements below can be used to describe the congruent triangles in ercises 0 4? (here ma be more than one answer.). UV. VU. UV. VU I OU IU Identif an figures that can be proved congruent. plain our reasoning. or those that can be proved congruent, write a congruence statement H K J X W Y K V Z J K J H K U H HOWOK H ample : s ample 2: s. 4, 24, 25 ample 3: s ample 4: s. 6 2, 23 ample 5: IIYI OOI Use the triangles shown in ercise 7 above. Identif all pairs of congruent corresponding angles and corresponding sides. 23. II HIKI Use the triangles shown at the right. How man pairs of angles are congruent? re the triangles congruent? plain our reasoning. V W X 206 hapter 4 ongruent riangles

7 UI Use the given information to find the indicated values. 24. iven H, 25. iven XYZ, find the values of and. find the values of a and b (4 4) H (0 65) UI Use the given information to find the indicated value. 26. iven and H, 27. iven and Q, find the value of. find the value of m. X Z 70 (5b 3) (4a 4) Y J (2 50) H m U 28. iven K and J, 29. iven X and Z, find the value of s. find the value of r. K 35 Z r (3s 20) J Y 78 X O I Use the diagram based on the photo. he small triangles,,, and, are congruent. 30. plain wh is euilateral. his pattern was made b mowing a field in ngland. 3. he sum of the measures of,, and is 360. ind m. 32. ach of the small isosceles triangles has two congruent acute angles. ind m and m. 33. OI OI plain wh is euiangular. 4.2 ongruence and riangles 207

8 OU O O 34. UU he sculpture shown in the photo is made of congruent triangles cut from transparent plastic. uppose ou use one triangle as a pattern to cut all the other triangles. Which propert guarantees that all the triangles are congruent to each other? HI IO is an artist who has created man works of art that rel on or epress mathematical principles. he pattern used to arrange the triangles in her sculpture shown at the right can be etended indefinitel. I 35. VOI OO omplete the proof of the hird ngles heorem. IV, OV tatements., 2. m? = m?, m? = m? 3. m + m + m = 80, m + m + m = m + m + m = m + m + m 5. m + m + m = m + m + m 6. m = m 7.? easons.? 2.? 3.? 4.? 5.? 6.? 7. ef. of. OII Origami is the art of folding paper into interesting shapes. ollow the directions below to create a kite. Use our kite in ercises old a suare piece of paper in half Æ diagonall to create. Æ et fold the paper so that side Æ lies directl on. Æ hen fold the paper so that side Æ lies directl on. 36. Is congruent to? Is congruent to? plain. 37. OI OI rom folding, ou know that Æ bisects and Æ bisects. iven these facts and our answers to ercise 36, which triangles can ou conclude are congruent? plain. 38. OO Write a proof. Æ IV fi, is the midpoint of,, and Æ bisects. OV 208 hapter 4 ongruent riangles

9 est reparation hallenge X H UI- O Use the diagram, in which. a. plain how ou know that. b. plain how ou know that. c. plain how ou know that. d. plain how ou know that. e. Writing o ou have enough information to prove that? plain. 40. OII VII ook back at ercises on page 208. uppose the following statements are also true about the diagram. Æ bisects and Æ bisects. and are right angles. ind all of the unknown angle measures in the figure. Use a sketch to show our answers. IX VIW I OU ind the distance between each pair of points. (eview.3 for 4.3) 4. (3, 8) 42. (3, º8) 43. (º2, º6) (º, º4) (º3, 7) (3, º5) 44. (0, 5) 45. J(0, º4) 46. (7, º2) H(º5, 2) K(9, 2) (0, 9) II H IOI ind the coordinates of the midpoint of a segment with the given endpoints. (eview.5) 47. (º, 5) 48. Q(5, 7) 49. (º6, º2) (º3, º9) (º, 4) (8, 2) 50. U(0, º7) 5. W(2, 0) 52. (º5, º7) V(º6, 4) Z(8, 6) (0, 4) II OY In ercises 53 55, and 2 are complementar. ind m 2. (eview.6) 53. m = m = m = 62 m 2 =? m 2 =? m 2 =? IIYI ind the slope of each line. re the lines parallel? (eview 3.6) (, 2) (6, ) ( 3, 3) (2, 3) ( 3, ) (2, 2) (, 2) (4, ) 4.2 ongruence and riangles 209

