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Ch 4 Note Sheets L2 Key 3 Name ___________________________ Chapter 4: Discovering and Proving Triangle Properties Note Sheets S. Stirling Page 1 of 13 Ch 4 Note Sheets L2 Key 3 4.1 Triangles Sum Conjectures Name ___________________________ Read the top of page 200. These notes replace pages 200 – 202 in the book. Rigidity is a property that triangles have. They cannot be shifted, like a quadrilateral can. They retain their shape. Triangle Sum Conjecture The sum of the measures of the angles in every triangle is 180°. Numeric Example: Find x. Find y. x + x + y = 180 2x + y = 180 y = 180 – 2x B 101 56 x A C F x E x y D 101 + 56 + x = 180 x = 180 – 157 = 23 Third Angle Conjecture If two angles of one triangle are congruent to two angles of another triangle, then the third angles of the triangles are congruent. Numeric Example: If m∠ K = m∠ B and m∠ I = m∠ C , find y. Give reasons for your answer! A K y B 101 56 J 23 I C y = 56 because the third angles are equal also. Easier than 180 – 101 – 23 = 56. S. Stirling Page 2 of 13 Ch 4 Note Sheets L2 Key 3 Name ___________________________ Lesson 4.2 Properties of Isosceles Triangles Remember! With If..then.. statements: If a, then b. The converse of the statement is If b, then a. Read top of page 206. This vocabulary is review! Know it!! Label and define the special vocabulary for the isosceles triangle ∆ VBS . Base angles B sides. VB and VS Base S leg leg A leg (of an isosceles triangle) is one of the congruent The base (of an isosceles triangle) is the side that is not a leg. BS The vertex angle is the angle between the two legs. ∠ V V Vertex angle The base angles are the pair of angles whose vertices are the endpoints of the base. ∠B and ∠ S Do group Investigation 1, Base angles of an Isosceles Triangle on WS page 2. Isosceles Triangle Conjecture If a triangle is isosceles, then its base angles are congruent. V Given ∆VBS with VB = VS , then ∠ B ≅ ∠ S or If VB ≅ VS , then ∠ B ≅ ∠ S S B ***Note: Since equilateral triangles are a special case of isosceles triangles, any property that applies to isosceles triangles also applies to equilateral triangles. Do group Investigation 2, Is the Converse True? on WS page 3. Use a protractor. Converse of the Isosceles Triangle Conjecture If a triangle has two congruent angles, then it is an isosceles triangle. If ∠ B ≅ ∠ S , then ∆VBS is isosceles B V or If ∠ B ≅ ∠ S , then VB ≅ VS . S. Stirling Page 3 of 13 S Ch 4 Note Sheets L2 Key 3 Lesson 4.3 Triangle Inequalities Name ___________________________ Read of page 215. Do group Investigation 1 on WS page 5. Triangle Inequality Conjecture The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Is a triangle measuring 5 cm, 4 cm and 2 cm possible? Sketch it then explain! Is a triangle measuring 5 cm, 3 cm and 2 cm possible? Sketch it then explain! Since 2 + 4 > 5, it makes a triangle. Since 2 + 3 = 5, it makes segment NOT a triangle. Do group Investigation 2, Largest and Smallest Angles in a Triangle on WS page 6. Side-Angle Inequality Conjecture (This only applies to triangles!) 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.) Which is the largest angle? The smallest angle? Why? Which is the largest side? Why? D K 5.34 2.48 J 6.3 ∠K largest, its opposite the longest side, JI . ∠I smallest, its opposite the shortest side, JK . S. Stirling I In ∆DAQ , ∠ D largest, so AQ largest in ∆DAQ . In ∆QAU , A m∠QAU = 100 Q 36 90 23 54 100 which makes UQ the largest in ∆QAU since AQ < UQ . Overall, UQ is the longest. Page 4 of 13 57 U Ch 4 Note Sheets L2 Key 3 Name ___________________________ Label the drawing with the following terms and define the terms: Exterior angle is an angle that forms a linear pair with one of the interior angles of a polygon. Adjacent interior angle is the angle of a polygon that forms a linear pair with a given exterior angle of a polygon. Remote interior Exterior angle Adjacent interior The remote interior angles (of a triangle) are the interior angles of a triangle that do not share a vertex with a given exterior angle. Triangle Exterior Angle Conjecture The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Given: ∆ABC with exterior angle labeled as shown. Show: ∠ BCD and angles x = a+b x + c = 180 Linear pairs are supplementary a + b + c = 180 Triangle’s angles add to 180° Since they both equal 180, x + c = a + b + c now subtract measure c from both sides x = a+b Given ∆ ABC above. If a = 50 and b = 60, what is the measure of ∠ BCD ? Explain. Given ∆ ABC above. If x = 80 and b = 30, what is the measure of ∠ A ? Explain. a+b = x 50 + 60 = 110 = x 80 = a + 50 80 – 50 = 30 = a The exterior angle equals the sum of the two remote interior angles. The exterior angle equals the sum of the two remote interior angles. S. Stirling Page 5 of 13 Ch 4 Note Sheets L2 Key 3 Name ___________________________ Before starting Lesson 4.4: How do you get equal parts (for congruent triangles)? How can you use given information to get equal sides and angles in a triangle. You will need to use your definitions and conjectures that you already know to be true. Complete the following to help you review these properties. Remember, to mark your diagrams (tic marks and arc marks) to show which parts are equal. Also never assume things are congruent! You must have a definitions or conjecture to back you up!! Ways to Get Equal Segments Converse of the Isosceles Triangle Conj. Def. Isosceles Triangle If a triangle has two congruent angles, then it is an isosceles triangle. V If a triangle is isosceles, then its legs are congruent. S If ∠ B ≅ ∠ S , V ∆VBS isosceles then VB ≅ VS . B If then VB ≅ VS B Def. Midpoint Def. segment bisector S If a point is a midpoint, then it divides the segment into two equal segments. G If M is the midpoint of SG , then SM ≅ MG . “Same Segment” When two triangles share the exact same segment, you get a pair of equal segments. ∆BET and ∆WTE share ET ≅ TE B W O E C If a line (or part of a line) is a bisector, then it passes through the midpoint of the segment. M T CM bisects AB , then AM ≅ MB . If M A If AM is median in ∆ABC , then BM ≅ MC . Ways to Get Equal Angles and Segments CD is the perp. bisector of AB , then AM ≅ MB and ∠ CMA = ∠ CMB = 90° . If S. Stirling M B If a segment is a median, then it connects the vertex to the midpoint of the opposite side. Def. perpendicular bisector B A Def. Median A line (or part of a line) that passes through the midpoint of a segment and is perpendicular to the segment. S C B M A D Page 6 of 13 C Ch 4 Note Sheets L2 Key 3 Name ___________________________ Ways to Get Equal Angles Isosceles Triangle Conjecture Vertical Angle Conjecture If a triangle is isosceles, then its base angles are congruent. V If two angles are vertical, then they are congruent. ∆VBS isosceles or if VB ≅ VS , then ∠ B ≅ ∠ S . V E R S C If ∠ VET and ∠ CER are vertical, then ∠ VET ≅ ∠ CER . If B Definition of Angle Bisector If you have an angle bisector, then the ray cuts the angle into two equal angles. Def. perpendicular lines A D B B M A C BD bisects ∠ ABC then ∠ ABD ≅ ∠ DBC . C If two lines are perpendicular, then they intersect to form equal 90° angles. CM ⊥ AB , then m∠ CMA = m∠ CMB = 90° . If If R Def. Altitude If a segment is an altitude, then it goes from a vertex perpendicular to the line that contains the opposite side. T Corresponding Angles Conjecture T I A If two parallel lines are cut by a transversal, then corresponding angles are congruent. A RA is an altitude of ∆TRI then m∠ RAT = m∠ RAI = 90° . E If ∆TIA and ∆NIR share ∠ TIA ≅ ∠ RIN . AD EK , then ∠ MDA ≅ ∠ K A R K D If I “Same Angle” When two triangles share the exact same angle, you get a pair of equal angles. M . G T N Alternate Interior Angles Conjecture If two parallel lines are cut by a transversal, then alternate interior angles are congruent. C A CB AD , then ∠ B ≅ ∠ A . If S. Stirling Page 7 of 13 B D Ch 4 Note Sheets L2 Key 3 Name ___________________________ Before starting Lesson 4.4: Vocab for triangles Triangles have 6 “parts”: 3 angles and 3 sides. angle “is opposite” means A ∠A is opposite BC and AC is opposite ∠B side ∠A is between BA and AC (angles are between sides) side angle B side BC is between ∠B and ∠ C (sides are between angles) angle C An included angle is an angle formed between two consecutive sides of a polygon. A For sides AC and BC , the included angle is ∠ C For sides AC and BC , a non-included angle is ∠ A or ∠ B B For sides AB and AC , the included angle is ∠ A C For sides AB and AC , a non-included angle is ∠ C or ∠ B An included side is a side of a polygon between two consecutive angles. For angles ∠ B and ∠ A , the included side is AB A For angles ∠ B and ∠ A , a non-included side is AC or CB B For angles ∠ C and ∠ A , the included side is AC C For angles ∠ C and ∠ A , a non-included side is AB or CB Lesson 4.4: Read page 221, then complete the Triangle Congruence Shortcut Investigation, page 1 – 2. results on page 9 and 10 of these notes. Summarize your Whole Class: Complete page 4 of the Triangle Congruence Shortcut Investigation. Remember you must use the properties you know to be true!! Use your notes pages 6 and 7. Lesson 4.5: Complete the Triangle Congruence Shortcut