4.1 Triangles Sum onjectures uxillary line: an extra line or segment that helps you with your proof. Page 202 Paragraph proof explaining why the Triangle Sum onjecture is true. onjecture: The sum of the measures of the angles in every triangle is 180. with auxiliary line E Given: angles labeled as shown. Show: m 2 + m 4 + m 5 = 180 and m 1+ m 2 + m 3 = 180 Linear pair conjecture and form transversals between parallel lines E and m 1 = m 4 and m 3 = m 5 because I are congruent Substituting into the first equation above m 2 + m 4 + m 5 = 180 Therefore, the sum of the measures of the angles in every triangle is 180. Page 204 #18 Prove Third ngle onjecture onjecture: If two angles of one triangle are congruent to two angles of another triangle, then the third angle in each triangle is congruent to the third angle in the other triangle. Given: m = m E and m = m F Show: m = m D E F D m + m + m = 180 and m E + m F + m D = 180 by the triangle sum conjecture. Since they both equal 180, m + m + m = m E + m F + m D Now subtract equal measures m = m E and m = m F. m = m D Therefore, the third angles are always congruent. S. Stirling Page 1 of 20
4.1 Page 203 Exercise #8 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 look 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 2 of 20
4.2 Group Investigation 1: ase ngles of an Isosceles Triangle Each of the triangles below is isosceles. arefully 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? 20 140 28 20 76 45 45 76 90 Finish the following conjecture using the vocabulary you learned about isosceles triangles. Isosceles Triangle onjecture If a triangle is isosceles, then its base angles are congruent. omplete the conjecture in the notes. S. Stirling Page 3 of 20
4.2 Group Investigation 2: Is the onverse True? Write the converse of the Isosceles Triangle onjecture below. onverse of the Isosceles Triangle onjecture 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. Then measure the sides to see if is isosceles. 6.1 cm 6.1 cm 35 10 cm 35 70 8.4 cm 5.7 cm 8.4 cm 70 Is the converse of the Isosceles Triangle onjecture true? YES omplete the conjecture in the notes. S. Stirling Page 4 of 20
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 ll of the triangles are isosceles. S. Stirling Page 5 of 20
4.3 Group Investigation 1: Lengths of the Sides of a Triangle For each of the following, construct the triangle given the three sides. ompare 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? onstruct T from T T T onstruct FSH from F S H H F S F S T but not able to construct FSH Why were you able to construct? 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 onjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. omplete the conjecture in the notes and the example problems. S. Stirling Page 6 of 20
4.3 Group Investigation 2: Largest and Smallest ngles 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. L? M? S Which side is opposite? Write a conjecture that states where the largest and smallest angles are in a triangle, in relation to the longest and shortest sides. S 19 128 m l L 33 s S M 35 m l L 75 70 s M Side-ngle Inequality onjecture In a triangle, if one side is the longest side, then the angle opposite the longest side is the largest angle. (nd visa versa.) Likewise, if one side is the shortest side, then the angle opposite the shortest side is the smallest angle. (nd visa versa.) Does this property apply to other types of polygons? Test it out! Would you really need to measure these? P E T N E is the largest angle but it is opposite the shortest side T. omplete the conjecture in the notes and the example problems. Q U D an t be true for polygons with an even number of sides because angles are opposite angles and sides are opposite sides. S. Stirling Page 7 of 20
EXERISES Lesson 4.4 Page 224-225 #3 10, 12 17 Mark diagrams! If congruence cannot be determined, draw a counterexample. S. Stirling Page 8 of 20
For Exercises 12 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS or SS). 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. S. Stirling Page 9 of 20
EXERISES Lesson 4.5 Page 229-230 #3 18 Mark diagrams! If congruence cannot be determined, draw a counterexample. S. Stirling Page 10 of 20
For Exercises 10 17, if possible, name a triangle congruent to the given triangle and state the congruence conjecture (SSS, SS, S or S). 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. S. Stirling Page 11 of 20
S. Stirling Page 12 of 20
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 13 of 20
4.6 orresponding Parts of ongruent Triangles 4.6 Page 232 Example Given: M Prove: D M and m = m 2 1 M D M M m = m given given MD M S ongruence m 1 = m 2 Vertical angles = D PT or Def. ongruence Example Given: D and D bisects m Prove: D given m D = m D = def. of perpendicular 90 D D bisects m given m D = m D def. of angle bisector D D Shared side PT or Def. ongruence D D S ongruence S. Stirling Page 14 of 20
4.8 Proving Special Triangle onjectures Prove: The bisector of the vertex angle of an isosceles triangle is also the median and altitude to the base. Given: Isosceles with = ; D bisects Prove: D is a median. D is an altitude. given = D bisects m given m D = m D def. of angle bisector D D D Shared side D D is a median Def. of a Median = D PT or Def. ongruence D D S ong. D is an altitude Def. of an ltitude m D = m D = 90 PT or Def. ongruence Prove: The bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector to the base. with = ; D bisects Given: Isosceles Prove: D is a perpendicular bisector. given = D = D Shared side D PT or Def. ongruence D D D bisects m given D S ongruence m D = m D def. of angle bisector D is a perpendicular bisector Def. of a perpendicular bisector m D = m D = Linear pair supp. and angles equal. D 90 D Def. Perpendicular S. Stirling Page 15 of 20
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 16 of 20
EXERISES h 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, SS, S or S). 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 SS does not guarantee congruence. OPT PZ vertical angles = TOP ZP by S ong. or if you state TO Z because alternate interior angles =, then T Z the lines are parallel. now TOP ZP by S ong. MSE OSU by SSS ong. or if you state ESM USO because vertical angles =, now MSE OSU by SS ong. an t be determined because you can not get more equal sides nor angles and SS does not guarantee congruence. TRP PR by SS ong. Since GHI HIG, HG GI because if base angles =, then isosceles. HG IGN because vertical angles = GH NGI by SS ong. S. Stirling Page 17 of 20
If isosceles, then base angles =. So O T. WH WH a shared side an t be determined because you can not get more equal sides nor angles and SS does not guarantee congruence. Since D D and because lines, so alternate interior angles =. lso E ED because vertical angles = E DE by S ong. or S ong. Since it is a regular polygon all sides and angles are =: and N = = O = R So N OR or N RO by SS ong. MD UMT by SS ong. D UT Def. of ongruent Triangles or PT an t be determined because you can not get more equal sides nor angles and does not guarantee congruence. an t be determined because you can not get more equal sides nor angles and SS does not guarantee congruence. S. Stirling Page 18 of 20
Since L TR, T because lines, so alternate interior angles =. SL IRT by S ong. TR L Def. of ongruent Triangles or PT INK VSE by SSS ong. ut not needed because EV = IK and EV + VI = IK + VI EI = VK by addition. Parts do not match. oth triangles are S but the angles do not match. Since MN T, MNT NT because lines, so alternate interior angles =. NT NT a shared side an t be determined because you can not get more equal sides nor angles and SS does not guarantee congruence. Overlapping triangles: LZ IR by S ong. because same angle. Since SPT PTO, the alternate interior angles = and lines. SP TO Since OPT PTS, the alternate interior angles = and lines. OP TS. Since the opposite sides are parallel, STOP is a parallelogram. S. Stirling Page 19 of 20
4.R Page 253 Exercise #27 X In PX : m PX = 30 triangle sum 180 30 120 = 30 So f larger than a = g In PXM : m PXM = 60 straight angle 180 30 90 = 60 m PMX = 60 triangle sum 180 60 60 = 60 So all sides of PXM are equal f = e = d In XM : m XM = 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 20 of 20