10 QUIZ elf-est for essons 4. and 4.2 lassif the triangle b its angles and b its sides. (esson 4.) ind the value of in the figure at the right. hen give the measure of each interior angle and the measure of the eterior angle shown. (esson 4.) (6 20) 77 (7 6) Use the diagram at the right. (esson 4.2) 5. Write a congruence statement. Identif all pairs of congruent corresponding parts. 6. You are given that m = 46 and m Q = 27. ind m. riangles In rchitecture I IIO IK H OW OU , construction of the reat ramid of Khufu began. It took the ancient gptians about 30 ears to transform 6.5 million tons of stone into a pramid with a suare base and four congruent triangular faces. OY, triangles are still used in architecture. he are even being used in structures designed to house astronauts on long-term space missions.. he original side lengths of a triangular face on the reat ramid of Khufu were about 29 meters, 230 meters, and 29 meters. he measure of one of the interior angles was about 63. he other two interior angles were congruent. ind the measures of the other angles. hen classif the triangle b its angles and sides. onstruction on the reat ramid of Khufu begins. c s oscow s olshoi heater uses triangles in its design. rchitect onstance dams uses triangles in the design of a space module. 20 hapter 4 ongruent riangles

11 UI I Vocabular heck oncept heck. ketch a triangle and label its vertices. ame two sides and the included angle between the sides. 2. O YI Henr believes he can use the information given in the diagram and the ongruence ostulate to prove the two triangles are congruent. plain Henr s mistake. kill heck OI OI ecide whether enough information is given to prove that the triangles are congruent. If there is enough information, tell which congruence postulate ou would use. 3., 4. H, JKH 5. Q, Q H J K I IIO U H tra ractice to help ou master skills is on p I I IU Use the diagram. ame the included angle between the pair of sides given. 6. JK and K 7. K Æ and Æ K J 8. and K 9. J Æ and JK Æ 0. K and J. K and OI OI ecide whether enough information is given to prove that the triangles are congruent. If there is enough information, state the congruence postulate ou would use. 2. UV, WV 3., 4. YZW, YXW Z K Y W U H HOWOK H ample : s. 8, ample 2: s ample 3: s. 2 7 ample 4: s ample 5: s. 30, 3 ample 6: s X W V U 5., 6., WVU 7. JH, HK U J V W H K 26 hapter 4 ongruent riangles

12 VOI OO In ercises 8 and 9, use the photo of the avajo rug. ssume that and. 8. What other piece of information is needed to prove that using the ongruence ostulate? 9. What other piece of information is needed to prove that using the ongruence ostulate? 20. VOI OO omplete the proof b suppling the reasons. IV H, H OV H H tatements. H 2. H H easons.? 2.? 3.? 4.? WO-OU OO Write a two-column proof. 2. IV Q, 22. IV, Q OV OV Q 2 U H HOWOK H Visit our Web site for help with paragraph proofs. I H OO Write a paragraph proof. 23. IV Q Æ bisects, 24. IV, Q OV Q OV Q Q 4.3 roving riangles are ongruent: and 27

13 OO Write a two-column proof or a paragraph proof. 25. IV Æ Æ, 26. IV,, Æ is the midpoint of. OV OV Æ 27. IV, OV Æ 28. IV, Q fi Q fi OV Q Q Æ, U H OW H Visit our Web site to see instructions for several software applications. I 29. HOOY Use geometr software to draw a triangle. raw a line and reflect the triangle across the line. easure the sides and the angles of the new triangle and tell whether it is congruent to the original one. Writing plain how triangles are used in the object shown to make it more stable OUIO raw an isosceles triangle with vertices,, and. Use a compass and straightedge to construct so that. UI Use the istance ormula and the ongruence ostulate to show that hapter 4 ongruent riangles

14 est reparation hallenge X H UI HOI In and,,, and. Which angle is congruent to? cannot be determined 37. UI HOI In euilateral, a segment is drawn from point Æ to, the midpoint of. Which of the statements below is not true? Æ Æ Æ Æ Æ Æ 38. HOOI HO escribe how to show that O using the ongruence ostulate. hen find a wa to show that the triangles are congruent using the ongruence ostulate. You ma not use a protractor to measure an angles. ompare the two methods. Which do ou prefer? Wh? O IX VIW I OIO ind an important angle in the photo. op the angle, etend its sides, and use a protractor to measure it to the nearest degree. (eview.4) UI I ind m and m 2. plain our reasoning. (eview 3.3 for 4.4) I IOHI ind the slope of each line. Identif an parallel or perpendicular lines. (eview 3.7) œ 2 H 4.3 roving riangles are ongruent: and 29