Investigation, page 3. of these notes. Summarize your results on page 9 and 10 Whole Class: Complete page 5 of the Triangle Congruence Shortcut Investigation. Remember you must use the properties you know to be true!! Use your notes pages 6 and 7. S. Stirling Page 8 of 13 Ch 4 Note Sheets L2 Key 3 Name ___________________________ The following properties to guarantee congruent triangles: SSS Congruence Conjecture If the three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. A C D T SAS Congruence Conjecture If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. O A G C I T Z P ∆CAT ≅ ∆DGO ∆PIZ ≅ ∆TAC By SSS Congruence Conjecture. By SAS Congruence Conjecture. ASA Congruence Conjecture If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. SAA or AAS Congruence Conjecture If two angles and a non-included side of one triangle are congruent to the two angles and the same non-included side of another triangle, then the triangles are congruent. I D O G D O I D O I G Z P G Z P Z P ∆DOG ≅ ∆ZPI ∆DOG ≅ ∆ZPI By ASA Congruence Conjecture. By AAS Congruence Conjecture. S. Stirling Page 9 of 13 Ch 4 Note Sheets L2 Key 3 Name ___________________________ The following properties DO NOT guarantee congruent triangles: SSA or ASS Congruence? If two sides and the non-included angle of one triangle are congruent to two sides and the non-included angle of another triangle, then the triangles are NOT necessarily congruent. A Draw a counterexample. I I I C T Z P ∆ZIP ≅ ∆_______ Z Z P P By NOT necessarily congruent. AAA Congruence Conjecture If three angles of one triangle are congruent to the corresponding angles of another triangle, then the triangles are NOT necessarily congruent. A C Draw a counterexample. I T Z I P ∆ZIP ≅ ∆_______ By NOT necessarily congruent. S. Stirling Z P I Z P Page 10 of 13 Ch 4 Note Sheets L2 Key 3 Name ___________________________ 4.6 Corresponding Parts of Congruent Triangles Note: Once you can prove that two triangles are congruent, by using the conjectures above, then any of the corresponding parts will be equal. Definition of Congruent Triangles: If two triangles are congruent, then all of their corresponding parts (sides and angles) are congruent. Can also be stated “Corresponding Parts of Congruent Triangles are Congruent” or CPCTC. When you write congruent triangles you must match the corresponding vertices. (To show how the match up.) So if you know that ∆ CAT ≅ ∆ DOG then you can say any of the following: ∠C ≅ ∠D ∠A ≅ ∠O ∠T ≅ ∠G CA ≅ DO AT ≅ OG CT ≅ DG O A T D C G by Def. of Congruent Triangles or by CPCTC. How to prove (show deductively) that parts are equal. P Example 1: ∠ PAC ≅ ∠ TAC and PA ≅ AT ∠ PCA ≅ ∠ TCA ? Know: Is ∠ PAC ≅ ∠ TAC given PA ≅ AT given A T . C CA ≅ AC Shared side ∆PAC ≅ ∆TAC SAS Congruence ∠ PCA ≅ ∠ TCA CPCTC or Def. Congruence S. Stirling Page 11 of 13 Ch 4 Note Sheets L2 Key 3 Name ___________________________ Example 2: Given: Z is midpoint of I P DI and EP . PI ≅ DE Prove: Z IZ = ZD Z is midpoint of DI and EP D Def. of Midpoint ∠PZI ≅ ∠EZD given PZ = ZE Vertical angles = Def. of Midpoint ∆ZIP ≅ ∆ZDE SAS Congruence PI ≅ DE CPCTC or Def. Congruence Example 3: Know: Is R E RE TC and TR CE . RE ≅ EC ? What is congruent? RE TC given ∠ RET ≅ ∠ ETC ||, so alt. int. ∠ s = T C TR CE given TE = TE ∠ RTE ≅ ∠ CET ||, so alt. int. ∠ s = Shared side ∆TRE ≅ ∆ECT and EC are NOT corresponding sides. So, probably not equal! RE ASA Congruence RE ≅ TC , RT ≅ EC or ∠R ≅ ∠C Try Example A and Example B on Ch 4 Worksheet page 13. CPCTC or Def. Congruence S. Stirling E Page 12 of 13 Ch 4 Note Sheets L2 Key 3 4.8 Proving Special Triangle Conjectures Name ___________________________ Equilateral/Equiangular Triangle Conjecture Every equilateral triangle is equiangular. Conversely, every equiangular triangle is equilateral. Vertex Angle Bisector Conjecture In an isosceles triangle, the bisector of the vertex angle is also the altitude and the median to the base and the perpendicular bisector of the base. Isosceles Triangle with vertex angle A. A Bisect vertex angle A, label the intersection X. Measure all parts of the resulting figure. List all possible congruent parts: m∠ C = m ∠ B base angles congruent m∠ CAX = m∠ BAX C angle bisector m∠ CXA = m∠ BXA = 90 AX is an altitude of ∆CAB B X CA = BA legs of isosceles triangle CX = XB AX is a median of ∆CAB AX is a perpendicular bisector of ∆CAB Isosceles Triangle with vertex angle A, and base angle C. Bisect base angle C, label the intersection X. Z Draw median CM . Draw perpendicular bisector MY . Draw altitude CZ . A Y X M B C Are any of these the same segment? NO S. Stirling Page 13 of 13