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Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.1 Triangles Sum Conjectures 4.1 Page 203 Exercise #8 Look for overlapping triangles! Hint: look for large overlapping triangles (ie. The one with the 40°, 71° and a.) a = 69, b = 47, c = 116, d = 93, e = 86 4.1 Page 203 Exercise #9 Hint: Fill in angles that do not have a variable and l ook for large overlapping triangles! There are many!! m = 30, n = 50, p = 82, q = 28, r = 32, s = 78, t = 118, u = 50 S. Stirling Page 1 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.2 Group Investigation 1, Base Angles of an Isosceles Triangle Each of the triangles below is isosceles. Carefully measure the angles of each triangle. (Make sure the triangles’ angles sum is 180° right?) If you disregard measurement error, are there any patterns for all isosceles triangles? A 140 20 B 20 C A 28 76 B 45 76 45 C C 90 A Finish the following conjecture using the vocabulary you learned about isosceles triangles. Isosceles Triangle Conjecture If a triangle is isosceles, then its base angles are congruent. Complete the conjecture in the notes and the example problem. S. Stirling Page 2 of 18 B Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.2 Group Investigation 2, Is the Converse True? Write the converse of the Isosceles Triangle Conjecture below. Converse of the Isosceles Triangle Conjecture If a triangle has two congruent angles, then it is an isosceles triangle. Is this converse true? In this investigation, you are going to make congruent angles and then measure the sides to see if the triangle is isosceles. For each of the following, make Extend the sides to form ∠A . Then measure the sides to see if ΔABC is isosceles. 6.1 cm 6.1 cm 35 35 B 10 cm C ∠B ≅ ∠C . C 8.4 cm 70 5.7 cm 8.4 cm 70 B Is the converse of the Isosceles Triangle Conjecture true? YES Complete the conjecture in the notes and the example problem. S. Stirling Page 3 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.2 Page 209 Exercise #10 Hint: Look for the overlapping triangle involving e, d and 66°. Do you see 3 equal angles? a = 124, b = 56, c = 56, d = 38, e = 38, f = 76, g = 66, h = 104, k = 76, n = 86, p = 38 4.2 Page 209 Exercise #11 In the problem, they state that the angles around the center are congruent. Note: In order for the pattern of tiles to look symmetric, all of the triangles of the same size must be congruent! How many of the tiles are isosceles triangles? a = 36, b = 36, c = 72, d = 108, e = 36 All of the triangles are isosceles. S. Stirling Page 4 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.3 Group Investigation 1, Lengths of the Sides of a Triangle For each of the following, construct the triangle given the three sides. Compare your results with your group members. When is it possible to construct a triangle from 3 sides and when is it not possible? Measure the three sides in centimeters. How do the numbers compare? Construct ΔCAT C from A A T C T C T Construct ΔFSH from F S H H F F S S Why were you able to construct ΔCAT but not able to construct ΔFSH ? Give more examples of three side lengths that will NOT make a triangle. Will sides of 4 cm, 6 cm and 10 cm make a triangle? Various examples: 2, 5, 9 because 2 + 5 < 9 4, 6 and 10? No because 4 + 6 = 10 NOT a triangle, it’s a segment. State your observations in the conjecture. Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Complete the conjecture in the notes and the example problems. S. Stirling Page 5 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.3 Group Investigation 2, Largest and Smallest Angles in a Triangle For each of the following triangles, carefully measure the angles. Label the angle with the greatest measure ∠ L , the angle with the second largest measure ∠ M , and the smallest angle ∠ S . Now measure the sides in centimeters. . Label the side with the greatest measure l, the side with the second largest measure m, and the shortest side s. Which side is opposite ∠ L ? ∠ M ? ∠ S ? Write a conjecture that states where the largest and smallest angles are in a triangle, in relation to the longest and shortest sides. M L 33 l 75 s S m 128 19 m s L S 35 70 M l Side-Angle Inequality Conjecture In a triangle, if one side is the longest side, then the angle opposite the longest side is the largest angle. (And visa versa.) Likewise, if one side is the