15 UI I Vocabular heck oncept heck. ame the four methods ou have learned for proving triangles congruent. Onl one of these is called a theorem. Wh is it called a theorem? Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem ou would use. plain our reasoning. 2. and Q 3. JK and 4. and JH K kill heck tate the third congruence that must be given to prove that using the indicated postulate or theorem. 5. ongruence ostulate 6. ongruence heorem J J H 7. Y course for a rela race is marked on the gmnasium floor. Your team starts at, goes to, then, then returns to. he other team starts at, goes to, then, then returns to. iven that Æ Æ and and are right angles, eplain how ou know the two courses are the same length. I IIO U H tra ractice to help ou master skills is on pp. 809 and 80. OI OI Is it possible to prove that the triangles are congruent? If so, state the postulate or theorem ou would use. plain our reasoning V U U H HOWOK H ample : s. 8 3 ample 2: s ample 3: s , 28. H 2. K 3. J J W Y X Z 4.4 roving riangles are ongruent: and 223

16 U H tud ip When a proof involves overlapping triangles, such as the ones in s. 8 and 22, ou ma find it helpful to sketch the triangles separatel. VOI OO tate the third congruence that must be given to prove that Q U using the indicated postulate or theorem. (Hint: irst sketch Q and U. ark the triangles with the given information.) 4. IV Q, Q Use the ongruence heorem. 5. IV U, U Use the ongruence ostulate. 6. IV U, Use the ongruence ostulate. 7. IV U, U Use the ongruence ostulate. 8. VOI OO omplete the proof that XWV ZWU. IV VW Æ UW Æ X Z OV XWV ZWU tatements. VW Æ UW Æ 2. X Z 3.? 4. XWV ZWU Z V easons.? 2.? 3. efleive ropert of ongruence 4.? W Y U X OO Write a two-column proof or a paragraph proof. 9. IV H Æ K Æ, OV H K K 20. IV fi, fi, Æ Æ OV H Æ 2. IV VX XY, XW YZ XW YZ OV VXW XYZ Æ, 22. IV Q Q, OV Q Q V X W Y Z 224 hapter 4 ongruent riangles

17 I Use the information about bearings in ercises In surveing and orienteering, bearings conve information about direction. or eample, the bearing W 53. means 53. to the north of west. o find this bearing, face west. hen turn 53. to the north. 23. You want to describe the boundar lines of a triangular piece of propert to a friend. You fa the note and the sketch below to our friend. Have ou provided enough information to determine the boundar lines of the propert? plain. he southern border is a line running east from the apple tree, and the western border is the north-south line running from the cherr tree to the apple tree. he bearing from the easternmost point to the northernmost point is W 53.. he distance between these points is 250 feet. cherr tree 250 ft 53. apple tree OII In the sport of orienteering, participants use a map and a compass to navigate a course. long the wa, the travel to various points marked on the map. OU O IIO I 24. surveor wants to make a map of several streets in a village. he surveor finds that reen treet is on an east-west line. lain treet is at a bearing of 55 from its intersection with reen treet. It runs 20 ards before intersecting llis venue. llis venue runs 00 ards between reen treet and lain treet. ssuming these measurements are accurate, what additional measurements, if an, does the surveor need to make to draw llis venue correctl? plain our reasoning. 25. You are creating a map for an orienteering race. articipants start out at a large oak tree, find a boulder that is 250 ards east of the oak tree, and then find an elm tree that is W 50 of the boulder and 35 of the oak tree. Use this information to sketch a map. o ou have enough information to mark the position of the elm tree? plain. UI raph the euations in the same coordinate plane. abel the vertices of the two triangles formed b the lines. how that the triangles are congruent. 26. = 0; = ; = º + 3; = = 2; = 6; = 3; = 5; = 2 º 4 lain t. 20 d 55 reen t. llis ve. 00 d 4.4 roving riangles are ongruent: and 225

