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All Textbook Solutions for Elementary Geometry for College Students

a What is true of any pair of corresponding angles of two similar polygons? b What is true of any pairs of corresponding sides of two similar polygons?a Are any two quadrilaterals similar? b Are any two squares similar?a Are any two regular pentagons similar? b Are any two equiangular pentagons similar?a Are any two equilateral hexagons similar? b Are any two regular hexagons similar?In Exercises 5 and 6, refer to the drawing. a Given that AX,BT, and CN, write a statement claiming that the triangles shown are similar. b Given that AN,CX, and BT, write a statement claiming that the triangles shown are similar. Exercises 5, 6 In Exercises 5 and 6, refer to the drawing. a If ABCXTN, which angle of ABC corresponds to N of XTN? b If ABCXTN, which side of XTN corresponds to side AC of ABC?7E Given that rectangle ABCE is similar to rectangle MNPR and that CDEPQR, what can you conclude regarding pentagon ABCDE and pentagon MNPQR?Given: MNPQRS,mM=56,mR=82,MN=9,QR=6,RS=7,MP=12 Find: a mN c NP b mP d QS Given: ABCPRC,mA=67,PC=5,CR=12,PR=13,AB=26 Find: a mB b mRPC c AC d CB a Does the similarity relationship have a reflexive property for triangles and polygons in general? b Is there a symmetric property for the similarity of triangles and polygons? c Is there a transitive property for the similarity of triangles and polygons?Using the names of property from Exercise 11, identify the property illustrated by each statement: a If 12, then 21. b If 12, 23, and 34, then 14. c 11.If the drawing, HJKFGK. If HK=6,KF=8, and HJ=4, find HJ.If the drawing, HJKFGK. If HK=6,KF=8, and FG=5, find HJ.Quadrilateral ABCD quadrilateral HJKL. If mA=55, mJ=128 and mD=98, find mK.Quadrilateral ABCD quadrilateral HJKL. If mA=x,mJ=x+50,mD=x+35 and mK=2x45, find x.Quadrilateral ABCD quadrilateral HJKL. If AB=5,BC=n,HJ=10, and JK=n+3, find n.Quadrilateral ABCD quadrilateral HJKL. If mD=90,AD=8,DC=6, and HL=12, find the length of diagonal HK not shown.Quadrilateral ABCD quadrilateral HJKL. If mA=2x+4,mH=68, and mD=3x6, find mL.Quadrilateral ABCD quadrilateral HJKL. If mA=mK=70, and mB=110,.What types of quadrilaterals are ABCD HJKL?In Exercises 21 to 24, ADEABC. Given: DE=4,AE=6,EC=BC Find: BC In Exercises 21 to 24, ADEABC. Given: DE=5,AD=8,DB=BC Find: AB HINT: Find DB first.In Exercises 21 to 24, ADEABC. Given: DE=4,AC=20,EC=BC Find: BC In Exercises 21 to 24, ADEABC. Given: AD=4,AC=18,DB=AE Find: AE Pentagon ABCDE pentagon GHJKL not shown, AB=6, and GH=9. If the perimeter of ABCDE is 50, find the perimeter of GHJKL.Quadrilateral MNPQ quadrilateral WXYZ not shown, PQ=5, and YZ=7. If the longest side of MNPQ is of length 8, find the length of the longest side of WXYZ.27E A technical drawing shows the 312 ft lengths of the legs of a babys swing by line segments 3 in. long. If the diagram should indicate the legs are 212 ft apart at the base, what length represents this distance on the diagram?In Exercises 29 to 32, use the fact that triangles are similar. A person who is while walking away from a 10-ft lamppost casts a shadow 6 ft long. If the person is at a distance of 10 ft from the lamppost at that moment, what is the persons height?In Exercise 29 to 32, use the fact that triangles are similar. With 100 ft of string out, a kite is 64 ft above ground level. When the girl flying the kite pulls in 40 ft of string, the angle formed by the string and the ground does not change. What is the height of the kite above the ground after the 40 ft of string have been taken in?In Exercise 29 to 32, use the fact that triangles are similar. While admiring a rather tall tree, Fred notes that the shadow of his 6-ft frame has a length of 3 paces. On the level ground, he walks off the complete shadow of the tree in 37 paces. How