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Name__ Class _ Test 3 Series 1 CONGRUENT TRIANGLES Part I (10 points) 1 A triangle in which no two sides are congruent is called a(n) __ triangle. A In the diagram, if ~-~ ---- ~, then in order to prove ~ABC ~ ~EDC by HL, what additional two sides must be congruent? C In a triangle, what name is given to a line segment drawn from a vertex to the midpoint of the opposite side? 4 2 3 H If ~ ~ ~J, name the base angles. 4 5 If ~ ------ ~J and FO ~ FM, then what property justifies that ~ M 5 In problems 6-10, write A for always, S for sometimes, or N for never. triangle has 3 sides. 6 7 During the 5 minutes from 3:05 to 3:10, the minute and hour hands always form acute angles. 8 If a median of a triangle is also an altitude of the triangle, the triangle is scalene. 9 If an angle is selected at random from a triangle, the angle is obtuse. 10 9 If A -- (0, o), B = (10, 0), and ~g is rotated 90° ~vith respect to the origin, then B will rotate to the point (0, -- 10). y A (0, 0) Copyright © McOougal, Littell & Company x lO B (~0, o)~ ~ o~ Test 3 [ 9 Date Class Name CONGRUENT TRIANGLES Test 3 Series 1 (continued) Part II (20 points) In problems 11-13, write a two-column proof. Use a separate sheet of paper. 11 A Given: Two triangles, ~ABC and ~ABD, standing on a desktop called f BC ~ BD LABC ~- LABD Prove: F 12 Given: ~ = ~ G H KG ~ JF Prove: LE=LH 13 Given: @O LR~L_V PO --- OW Prove: TP- TW R V \ PoW Littell & Company Name Class__ CONGRUENT TRIANGLES Date. Test 3 Series 1 (continued) Supply the missing reasons in the proof for problem 14. 14 F E Given: LEBC ~LFCB LABF ~- LDCE Prove: ~EHC is isosceles. Statements 1 !_EBC ~ LFCB 2 LABF is supp. to LFBC. !_DCE is supp. to LECB.. 3 LABF ~ LDCE 4 LFBC m/ECB A B D C Reasons 1 Given 2 Iftwo angles forma straight angle, then they are supplementary. 3 Given 4 5 6 ~FBC ~ ~ECB 7 FB~-EC 8 CH~FB 9 EC~CH 10 AEHC is isosceles. 6 7 CPCTC 8 Given 9 10 Part II][ (15 points) 15 16 The perimeter of ~BAG is 43. AG=I6, AB=x +4, BG = 2x + 2 By solving for x, determine whether ~BAG is scalene, isosceles, or equilateral. A 2x+ 2 15 The circle has its center at the origin and passes through (0, To the nearest hundredth, find 5).4 the area of the circle. Copyright © McDougal, Littell & Company Test 3 I 11 Date Class__ Name _ CONGRUENT TRIANGLES 17 18 Test 3 Series 1 (continued) EC=12, ET=3x-5, VE= IO, ER=x +4 v mLVEC=5x-2 mLRET = 3x + 10 On the basis of the given, what c must be the value of x? Is AVEC ~ ARET? R T 17__ LB -~ L C AB=3x +I, AC=2x +5, BC=x +y Solve for x. If y < 2.97, then BC must be less than what number? 19 ~RGH ~ ~ANE GH = 10 mLG = 2w+ 2, mLN = 17w - 658 By solving for w, tell whether or not AN is an altitude of ~ANE. A C R G 18__ A HN 19 Part IV (5 points) 2O RECT is a rectangle with EC > RE. E = (4, 1) and C = (x, 1~, If the area of RECT is 24 and all sides of RECT have whole numbers as lengths, find the 4 possible coordinate values for x. (Hint: Find all possible lengths for ~ first.) Copyright © McDougal, Littell & Company Name Class Date Test 3 Series 3 CONGRUENT TRIANGLES Part I (24 points) B c In problems 1-6, choose the best answer. I If ABEC is isosceles, which of the following must be given in order to prove ABAE ~ ~CDE by ASA? a L2~L3 b AB~CD c AE~ED d LABC~LDCB 1 Given AB ~ CD and AC ~ BD. In order to prove the pair of overlapping triangles congruent, what additional fact must we use? a Vertical angles are congruent. b The reflexive property c The addition property d If A ,then Given AE ~ ED and BE ~- CE, what additional given is required in order to ]Stove ~ABE ~ .