18 28. QUII You are making a uilt block out of congruent right triangles. efore cutting out each fabric triangle, ou mark a right angle and the length of each leg, as shown. What theorem or postulate guarantees that the fabric triangles are congruent? 3 2 in. 3 in. 2 est reparation 29. UI- O You can use the method described below to approimate the distance across a stream without getting wet. s shown in the diagrams, ou need a cap with a visor. tand on the edge of the stream and look straight across to a point on the other edge of the stream. djust the visor of our cap so that it is in line with that point. Without changing the inclination of our neck and head, turn sidewas until the visor is in line with a point on our side of the stream. easure the distance between our feet and that point. hallenge X H a. rom the description of the measuring method, what corresponding parts of the two triangles can ou assume are congruent? b. What theorem or postulate can be used to show that the two triangles are congruent? c. Writing plain wh the length of Æ is also the distance across the stream. OO Use the diagram. 30. licia thinks that she can prove that Q Q based on the information in the diagram. plain wh she cannot. 3. uppose ou are given that XQ XQ and that. rove that Q Q. X 226 hapter 4 ongruent riangles

19 IX VIW II OI ind the coordinates of the other endpoint of a segment with the given endpoint and midpoint. (eview.5) 32. (5, 7), (º, 0) 33. (0, 9), (6, º2) 34. (8, º5), (º, º3) UI IO Æ is the angle bisector of. ind the two angle measures not given in the diagram. (eview.5 for 4.5) OO You are making a brace for a barn door, as shown. he top and bottom pieces are parallel. o make the middle piece, ou cut off the ends of a board at the same angle. What postulate or theorem guarantees that the cuts are parallel? (eview 3.4) cut cut QUIZ 2 elf-est for essons 4.3 and 4.4 In ercises 6, decide whether it is possible to prove that the triangles are congruent. If it is possible, state the theorem or postulate ou would use. plain our reasoning. (essons 4.3 and 4.4) U H V 4. J K 5. Z 6. V U 7. OO Write a two-column proof. (esson 4.4) Æ, IV is the midpoint of fi Q, fi, Q OV Q 4.4 roving riangles are ongruent: and 227

20 UI I oncept heck kill heck In ercises 3, use the photo of the eagle ra.. o prove that Q Q, which triangles might ou prove to be congruent? 2. If ou know that the opposite sides of figure Q are parallel, can ou prove that Q? plain. 3. he statements listed below are not in order. Use the photo to order them as statements in a two-column proof. Write a reason for each statement. IV Q fi, OV. Q fi and are right angles. I IIO U H tra ractice to help ou master skills is on p. 80. I WIOW he eight window panes in the diagram are isosceles triangles. he bases of the eight triangles are congruent. 4. plain how ou know that U UQ. 5. plain how ou know that U QU. 6. o ou have enough information to prove that all the triangles are congruent? plain. 7. plain how ou know that U UQ. VOI OO tate which postulate or theorem ou can use to prove that the triangles are congruent. hen eplain how proving that the triangles are congruent proves the given statement. 8. OV Q 9. OV V UV 0. OV K = U K U H HOWOK H ample : s. 4 4, 7, 8 ample 2: s. 4, 7, 8 ample 3: s. 5, 6 ample 4: s. 9 2 V U J 232 hapter 4 ongruent riangles

21 Use the diagram of the string game at s radle and the information given below. IV H. OV Æ H Æ 2. OV H H 3. OV H 4. VOI OO omplete the proof that. IV is the midpoint of Æ,, OV tatements. is the midpoint of Æ.. iven 2. 2.? iven Æ easons 4. Æ 4. iven 5. 5.? 6.? 6. ongruence heorem 7. 7.? 5. VOI OO omplete the proof that. 2 IV OV 4 3 tatements. 2.? ? 3.? easons 3. efleive ropert of ongruence? ? 6.? 6. Vertical ngles heorem 7. 7.? 4.5 Using ongruent riangles 233

22 OU O 6. I he diagram represents a section of the framework of the Kap hui un ridge shown in the photo on page 229. Write a two-column proof to show that KJ Q. IV is the midpoint of J Æ, J Q, Q, KJ and Q are right angles. OV KJ Q J K OUIO construction manager plans and directs the work at a building site. mong other things, the manager reviews engineering specifications and architectural drawings to make sure that a project is proceeding according to plan. IK I I OO Write a two-column proof or a paragraph proof. 7. IV U, 8. IV fi, and are Æ bisects Æ. right angles. OV and are OV U U complementar angles. U 9. OVI OUIO he diagrams below summarize the construction used to bisect. construction, ou can assume that Æ and. Write a proof to verif that bisects. irst draw an arc with center. abel the points where the arc intersects the sides of the angle points and. 2 raw an arc with center. Using the same compass setting, draw an arc with center. abel the intersection point. 3 raw Æ. U H ook ack or help with bisecting an angle, see p. 36. OVI OUIO Use a straightedge and a compass to perform the construction. abel the important points of our construction. hen write a flow proof to verif the results. 20. isect an obtuse angle. 2. op an obtuse angle. 234 hapter 4 ongruent riangles