tall is the tree?In Exercise 29 to 32, use the fact that triangles are similar. As a garage door closes, light is case 6 ft beyond the base of the door as shown in the accompanying drawing by a light fixture that is set in the garage ceiling 10 ft back from the door. If the ceiling of the garage is 10 ft above the floor, how far is the garage door above the floor at the time that light is cast 6 ft beyond the door?In the drawing, ABDCEF with transversals l and m. If D and C are the midpoints of AE and BF, respectively, then is trapezoid ABCD similar to trapezoid DCFE?In the drawing, ABDCEF. Suppose that transversals l and m are also parallel. D and C are the midpoints of AE and BF, respectively. If parallelogram ABCD similar to parallelogram DCFE?35E Given RST, a second triangle (UVW) is constructed so that UV=2(RS),VW=2(ST), and WU=2(RT). a What is the constant value of the ratios UVRS,VWST,WURT? b Using intuition appearance, does it seem that UVW is similar to RST?37E 38E 39E For Exercises 39 to 40, use intuition to from a proportion based on the drawing shown. A square with sides of length 2 in. rests as shown on a square with sides of length 6 in. Find the perimeter of trapezoid ABCD.What is the acronym that is used to represent each statement. Corresponding angles of similar triangles are congruent?2E Classify as true or false: a If the vertex angles of two isosceles triangles are congruent, the triangles are similar. b Any two equilateral triangles are similar.Classify as true or false: a If the midpoints of two sides of a triangle are joined, the triangle formed is similar to the original triangle. b Any two isosceles triangles are similar.In Exercises 5 to 8, name the method AA,SSS~,orSAS~ that is used to show that the triangles are similar. WU=32TR,WV=32TS,andUV=32RS In Exercises 5 to 8, name the method AA,SSS~,orSAS~ that is used to show that the triangles are similar. TW and RU In Exercises 5 to 8, name the method AA,SSS~,orSAS~ that is used to show that the triangles are similar. TW and TRWU=TSWV In Exercises 5 to 8, name the method AA,SSS~,orSAS~ that is used to show that the triangles are similar. TRWU=TSWV=RSUV 9E In Exercises 9 to 12, name the method that explains why DGH~DEF. See the figure in the right column. DE=3DG and DF=3DH 11E In Exercises 9 to 12, name the method that explains why DGH~DEF. See the figure in the right column. DGHDEF In Exercises 13 to 16, provide the missing reasons. Given: RSTV;VWRS;VXTS Prove: VWRVXT PROOF Statements Reasons 1. RSTV;VWRS;VXTS 1. ? 2. VWR and VXT are rt s 2. ? 3. VWRVXT 3. ? 4. RT 4. ? 5. VWRVXT 5. ?In Exercises 13 to 16, provide the missing reasons. Given: DETandABCD Prove: ABECTB PROOF Statements Reasons 1. DETandABCD 1. ? 2. ABDT 2. Opposite sides of a are 3. EBAT 3. ? 4. EDCB 4. ? 5. ECBT 5.? 5. ABECTB 6. ?In Exercises 13 to 16, provide the missing reasons. Given: ABC;MandN are midpoints of AB and AC, respectively Prove: AMNABC PROOF Statements Reasons 1. ABC;MandN are the midpoints of AB and AC, respectively. 1. ? 2. AM=12(AB) and AN=12(AC) 2. ? 3. MN=12(BC) 3. ? 4. AMAB=12,ANAC=12, andMNBC=12 4. ? 5. AMAB=ANAC=MNBC 5.? 6. AMNABC 6. ?In Exercises 13 to 16, provide the missing reasons. Given: XYZ with XY trisected at P and Q and YZ trisected at R and S Prove: XYZPYR PROOF Statements Reasons 1. XYZ;XY trisected at P and Q; YZ trisected at R and S 1. ? 2. YRYZ=13 and YPYX=13 2. Definition of trisect 3. YRYZ=YPYX 3. ? 4. YY 4. ? 5. XYZPYR 5.?In Exercises 17 to 24, complete each proof. Given: MNNP,QRRP Prove: MNPQRP Exercises 17, 18 PROOF Statements Reasons 1. ? 1. ? 2. sN and QRP are right s 2. ? 3. ? 3. All right s are 4. PP 4. ? 5. ? 5.?In Exercises 17 to 24, complete each proof. Given: MNQR See figure for Exercise 17. Prove: MNPQRP Exercises 17, 18 PROOF Statements Reasons 1. ? 1. Given 2. MRQP 2. ? 3. ? 