&DCE by H.L.? a LI~L4 b ABmCD c L 1 and L4 are right angles, d BD ± AC 4 If ~PQR ~ ATWV, which of the following is not necessarily true? a LP~LT b VT~-RP c QR~WT d LW~LQ 5 Complete the correspondence: AGFP ~ __ a ~HPF b ~HFL c APFH d ~HFP H 6 Which of the following is not necessarily true? b APGL ~ ~PHL a AGMF~- AGMH d c AKPF ~ AMFP Copyright © McDouga!, Littell & Company Test 3 I 185 Date _ Class _ Name. Test 3 Series 3 CONGRUENT TRIANGLES (continued) In problems 7-12, write A for always, S for sometimes, or N for never. 7 If ~-g ~ ~-~ in AABC, then LBAC ~ LABC 7 8 Two triangles are congruent if two sides and an angle of one triangle are congruent to two sides and an angle of the other. .9 If three sides of one triangle are congruent to three sides of another triangle, then each pair of corresponding angles is congruent. 10 If APQR = AQRP, then ~PQR is equilateral. 11 If two altitudes of a triangle are congruent then the medians to the same sides are congruent. 12 An obtuse scalene triangle is congruent to an acute isosceles triangle. 11_ 12_ Copyright © McDougal, Littetl & Company Date Class _ Name _ CONGRUENT TRIANGLES Test 3 Series 3 (continued) Part II (10 points) In problems 13-17, give all answers accurate to two decimal places. 13 Find the area of the shaded region. (Use ~ ~ 3.1416.) ~ 14 For what nonzero value of r, the radius, is the circumference of a circle numerically equal to its area? 14__ B 15 Given mLA is greater than mLB. What are the restrictions on the value of x ? x+ 10 15 A~ 2x+4 c 16 ~PQR ~ ASTV pQ = x2 SV = 6 ST=x+6 TV = 3-x Find all possible values for x. Then find the perimeter of ~PQR. 16 17 The sides Of an isosceles triangle are whole numbers. If the perimeter is 24, what is the probability that the triangle is equilateral. (Hint: list all valid possibilities first.) Copyright © McDougaL Uttell & Company 17 Test 3 1 187 Date _ Class __ Name Test 3 Series 3 CONGRUENT TRIANGLES (continued) (5, 11) Part III (9 points) W ~ 18 The vertices of ~MRW are (5, 11), (2, 4), and (9, 7) respectively. Explain why ~MRW is isosceles. (9, 7). 4) X 18_ In problem 19, draw a diagram, state the given and the conclusion, and write the proof. 19" If two points on the legs of an isosceles h’iangle are equidistant from the vertex, the segments joining each to the endpoint of the base on the opposite leg are congruent. 19_ Part IV (7 points) ¯ f In problem 20, wr~te a paragraph proo. 2O Given: ~)P BC ---- FE AB ± FC PA ~ PD Prove: LC ~LF Copyright © McDougal, Littell & Company are congruent. 15 1 A--~ ~- ~ / Given 2 B is the midpoint of ~-~. Given 3 D is the midpoint of i~7~. / Given 4 ~ ~ ~ / If two segments are congruent, then their halves are congruent (Division Property). 