23 est reparation hallenge 22. UI HOI uppose Q. You want to prove that Q. Which of the reasons below would not appear in our two-column proof? ongruence ostulate efleive ropert of ongruence ongruence heorem ight ngle ongruence heorem lternate Interior ngles heorem 23. UI HOI Which statement correctl describes the congruence of the triangles in the diagram in ercise 22? Q Q Q Q Q Q Q Q 24. OVI OUIO Use a straightedge and a compass to bisect a segment. (or help with this construction, X X H look back at page 34.) hen write a proof to show that the construction is valid. Y IX VIW II I, IU, ind the perimeter (or circumference) and area of the figure. (Where necessar, use π 3.4.) (eview.7) m 43.5 m 30.8 m 55 m 53.3 m 2 cm OVI QUIO olve the euation and state a reason for each step. (eview 2.4) 28. º 2 = = = = ( º ) = (2 º ) + 5 = 69 IIYI O I lassif the triangle b its angles and b its sides. Identif the legs and the hpotenuse of an right triangles. Identif the legs and the base of an isosceles triangles. (eview 4. for 4.6) X 62 Y Z 4.5 Using ongruent riangles 235

24 UI I Vocabular heck oncept heck kill heck. escribe the meaning of euilateral and euiangular. ind the unknown measure(s). ell what theorems ou used ? 5 cm 50? etermine whether ou are given enough information to prove that the triangles are congruent. plain our answer U V 7.? H? J W I IIO U H UI olve for and. tra ractice to help ou master skills is on p OI OI ecide whether enough information is given to prove that the triangles are congruent. plain our answer U W H V U H HOWOK H ample : s ample 2: s. 8 0, 7 25 ample 3: s. 3, 33, 34, K 6. J 4.6 Isosceles, uilateral, and ight riangles 239

25 UI ind the value of ( + 3) ft 24 ft 2 in. 56 ft 8 ft 2 in. UI ind the values of and OO In ercises 26 28, use the diagrams that accompan the theorems on pages 236 and he onverse of the ase ngles heorem on page 236 states, If two angles of a triangle are congruent, then the sides opposite them are congruent. Write a proof of this theorem. 27. he orollar to heorem 4.6 on page 237 states, If a triangle is euilateral, then it is euiangular. Write a proof of this corollar. 28. he orollar to heorem 4.7 on page 237 states, If a triangle is euiangular, then it is euilateral. Write a proof of this corollar. HIU he diagram represents part of the eterior of the building in the photograph. In the diagram, and are congruent euilateral triangles. 29. plain wh is isosceles. 30. plain wh. 3. OO rove that and are congruent right triangles. 32. ind the measure of. 240 hapter 4 ongruent riangles

26 OO Write a two-column proof or a paragraph proof. Æ 33. IV is the midpoint of, 34. IV VW ZY, and are UV XW, UZ XY, right angles, and Æ Æ. VW fi VZ, VW fi WY OV OV U X V W U Z Y X I WO he nglish scientist Isaac ewton ( ) observed that light is made up of a spectrum of colors. ewton was the first person to arrange the colors of the spectrum in a color wheel. IIO IK I I OU O O OO WH rtists use a color wheel to show relationships between colors. he 2 triangles in the diagram are isosceles triangles with congruent verte angles. 35. omplementar colors lie directl opposite each other on the color wheel. plain how ou know that the ellow triangle is congruent to the purple triangle. 36. he measure of the verte angle of the ellow triangle is 30. ind the measures of the base angles. 38. orm other triangles that are congruent to the triangle in ercise 37. he colors of the vertices are called triads. What are the possible triads? HYI Use the information below. When a light ra from an object meets a mirror, it is reflected back to our ee. or eample, in the diagram, a light ra from point is reflected at point and travels back to point. he law of reflection states that the angle of incidence is eual to the angle of reflection. 39. IV fi OV 40. Verif that is isosceles. 4. oes moving awa from the mirror have an effect on the amount of his or her reflection the person sees? green blue redpurple 37. race the color wheel. hen form a triangle whose vertices are the bluepurple purple midpoints of the bases of the red, ellow, and blue triangles. (hese colors are the primar colors.) What tpe of triangle is this? bluegreen ellowgreen ellow orange red elloworange redorange or a person to see his or her complete reflection, the mirror must be at least one half the person s height. 4.6 Isosceles, uilateral, and ight riangles 24