3. If two lines are cut by a transversal, the corresponding s are 4. ? 4. ?19E In Exercises 17 to 24, complete each proof. Given: HJJF,HGFG See figure for Exercise 19. Prove: HJKFGK Exercises 19, 20 PROOF Statements Reasons 1. ? 1. Given 2. sG and J are right s 2. ? 3. GJ 3. ? 4. HKJGKF 4. ? 5. ? 5. ?21E In Exercises 17 to 24, complete each proof. Given: DGDE=DHDF Prove: DGHE PROOF Statements Reasons 1. ? 2. DD 3. DGHDEF 4. ? 1. ? 2. ? 3. ? 4. ?In Exercises 17 to 24, complete each proof. Given: RSUV Prove: RTVT=RSVU PROOF Statements Reasons 1. ? 2. RV and SU 3. ? 4. ? 1. ? 2. ? 3. AA 4. ?24E In Exercises 25 to 28, ABCDBE Exercises 25-28 Given AC = 8, DE = 6, CB = 6. Find: EB HINT: Let EB = x, and solve an equation.In Exercises 25 to 28, ABCDBE Exercises 25-28 Given: AC = 10, CB = 12. E is the midpoint of CB Find: DE In Exercises 25 to 28, ABCDBE Exercises 25-28 Given: AC = 10, DE = 12, AD = 4. Find: DB In Exercises 25 to 28, ABCDBE Exercises 25-28 Given: CB = 12, CE = 4, AD = 5. Find: DB CDECBA with CDEB. If CD=10,DA=8 and CE=6, find EB. Exercises 29, 30 CDECBA with CDEB. If CD=10,CA=16 and EB=12, find CE.See the figure for exercises Exercises 29, 30 ABFCBD with obtuse angles at vertices D and F as indicated. If mB=45, mC=x and mAFB=4x, find x. Exercises 31, 32 ABFCBD with obtuse angles at vertices D and F as indicated. If mB=44, and mA:mCDB=1:3, find mA. Exercises 31, 32 In Exercise 33, provide a two-column proof. Given:ABDF,BDFG Prove:ABCEFG In Exercise 34, provide a paragraph proof. Given: RSAB,CBAC Prove: BSRBCA Use a two-column proof to prove the following theorem: The lengths of the corresponding altitudes of similar triangles have the same ratio as lengths of any pair of corresponding sides. Given: DEFMNP; DG and MQ are altitudes Prove: DGMQ=DEMN 36E Use the result of Exercise 13 to do the following problem. In MNPQ, QP=12 and QM=9. The length of altitude QR to side MN is 6. Find the length of altitude QS from Q to PN.Use the result of Exercise 13 to do the following problem. In ABCD, AB=7 and BC=12. The length of altitude AF to side BC is 5. Find the length of altitude AE from A to DC.The distance across a pond is to be measured indirectly by using similar triangles .If XY=160ft, YW=40ft, TY=120ft, and WZ=50ft, find XT.In the figure, ABCADB. Find AB if AD=2 and DC=6.Prove that the altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.Prove that the line segment joining the midpoints of two sides of a triangle determines a triangle that is similar to the original triangle.43E By naming the vertices in order, state three different triangles that are similar to each other. Exercises 1-6 Use theorem 5.4.2 to form a proportion in which SV is a geometric mean. Hint: SVTRVS Exercises 1-6 Use theorem 5.4.3 to form a proportion in which RS is a geometric mean. Hint RVSRST Exercises 1-6 Use theorem 5.4.3 to form a proportion in which TS is a geometric mean. Hint: TVSTSR Exercises 1-6 Use theorem 5.4.2 to find RV if SV=6 and VT=8. Exercises 1-6 6E Find the length of DF if: a DE=8 and EF=6. b DE=5 and EF=3. Exercises 7-10 Find the length of DE if: a DF=13 and EF=5. b DE=12 and EF=63. Exercises 7-10 Find EF if: a DF=17 and DE=15. b DF=12 and DE=82. Exercises 7-10 Find DF if: a DE=12 and EF=5 b DF=12 and EF=6. Exercises 7-10 Determine whether each triple (a,b,c) is a Pythagorean triple. a)(3,4,5)c)(5,12,13)b)(4,5,6)d)(6,13,15)12E 13E Determine the type of triangle represented if the lengths of its sides are: a a=1.5, b= 2 and c = 2.5 b a= 20, b= 21 and c = 29 c a=10, b= 12, and c =16 d a= 5, b= 7 and c = 9 A guy wire 25 ft long supports an antenna at a point that is 20 ft above the base of the antenna. How far from the base of the antenna is the guy wire secured?A strong wind holds a kite 30 ft above the earth in a position 40 ft across the ground. How much string does the girl let out to the kite?A boat is 6 m below