shortest side, then the angle opposite the shortest side is the smallest angle. (And visa versa.) Does this property apply to other types of polygons? Test it out! Would you really need to measure these? ∠E is the E P N A T largest angle but it is opposite the shortest side AT U Q A D . Can’t be true for polygons with an even number of sides because angles are opposite angles and sides are opposite sides. Complete the conjecture in the notes and the example problems. S. Stirling Page 6 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ EXERCISES Lesson 4.4 Page 224-225 #3 – 10, 12 – 17 I I Z P Z P For Exercises 4 – 9, decide whether the triangles are congruent, and name the congruence shortcut you used (SSS or SAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible. SSS Cong. FD ≅ FD shared side SSS Cong. CV ≅ CV shared side KA ≅ KA shared side SSS Cong. SAS Cong. Not congruent, sides are not matched. But, ΔCOT ≅ ΔNAP by SAS Cong. AY ≅ YR Def. midpoint BY ≅ YN If base angles =, ΔBNY isosceles. SSS Cong. Perimeter ΔABC = 180 x + x + 11 + x − 11 = 180 3 x = 180 x = 60 AC = 60 , AB = 60 + 11 = 71 m∠DAE = m∠BAC Vertical angles = So ΔABC ≅ ΔADE by SAS Cong. S. Stirling Page 7 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ For Exercises 12 – 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS or SAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible. ΔANT ≅ ΔFLE SSS Cong. Can’t be determined since SSA is not a congruence guarantee. ΔSAT ≅ ΔSAO Can’t be determined since the parts do not match up. One triangle is an ASA the other is an AAS. S. Stirling SAS Cong. Since AS ≅ AS a shared side. ΔGIT ≅ ΔAIN SSS Cong. Or if you state ∠GIT ≅ ∠AIN because vertical angles = then ΔGIT ≅ ΔAIN by SAS Cong. Even with WO ≅ WO a shared side. Can’t be determined since the parts do not match up. One triangle is an ASA the other is an AAS. Page 8 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ EXERCISES Lesson 4.5 Page 229-230 #3 – 18 I Z P I Z P For Exercises 4 – 9, determine whether the triangles are congruent, and name the congruence shortcut you used (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible. ∠AMD ≅ ∠RMC because vertical angles = then ΔAMD ≅ ΔRMC by ASA Cong. ∠HOW ≅ ∠FEW because vertical angles = Even with this there is not enough information to determine congruence. S. Stirling Even with FS ≅ FS a shared side. Can’t be determined. This would be a SSA which does not guarantee congruence. ΔBOX ≅ ΔCAR ASA Cong. ΔGAS ≅ ΔIOL AAS Cong. Or if you use the third angle conjecture: state ∠S ≅ ∠L , then ΔGAS ≅ ΔIOL by ASA Cong. ∠INT ≅ ∠TLA and ∠NIT ≅ ∠TAL because lines ||, so alternate interior angles =. ∠NTI ≅ ∠LTA because vertical angles = Even with all of this the triangles are not necessarily congruent because AAA does not guarantee congruence. Page 9 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ For Exercises 10 – 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible. ΔFAD ≅ ΔFED SSS Cong. Since FD ≅ FD a shared side. Overlapping Triangles ΔPOE ≅ ΔPRN by ASA Cong. and ∠OSN ≅ ∠RSE because vertical angles = ΔSON ≅ ΔSRE by ASA Cong. S. Stirling ∠H ≅ ∠T and ∠O ≅ ∠A because lines ||, so alternate interior angles =. ΔWHO ≅ ΔWTA by AAS Cong. or… ∠HWO ≅ ∠YWA because vertical angles = ∠H ≅ ∠T because lines ||, so alternate interior angles =. ΔWHO ≅ ΔWTA by ASA Cong. ∠DMA ≅ ∠RMC because vertical angles = Still can’t be determined since the parts do not match up. One triangle is an ASA the other is an AAS. ∠LAT ≅ ∠SAT def. of angle bisector Since TA ≅ TA a shared side, ΔLAT ≅ ΔSAT by SAS Cong. Overlapping Triangles: MR ≅ MR a shared side. ΔRMF ≅ ΔMRA by SAS Cong. Page 10 of 18 Ch 4 Worksheet L2 Key 3 post ∠B ≅ ∠K and ∠L ≅ ∠C because lines ||, so alternate interior angles =. ∠BAL ≅ ∠KAC because vertical angles = Even with all of this the triangles are not necessarily congruent because AAA does not guarantee congruence. Name ___________________________ WL ≅ WL a shared side. ΔLAW ≅ ΔWKL by ASA Cong. Perimeter ΔABC = 138 x + x + 4 + x − 4 = 138 3 x = 138 x = 46 AC = 46 m∠DAE = m∠BAC Vertical angles = ∠E ≅ ∠C and ∠D ≅ ∠B because lines ||, so alternate interior angles =. So ΔABC ≅ ΔADE by AAS Cong. or by ASA Cong. S. Stirling Page 11 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.4 Page 226 Exercise #23 a = 37, b = 143, c = 37, d = 58 e = 37, f = 53, g = 48, h = 84, k = 96, m = 26, p = 69, r = 