27 est reparation hallenge QUIIV OIO In ercises 42 and 43, refer to the figures below. hoose the statement that is true about the given values. he value in column is greater. he value in column is greater he two values are eual. he relationship cannot be determined from the given information. olumn olumn 44. OI OI regular heagon has si congruent sides and si congruent interior angles. It can be divided into si euilateral triangles. plain how the series of diagrams below suggests a proof that when a triangle is formed b connecting ever other verte of a regular heagon, the result is an euilateral triangle. 20 egular heagon X H IX VIW OU Use the istance ormula to decide whether Æ Æ. (eview.3 for 4.7) 45. (0, º4) 46. (0, 0) 47. (, º) (5, 8) (º6, º0) (º8, 7) (º2, ) (6, 0) (8, 7) II H IOI ind the coordinates of the midpoint of a segment with the given endpoints. (eview.5 for 4.7) 48. (4, 9), (0, 7) 49. (0, ), H(8, º3) 50. (, 7), (º5, º5) 5. (º2, 3), (5, 6) 52. (0, º3), H(2, º) 53. (º3, º5), (0, º20) WII QUIO ine j is perpendicular to the line with the given euation and line j passes through point. Write an euation of line j. (eview 3.7) 54. = º3 º 4; (, ) 55. = º 7; (0, 0) 56. = º 0 + 3; (5, º2) 57. = 2 + 4; (º3, 4) hapter 4 ongruent riangles

28 I IIO U H tra ractice to help ou master skills is on p. 80. I IU I OOI lace the figure in a coordinate plane. abel the vertices and give the coordinates of each verte unit b 8-unit rectangle with one verte at (0, 0) 7. n 8-unit b 6-unit rectangle with one verte at (0, º4) 8. suare with side length s and one verte at (s, 0) HOOI OO lace the figure in a coordinate plane. abel the vertices and give the coordinates of each verte. plain the advantages of our placement. 9. right triangle with legs of 3 units and 8 units 0. n isosceles right triangle with legs of 20 units. rectangle with length h and width k II UI OOI In the diagram, is isosceles. Its base is 60 units and its height is 50 units. 2. ive the coordinates of points and. 3. ind the length of a leg of. ound our answer to the nearest hundredth. ( 30, 0) 0 0 UI H I OU lace the figure in a coordinate plane and find the given information. 4. right triangle with legs of 7 and 9 units; find the length of the hpotenuse. 5. rectangle with length 5 units and width 4 units; find the length of a diagonal. 6. n isosceles right triangle with legs of 3 units; find the length of the hpotenuse unit b 3-unit suare; find the length of a diagonal. UI H IOI OU Use the given information and diagram to find the coordinates of H. 8. OH JH 9. OH H J(80, 80) (90, 70) U H HOWOK H ample : s. 6 ample 2: s. 2 7 ample 3: s. 8, 9 ample 4: s. 20, 2 ample 5: s ample 6: s. 26, 27 0 H O(0, 0) 40 (80, 0) 0 H (90, 35) O(0, 0) (45, 0) riangles and oordinate roof 247

29 VOI OO Write a plan for a proof. 20. IV O fi Æ OV O bisects O. Æ 2. IV is the midpoint of H. OV HJ O (0, 60) H(2, 6) J(6, 6) 0 O(0, 0) 0 (60, 0) O(0, 0) (4, 0) UI VI OOI ind the coordinates of an unlabeled points. hen find the reuested information. 22. ind. 23. ind O. h units k units h units O(0, 0) O(0, 0) 2h units 24. ind O and. 25. ind O. k units 2k units O(0, 0) (h, 0) (2h, 0) O U OOI OO Write a coordinate proof. 26. IV oordinates of O and O OV O O (0, 2h) 27. IV oordinates of O and OV O (h, 2k) (2h, 2k) (h, h) (h, k) O(0, 0) (2h, 0) O(0, 0) (h, 0) 248 hapter 4 ongruent riangles