the level of a pier and 12 m from the pier as measured across the water. How much rope is needed to reach the boat?A hot-air balloon is held in place by the ground crew at a point that is 21 ft from a point directly beneath the basket of the balloon. If the rope is of length 29 ft, how far above ground level is the basket?A drawbridge that is 104 ft in length is raised at its midpoint so that the uppermost points are 8 ft apart. How far has each of the midsections been raised?A drawbridge that is 136 ft in length is raised at its midpoint so that the uppermost points are 16 ft apart. How far has each of the midsections been raised? Hint: Consider the drawing for Exercise 19.A rectangle has a width of 16 cm and a diagonal of length 20 cm. How long is the rectangle?A right triangle has legs of lengths x and 2x2 and a hypotenuse of length 2x3. What are the lengths of its sides?A rectangle has base length x3, altitude length x1, and diagonals of length 2x each. What are the lengths of its base, altitude, and diagonals?The diagonals of a rhombus measure 6 m and 8 m. How long are each of the congruent sides?Each side of a rhombus measure 12 in. If one diagonal is 18 in. long, how long is the other diagonal?An isosceles right triangle has a hypotenuse of length 10 cm. How long is each leg?27E In right ABC with right C, AB=10 and BC=8. Find the length of MB if M is the midpoint of AC.29E 30E 31E 32E In quadrilateral RSTU, RSST and UT diagonal RT. If RS=6, ST= 8, and RU = 15, determine UT.34E If a=p2q2,b=2pq and c=p2+q2, show that c2=a2+b2.36E 37E When the rectangle in the accompanying drawing whose dimensions are 16 by 9 is cut into pieces and rearranged, a square can be formed. What is the perimeter of this square?A, C and F are three of the vertices of the cube shown in the accompanying figure. Given that each face of the cube is a square, what is the measure of angle ACF?40E In the figure, square RSTV has its vertices on the sides of square WXYZ as shown. If ZT=5 and TY=12 find TS. Also find RT.Prove that if (a,b,c) is a Pythagorean triple and n is a natural number, the (na,nb,nc) is also a Pythagorean triple.Use Figure 5.19 to prove Theorem 5.4.2. Theorem 5.4.2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.Use Figures 5.20 and 5.21 to prove Lemma 5.4.3. Lemma 5.4.3 The length of each leg of a right triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.45E For the 45-45-90 triangle shown, suppose that AC=a. Find: a BC b AB For the 45-45-90 triangle shown, suppose that AB=a2. Find: a AC b BC For the 30-60-90 triangle shown, suppose that XZ=a. Find: a YZ b XY For the 30-60-90 triangle shown, suppose that XY=2a. Find: a XZ b YZ In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right XYZ with mX=45 and XZ=8 Find YZ and XY In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right XYZ with XZYZ and XY=10 Find: XZ and YZ In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places Given: Right XYZ with XZYZ and XY=102 Find: XZ and YZ In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right XYZ with mX=45 and XY=122 Find: XZ and YZ In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right DEF with mE=60 and DE=5 Find: DF and FE In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right DEF with mF=30 and FE=12 Find: DF and DE In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right DEF with mE=60 and FD=123 Find: DE and FE In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right DEF with mE=2mF and EF=123 Find: DE and DF In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Rectangle HJKL with diagonals HK and JL mHKL=30 and LK=63 Find: HL, HK, and MK In Exercises 5 to 22, find the missing lengths. Give your answers in both simplest radical form and as approximations correct two decimal places. Given: Right RST with RT=62 and mSTV=150 Find: RS and ST 15E 16E 17E In