111, s = 69 4.6 Page 234 Exercise #18 a = 112, b = 68, c = 44, d = 44, e = 136, f = 68, g = 68, h = 56, k = 68, l = 56, m = 124 S. Stirling Page 12 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.6 Corresponding Parts of Congruent Triangles 4.6 Page 232 Example A Given: Prove: C AM ≅ MB and m∠A = m∠B AD ≅ BC B 2 1 M A D m∠ A = m∠ B AM ≅ MB given given m∠1 = m∠ 2 ΔAMD ≅ ΔBMC Vertical angles = ASA Congruence AD ≅ BC CPCTC or Def. Congruence B Example B Given: Prove: BD ⊥ AC and DB bisects m∠ABC ∠A ≅ ∠C m∠ADB = m∠BDC = 90 BD ⊥ AC given A D def. of perpendicular DB bisects m∠ABC given m∠ABD = m∠CBD BD ≅ BD Shared side def. of angle bisector ∠A ≅ ∠C ΔABD ≅ ΔCBD ASA Congruence CPCTC or Def. Congruence S. Stirling C Page 13 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.7 Page 241 Exercise #13 a = 72, b = 36, c = 144, d = 36 e = 144, f = 18, g = 162, h = 144, j = 36, k = 54, m = 126 4.8 Page 247 Exercise #12 a = 128, b = 128, c = 52, d = 76 e = 104, f = 104, g = 76, h = 52, j = 70, k = 70, l = 40, m = 110, n = 58 S. Stirling Page 14 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ EXERCISES Ch 4 Review Page 252 #7 – 24 For Exercises 10 – 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS, SAS, ASA or AAS). If not enough information is given, see if you can use the definitions and conjectures you have learned (all listed on the Note Sheets pages 6 & 7) to get more equal parts. Write what you know and the property you used. Mark diagrams with the parts you can deduce to be equal. If congruence still cannot be determined, write “cannot be determined” and draw a counterexample if possible. The triangles are not necessarily congruent because SSA does not guarantee congruence. Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence. S. Stirling ∠OPT ≅ ∠APZ vertical angles = ΔTOP ≅ ΔZAP by AAS Cong. Cong. or or if you state TO & AZ because alternate interior angles =, then ∠T ≅ ∠Z the lines are parallel. now ΔTOP ≅ ΔZAP by ASA Cong. if you state ΔTRP ≅ ΔAPR by SAS Cong. ΔMSE ≅ ΔOSU by SSS ∠ESM ≅ ∠USO because vertical angles = , now ΔMSE ≅ ΔOSU by SAS Cong. Since ∠GHI ≅ ∠HIG , HG ≅ GI because if base angles =, then isosceles. ∠HGC ≅ ∠IGN because vertical angles = ΔCGH ≅ ΔNGI by SAS Cong. Page 15 of 18 Ch 4 Worksheet L2 Key 3 post If isosceles, then base angles =. So ∠O ≅ ∠T . WH ≅ WH a shared side Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence. Name ___________________________ HJJG HJJG Since AB & CD ∠A ≅ ∠D and ∠B ≅ ∠C because lines ||, so alternate interior angles =. Also ∠BEA ≅ ∠CED because vertical angles = ΔABE ≅ ΔDEC by AAS Cong. or ASA Cong. Since it is a regular polygon all sides and angles are =: ∠C ≅ ∠B and CN = CA = OB = BR So ΔACN ≅ ΔOBR or ΔACN ≅ ΔRBO by SAS Cong. ΔAMD ≅ ΔUMT by SAS Cong. AD ≅ UT Def. of Congruent Triangles or CPCTC S. Stirling Can’t be determined because you can not get more equal sides nor angles and AAA does not guarantee congruence. Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence. Page 16 of 18 Ch 4 Worksheet L2 Key 3 post Since LA & TR , ∠A ≅ ∠T because lines ||, so alternate interior angles =. ΔSLA ≅ ΔIRT by AAS Cong. TR ≅ LA Def. of Congruent Triangles or CPCTC Since MN & CT , ∠MNT ≅ ∠NTC because Name ___________________________ ΔINK ≅ ΔVSE by SSS Cong. But not needed because EV = IK and Parts do not match. Both triangles are AAS but the angles do not match. EV + VI = IK + VI EI = VK by addition. Overlapping triangles: ΔALZ ≅ ΔAIR by ASA Cong. because Since ∠SPT ≅ ∠PTO , the alternate interior angles = and NT ≅ NT a shared side lines ||. SP & TO Since ∠OPT ≅ ∠PTS , the alternate interior angles = and Can’t be determined because you can not get more equal sides nor angles and SSA does not guarantee congruence. lines ||. OP & TS . Since the opposite sides are parallel, STOP is a parallelogram. lines ||, so alternate interior angles =. S. Stirling ∠A ≅ ∠A same angle. Page 17 of 18 Ch 4 Worksheet L2 Key 3 post Name ___________________________ 4.R Page 253 Exercise #27 X In ΔPCX : m∠CPX = 30 triangle sum 180 – 30 – 120 = 30 So f larger than a = g In ΔPXM : m∠PXM = 60 straight angle 180 – 30 – 90 = 60 m∠PMX = 60 triangle sum 180 – 60 – 60 = 60 So all sides of ΔPXM are equal f =e=d In ΔAXM : m∠XMA = 45 triangle sum 180 – 90 – 45 = 45 Since base angles =, triangle is isosceles. So So c larger than d = b So c is the largest overall! S. Stirling Page 18 of 18