30 28. You bu a tall, three-legged plant stand. When ou place a plant on the stand, the stand appears to be unstable under the weight of the plant. he diagram at the right shows a coordinate plane superimposed on one pair of the plant stand s legs. he legs are etended to form O. Is O an isosceles triangle? plain wh the plant stand ma be unstable. 6 O(0, 0) (2, 48) 6 (8, 0) HOOY Use geometr software for ercises ollow the steps below to construct. reate a pair of aes. onstruct point on the -ais so that the -coordinate is positive. onstruct point on the -ais. onstruct a circle with a center at the origin that contains point. abel the other point where the circle intersects the -ais. onnect points,, and to form. ind the coordinates of each verte. est reparation 29. What tpe of triangle does appear to be? oes our answer change if ou drag point? If ou drag point? 30. easure and compare and. What happens to these lengths as ou drag point? What happens as ou drag point? 3. ook back at the proof described in ercise 5 on page 246. How does that proof help eplain our answers to ercises 29 and 30? 32. UI HOI suare with side length 4 has one verte at (0, 2). Which of the points below could be a verte of the suare? (0, º2) (2, º2) (0, 0) (2, 2) 33. UI HOI rectangle with side lengths 2h and k has one verte at (ºh, k). Which of the points below could not be a verte of the rectangle? (0, k) (ºh, 0) (h, k) (h, 0) hallenge 34. OOI OO Use the diagram and the given information to write a proof. (0, 2k) X H IV oordinates of, H is the midpoint of is the midpoint of OV Æ H Æ Æ, Æ. H ( 2h, 0) O(0, 0) (2h, 0) 4.7 riangles and oordinate roof 249

31 IX VIW UI In the diagram, Æ bisects. (eview.5 for 5.) 35. ind the value of. 36. ind m. (4 55) 5 IU I IO Use the diagram to determine whether the statement is true or false. (eview.5, 2.2 for 5.) 37. Q is perpendicular to. 38. oints, Q, and are collinear. 39. Q bisects Æ. 40. Q and are supplementar. WII et p be two triangles are congruent and let be the corresponding angles of the triangles are congruent. Write the smbolic statement in words. ecide whether the statement is true. (eview 2.3) 4. p 42. p 43. ~p ~ QUIZ 3 elf-est for essons OO Write a two-column proof or a paragraph proof. (essons 4.5 and 4.6). IV, Æ H Æ 2. IV Æ U VU, U Æ V Æ OV H OV U UV V H U 3. OOI OO Write a plan for a coordinate proof. (esson 4.7) (3, 4) (8, 4) IV oordinates of vertices of O and O OV O and O are congruent isosceles triangles. O(0, 0) 3 (5, 0) 250 hapter 4 ongruent riangles

32 H 4 hapter ummar WH did ou learn? lassif triangles b their sides and angles. (4.) ind angle measures in triangles. (4.) Identif congruent figures and corresponding parts. (4.2) rove that triangles are congruent using corresponding sides and angles. (4.2) using the and ongruence ostulates. (4.3) using the ongruence ostulate and the ongruence heorem. (4.4) using the H ongruence heorem. (4.6) using coordinate geometr. (4.7) Use congruent triangles to plan and write proofs. (4.5) rove that constructions are valid. (4.5) Use properties of isosceles, euilateral, and right triangles. (4.6) WHY did ou learn it? a the foundation for work with triangles. ind the angle measures in triangular objects, such as a wing deflector. (p. 200) nalze patterns, such as those made b the folds of an origami kite. (p. 208) earn to work with congruent triangles. plain wh triangles are used in structural supports for buildings. (p. 25) Understand how properties of triangles are applied in surveing. (p. 225) rove that right triangles are congruent. lan and write coordinate proofs. rove that triangular parts of the framework of a bridge are congruent. (p. 234) evelop understanding of geometric constructions. ppl a law from phsics, the law of reflection. (p. 24) How does hapter 4 fit into the I IU of geometr? he was ou have learned to prove triangles are congruent will be used to prove theorems about polgons, as well as in other topics throughout the book. Knowing the properties of triangles will help ou solve real-life problems in fields such as art, architecture, and engineering. UY Y How did ou use our list of theorems? he list of theorems ou made, following the tud trateg on page 92, ma resemble this one. emembering heorems heorem 4.4 roperties of ongruent riangles. efleive 2. mmetric If, then. 3. ransitive If and JK, then JK. J K 25