Exercises 15 to 19, create drawings as needed. Given: XYZ with XYXZYZ ZWXY with W on XY YZ=6 Find: ZW In Exercises15 to 19, create drawing as needed. Given: Square ABCD with diagonals DB and AC intersecting at E DC=53 Find: DB Given: NQM with angles as shown in the drawing MPNQ Find: NM,MP,MQ, PQ, and NQ Given: XYZ with angles as shown in the drawing Find: XY HINT: Compare this drawing to the one for Exercise 20.Given: Rhombus ABCD not shown in which diagonals AC and DB intersect at point E; DB=AB=8 Find: AC 23E To unload groceries from a delivery truck at the Piggly Wiggly Market, an 8-ft ramp that rises 4 ft to the door of the trailer is used. What is the measure of the indicated angle D?A jogger runs along two sides of an open rectangular lot. If the first side of the lot is 200 ft long and the diagonal distance across the lot is 400 ft, what is the measure of the angle formed by the 200-ft and 400-ft dimensions? To the nearest foot, how much farther does the jogger run by travelling the two sides of the block rather than the diagonal distance across the lot?26E 27E In Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places. Given: In ABC, AD bisects BAC AB=20 and AC=10 Find: DC and DB 29E 30E In Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places. Given: Right ABC with mC=90 and mBAC=60; point D on BC; AD bisects BAC and AB=12 Find: BD In Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places. Given: Right ABC with mC=90 and mBAC=60; points D on BC;AD bisects BAC and AC=23 Find: BD In Exercises 27 to 33, give both exact solutions and approximate solutions to two decimal places. Given: ABC with mA=45, mB=30, and BC=12 Find: AB HINT: Use altitude CD from C to AB as an auxiliary line Note: Exercises preceded by an asterisk are of a more challenging nature. Given: Isosceles trapezoid MNPQ with QP=12 and mM=120; the bisectors of sMQP and Find: The perimeter of MNPQ In regular hexagon ABCDEF, AB=6 inches. Find the exact length of a diagonal BF b diagonal CF In regular hexagon ABCDEF, the length of AB is x centimeters. In terms of x find the length of a diagonal BF. b diagonal CF.37E Diagonal EC separates pentagon ABCDE into square ABCE and isosceles triangle DEC. If AB=8 and DC=5, find the length of diagonal DB. HINT: DrawDFAB.Note: Exercises preceded by an asterisk are of a more challenging nature. In preparing a certain recipe, a chef uses 5 oz of ingredient A, 4 oz of ingredient B, and 6 oz of ingredient C. If 90 oz of this dish are needed, how many ounces of each ingredient should be used?2E Given that ABEF=BCFG=CDGH, are the following proportions true? a ACEG=CDGH b ABEF=BDFH Given that XYTS, are the following proportion true? a TXXR=RYYS b TRXR=SRYR Given: l1l2l3l4,AB=5,BC=4,CD=3,EH=10 Find: EF,FG,GH See the figure for Exercise 6.Given: l1l2l3l4,AB=7,BC=5,CD=4,EF=6 Find: FG,GH,EH Given: l1l2l3,AB=4,BC=5,DE=x,EF=12x Find: x,DE,EF Given: l1l2l3,AB=5,BC=x,DE=x2,EF=7 Find: x,BC,DE Given: DEBC,AD=5,DB=12,AE=7 Find: EC Given: DEBC,AD=6,DB=10,AC=20 Find: EC 11E Given: DEBC,AD=5,DB=a+3,AE=a+1,EC=3(a1) Find: aandEC Given: RW bisects SRT Do the following equalities hold? a SW=WT b RSRT=SWWT Given: RW bisects SRT Do the following equalities hold? a RSSW=RTWT b mS=mT Given: UT bisects WUV,WU=8,UV=12,WT=6 Find: TV Given: UT bisects WUV,WU=9,UV=12,WV=9 Find: WT Given: NQ bisects MNP,NP=MQ,QP=8,MN=12 Find: NP 18E Exercises 18 and 19 are based on a theorem not stated that is the converse of Theorem 5.6.3. See the figure above. Given: NP=6,MN=9,PQ=4, and MQ=6;mP=62 and mM=36 Find: mQNM HINT: NPMN=PQMQ.Given: In ABC,AD bisects BAC AB=20 and AC=16 Find: DC and DB 21E In ABC,mCAB=80,mACB=60,andABC=40. With the angle bisectors as shown, which line segment is longer? a AE or EC? b CD or DB? c AF or FB?In ABC,AC=5.3,BC=7.2 and BA=6.7. With angle bisectors as shown, which line segment is longer? a