16 1 L 5 is supp. to/_ 6. / Given 2 /7 is supp. to/-6. / Given 3 L 5 ~ L 7 / If two angles are supplementary to the same angle, then they are congruent. Part III (15 points) 17 72 18 21.76° or 21°45’36" CT=5, EC= 10, RE = 5; area = 50 19 93 :~0 15 21 RT = 10, Part IV (5 points) 22 1 L3iscomp. toLl./Given 2 L2iscomp. toL1./Given 3 L2 ~/-3 / If two angles are complementary to the same angle, then they are congruent. 4 L4 ~ L5 / Given 5 LCOM ~- LPMO / If congruent angles are added to congruent angles, then the sums are congruent (Addition Property). Chapter 3 Congruent Triangles Part I Part II (10 points) 1 scalene 2 ~ and ~ 3 median 5 subtraction 6 A 7 A 8 N 4 LH and LJ 9 S 10 S (20 points) 11 1 ~ ~ ~ / Given 2 LABC ~ !_ABD / Given 3 ~--~ ~ ~-~ / Reflexive Property 4 AABC ~- z~ABD / SAS (1, 2, 3) 5 ~-~/CPCTC 12 1 E~J~JJ/Given 2 Given 3 ~]~ ~- ~-~ / Given 4 ~ ~ ~ / Addition Property 5 ~EKG---~HJF/SSS(1,2,4) 6 LE~LH/CPCTC 13 1 ~)O/ Given 20~ ~ OV / All radii of a circle are congruent. 3 ~-~ ~ OW / Given 4 ~VW/Subtraction Property 5 LR-~LV/Given 6 R-T~VT/If ~ ,then z~. 7 ARPT~AVWT/SAS(4,5,6) 8 TP ~ T~V / CPCTC 14 4 If two angles are supplementary to congruent angles, then they are congruent. 5 Reflexive Property 6 ASA (1, 5, 4) 9 Transitive Property 10 If at least two sides of a triangle are congruent, then the triangle is isosceles. Part III (15 points) 15 isosceles 16 78.54 19 w =44;yes 17 x = 6; no 18 x =4;BC<6.97 Part IV (5 points) ~0 28, 16, 12, 10 Chapter 4 Lines in the Plane Part I (10 points) 1T2T3F4F 8 d 9 right 472 I Answers to Tests 3-4 5 T 6 L6 7 LB andLE~D 10 1.31 Copyright © McDougal, Littell & Company 11 mL~QG ff mLHAT + m/HOG / Substitution 12 mLBOG = m/TAG / Substitution 13 LBOG ~- !TAG / Def. of congruence. Z0 To have a complement, the supplement of the angle must have a measure less than 90. If the supplement has measure less than 90, then the angle itself must be greater than 90 and hence does not have a complement. You cannot take the supplement of the complement because the angle does not have a complement. Chapter 3 Congruent Triangles Part I Part II (24 points) ld 2b 3c 4c 10 A 11 A 12 N 5d 6a 7S 8S 9A (10 points) 1385.84 14 2 15 -2<x<6 16 x = -2;P=15 17~ Part III (9 points) 18 The two ft. ~s have legs 7 and 3, and are ~ by SAS. Thus ~ ~ ~, so ~ is isos. A 19 Given: ~ABC is isos. ~ Prove: ~ ~ ~ B C A--~ ~ ~. ~ ~- ~, so ~-~ ~ ~ by subtraction. In an isos. ~, the base Ls are ~, so LABC ~ LACB. By the Reflexive Property, ~ ~ ~. Then ~DBC ~ ZhECB by SAS..’. D--~ ~ ~ by C15CTC. Part IV (7 points) 20 Since ~ 3_ ~-C and ~ 3_ ]TC, L BAC and L EDF are rt. L s. Then ~BAC and ~EDF are ft. ~s. ~ ~- ~-~. Since ~ and ~-ff are radii of (~)P, ~C ~ ~. P~ ~ P-~, so ~ m ~ by subtraction. Then ~ ~ ~ by addition. ~BAC ~ ~EDF by HL..’. LC ~ LF by CPCTC. Chapter 4 Lines in the Plane Part I (lO points) 1A2A 3N 92,3 102,5 4 S 5 A 6 A 71,5 82,4 Part II (~4 points) ¯ 11 ---~ 12 _3_4 13 (-18, 1) 14 ~ 15 (5, 7) 16 (-5, -7) 17 (-3,-2) Part III (8 points) o; oo 18 x =25orx = -4; forx =25,150°, 30;forx = -4, 92,88 19 6 = 2 +2 10_,21 5 +~_~, so D is the midpoint of ~-C. Then ~ bisects 1 + 3 1. The slopes are -5+3 AC, Or slope ~ = ~ = - 1, slope ~’~ = 6 -----~ = negativ~ reciprocals, so ~ 3_ ]~’C. Therefore, since ~ bisects ~ and is ± to ~, then B~ is the 3_ bisector of AC. : Copyrigh~ © McDougal, Littell & Company Answers to Tests 3-4 [ 497