33 H 4 hapter eview VOUY euilateral triangle, p. 94 isosceles triangle, p. 94 scalene triangle, p. 94 acute triangle, p. 94 euiangular triangle, p. 94 right triangle, p. 94 obtuse triangle, p. 94 verte of a triangle, p. 95 adjacent sides of a triangle, p. 95 legs of a right triangle, p. 95 hpotenuse, p. 95 legs of an isosceles triangle, p. 95 base of an isosceles triangle, p. 95 interior angle, p. 96 eterior angle, p. 96 corollar, p. 97 congruent, p. 202 corresponding angles, p. 202 corresponding sides, p. 202 base angles, p. 236 verte angle, p. 236 coordinate proof, p I amples on pp X You can classif triangles b their sides and b their angles. euilateral isosceles scalene acute euiangular right obtuse ote that an euilateral triangle is also isosceles and acute. You can appl the riangle um heorem to find unknown angle measures in triangles. m + m + m = 80 riangle um heorem = 80 ubstitute = 80 = 48 implif. ubtract 32 from each side. 40 m = 48 In ercises 4, classif the triangle b its angles and b its sides One acute angle of a right triangle measures 37. ind the measure of the other acute angle. 6. In, the measure of is 24. he measure of is five times the measure of. ind m and m. 252 hapter 4 ongruent riangles

34 4.2 OU I amples on pp X When two figures are congruent, their corresponding sides and corresponding angles are congruent. In the diagram, XYZ. X Z Y Use the diagram above of and XYZ. 7. Identif the congruent corresponding parts of the triangles. 8. iven m = 48 and m Z = 37, find m Y. 4.3 & 4.4 OVI I OU:,,, amples on pp , X You can prove triangles are congruent using congruence postulates and theorems. K J JK, K, J, so JK b the ongruence ostulate.,, and, so b the ongruence ostulate. ecide whether it is possible to prove that the triangles are congruent. If it is possible, tell which postulate or theorem ou would use. plain our reasoning U. V W Y Z X H J 4.5 UI OU I amples on pp X You can use congruent triangles to write proofs. IV Q, Q OV fi Q lan for roof Use the ongruence ostulate to show that Q. ecause corresponding parts of congruent triangles are congruent, ou can conclude that Q. hese angles form a linear pair, so fi Q. hapter eview 253

35 4.5 continued UVYI You want to determine the width of a river beside a camp. You place stakes so that fi, Q fi, and is the Æ midpoint of. 2. re and Q congruent? If so, state the postulate or theorem that can be used to prove the are congruent. 3. Which segment should ou measure to find the width of the river? 4.6 IO, QUI, IH I amples on pp X o find the value of, notice that is an isosceles right triangle. the ase ngles heorem,. ecause and are complementar, their sum is 90. he measure of each must be 45. o = 45. ind the value of I OOI OO amples on pp X You can use a coordinate proof to prove that OQ is isosceles. Use the istance ormula to show that O Q. O = (2 º 0 ) 2 + ( 3 º 0 ) 2 = 3 Q = (2 º 4 ) 2 + ( 3 º 0 ) 2 = 3 ecause O Q, OQ is isosceles. (2, 3) O(0, 0) œ(4, 0) 8. Write a coordinate proof. IV oordinates of vertices of O and (h, h) (2h, h) OV O O(0, 0) (h, 0) 254 hapter 4 ongruent riangles

36 H 4 hapter est In ercises 6, identif all triangles in the figure that fit the given description.. isosceles 2. euilateral 3. scalene 4. acute 5. obtuse 6. right 7. In, the measure of is 6. he measure of is three times the measure of. ind m and m. ecide whether it is possible to prove that the triangles are congruent. If it is possible, tell which congruence postulate or theorem ou would use. plain our reasoning H J 0. K. U 2. X 3. H V W Z Y K J ind the value of OO Write a two-column proof or a paragraph proof. 7. IV Æ Æ, Æ Æ OV 8. IV XY WZ, XZ WY OV X W X Y 2 lace the figure in a coordinate plane and find the reuested information. 9. right triangle with leg lengths of 4 units and 7 units; find the length of the hpotenuse. Z 20. suare with side length s and vertices at (0, 0) and (s, s); find the coordinates of the midpoint of a diagonal. W hapter est 255