AE or EC? b CD or DB? c AF or FB?In right RST not shown with right S,RV bisects SRT so that V lies on side ST. If RS=6,ST=63, and RT=12 find SV and VT.Given: RV bisects SRT, RS=x6,SV=3, RT=2x, and VT=x+2 Find: x HINT: You will need to apply the Quadratic Formula.Given: MR bisects NMP, MN=2x,NR=x, RP=x+1, and MP=3x1 Find: x Given point D in the interior of RST, which statements is are true? a RKKTTHHSGSRG=1 b TKKRRGGSSHHT=1 In RST, suppose that RH,TG, and SK are medians. Find the value of: a RKKT b THHS Given point D in the interior of RST, suppose that RG=3,GS=4,SH=4,HT=5, and KT=3. Find RK.30E 31E Given: RST, with XYRT and YZRS Prove: RXXS=ZTRZ 33E Use Exercise 33 and the following drawing to complete the proof of this theorem: The length of the median of a trapezoid is one-half the sum of the lengths of the two bases. Given: Trapezoid ABCD with median MN Prove: MN=12(AB+CD) 33. Use Theorem 5.6.1 and the drawing to complete the proof of this theorem: If a line is parallel to one side of a triangle and passes through the midpoint of a second side, then it will pass through the midpoint of the third side. Given: RST with M the midpoint of RS;MNST Prove: N is the midpoint of RT Use Theorem 5.6.3 to complete the proof of this theorem: If the bisector of an angle of a triangle also bisects the opposite side, then the triangle is an isosceles triangle. Given: XYZ;YW bisects XYZ;WXWZ Prove: XYZ is isosceles HINT: Use a proportion to show that YX=YZ.36E Given: ABC not shown is isosceles with mABC=mC=72;BD bisects ABC and AB=1 Find: BC Given: RST with right RST; mR=30 and ST=6; RST is trisected by SM and SN Find: TN,NM, and MR 39E 40E 1CR 2CR 3CR 4CR 5CR 6CR 7CR Find the values of x in each proportion: a x6=3x c 6x+4=2x+2 b x53=2x37 d x+35=x+57 e x2x5=2x+1x1 g x1x+2=103x2 f x(x+5)4x+4=95 h x+72=x+2x2 Use proportions to solve Review Exercises 9 to 11. Four containers of fruit juice cost 3.52. How much do six containers cost?10CR 11CR The ratio of the measures of sides of a quadrilateral is 2:3:5:7. If the perimeter is 68, find the length of each side.13CR The length of the sides of a triangle are 6, 8 and 9. The shortest side of a similar triangle has length 15. How long are its other sides?The ratio of the measure of the supplement of an angle to that of the complement of the angle is 5:2. Find the measure of the supplement.16CR Given: ABCD is a parallelogram. DB intersects AE at point F Prove: AFEF=ABDE 18CR 19CR 20CR 21CR 22CR 23CR For Review Exercises 24 to 26, GJ bisects FGH Given: FG=10,GH=8,FJ=7 Find JH.For Review Exercises 24 to 26, GJ bisects FGH Given: GF:GH=1:2,FJ=5 Find JH.For Review Exercises 24 to 26, GJ bisects FGH Given: FG=8,HG=12,FH=15 Find FJ.Given: EFGOHMJK,withtransversalsFJandEK;FG=2,GH=8,HJ=5,EM=6 Find EO, EK Prove that if a line bisects one side of a triangle and is parallel to a second side, then it bisects the third side.Prove that the diagonals of a trapezoid divide themselves proportionally.Given: ABCwithrightBACADBC a)BD=3,AD=5,DC=?b)AC=10,DC=4,BD=?c)BD=2,BC=6,BA=?d)BD=3,AC=32,DC=?Given: ABCwithrightBACADBC a)BD=12,AD=9,DC=?b)DC=5,BC=15,AD=?c)AD=2,DC=8,AB=?d)AB=26,DC=2,DC=?32CR Given: ABCDisarectangleEisthemidpointofBCAB=16,CF=9,AD=24 Find: AE,EF,AF,mAEF Find the length of a diagonal of a square whose side is 4 in. long.35CR Find the length of a side of a rhombus whose diagonals are 48cm and 14 cm long.Find the length of an altitude of an equilateral triangle if each side is 10 in. long.38CR The length of the three sides of a triangle are 13cm, 14 cm, and 15cm. Find the length of the altitude to the 14-cm side.40CR 41CR 42CR Reduce to its simplest form: a The ratio 12:20______ b The rate 200miles8gallons _________2CT 3CT 4CT 5CT In right triangle ABC,CD- is the altitude from C to hypotenuse AB-. Name three triangles that are similar to each other. ___________7CT 8CT 9CT 10CT In DEF,D is a right angle and m F=30. a Find DE if EF = 10m. ______________ b Find EF if DF=63 ft. ___________12CT 13CT For ABC, the three angle bisectors are shown. Find the product AEECCDDBBFFA. _________________15CT 16CT In Exercises 16 and 17, complete the statements and reasons in each proof. Given: ABC,P is the midpoint of AC-, and R is the midpoint of CB-. Prove: PRCB Statements Reasons 1.ABC 2. CC 3.P is the midpoint of AC-, and R is the midpoint of CB- 4. PCAC=12 and CRCB=12 5. PcAC=CRCB 6. CPR~CAB 7.______________________ 1.________________________ 2. ________________________ 3. ________________________ 4. Definition of midpoint 5._________________________ 6._________________________ 7. CASTC For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mAC=58,findmB.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mDE=46,findmO.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mDE=47.6,findmO.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mAC=56.4,findmB.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mB=28.3,findmAC.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mO=48.3,findmDE.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mDE=47, find the measure of the reflex angle that intercepts DBACE.For Exercises 1 to 8, use the figure provided. Exercises 1-8 If mECABD=312, find mDOE.Given: AOOBandOCbisectsACBinO Find: a mAB b mACB c mBC d mAOC Given: ST=12(SR)inQSRisadiameter Find: a mST b mTR c mSTR d mS HINT: Dra QT.Given: QinwhichmAB:mBC:mCA=2:3:4 Find: a mAB b mBC c mCA d m1(AQB) e m2(CQB) f m3(CQA) g m4(CAQ) h m5(QAB) i m6(QBC) HINT: Let mAB=2x,mBC=3x,andmCA=4x.Given: mDOE=76and mEOG=82inO EFisadiameter Find: a mDE b mDF c mF d mDGE e mEHG f WhethermEHG=12(mEG+mDF)Given: OwithABACand mBOC=72 Find: a mBC b mAB c mA d mABC e mABO In O not shown, OA is a radius, AB is a diameter, and AC is a chord. a How does OA compare to AB? b How does AC compare to AB? c How does AC compare to OA?Given: In O,OCABand OC=6 Find: a AB b BC Exercise 15 16E Given: Concentric circles with center Q TV=8andVW=2 RQTV Exercises 16, 17 Find: RQ HINT: Let RQ=x.AB is the common chord of O and Q. If AB=12 and each circle has a radius of length 10, how long is OQ? Exercises 18, 19 Circles O and Q have the common chord AB. If AB=6, O has a radius of length 4, and Q has a radius of length 6, how long is OQ? Exercises 18, 19 Suppose that a circle is divided into three congruent arcs by points A,B, and C. What is the measure of each arc? What type of figure results when A,B, and C are joined by line segments?Suppose that a circle is divided by points A,B,C, and D into four congruent arcs. What is the measure of each arc? If these points are joined in order, what type of quadrilateral results?22E 23E 24E 25E Five points are equally spaced on a circle. A five-pointed star pentagram is formed by joining nonconsecutive points two at a time. What is the degree measure of an arc determined by two consecutive points?A ceiling fan has equally spaced blades. What is the measure of the angle formed by two consecutive blades if there are a 5 blades? b 6 blades?A wheel has equally spaced lug bolts. What is the measure of the central angle determined by two consecutive lug bolts if there are a 5 bolts? b 6 bolts?An amusement park ride the Octopus has eight support arms that are equally spaced about a circle. What is the measure of the central angle formed by two consecutive arms?In Exercises 30 and 31, complete each proof. Given: Diameters AB and CD in E Prove: ACDB PROOF Statements Reasons 1. ? 1. Given 2. AECDEB 2. ? 3. mAEC=mDEB 3. ? 4. mAEC=mAC and mDEB=mDB 4. ? 5. mAC=mDB 5. ? 6. ? 6. If two arcs of a circle have the same measure, they are 31E In Exercises 32 to 37, write a paragraph proof. Given: RSandTVarediametersofW Prove: RSTVTS 33E 34E 35E

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