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Ch. 4 Note Sheet L1 Name:_____________________________ Triangulation is a procedure used by surveyors to locate Read top of pg 200. positions by using a network of triangles with their angles and distance measurements. Rigidity is a property of triangles, meaning they retain their shape and can’t be shifted, unlike quadrilaterals . 1 1 3 2 1 1 2 3 2 3 m∠1 + m∠2 + m∠3 = 180 Triangle Sum Conjecture: The sum of the measures of the angles in every triangle is 180 A.Simons Page 1 of 18 . Ch. 4 Note Sheet L1 Name:_____________________________ Numeric Example: Find x. Find y. F B 101 56 A C x x x + x + y = 180 2x + y = 180 y = 180 – 2x E x y D 101 + 56 + x = 180 x = 180 – 157 = 23 Proof: (What definitions and conjectures do we already know?) Given: Show: HJJG ΔABC with auxiliary line EC & AB . m∠ 2 + m∠ 4 + m∠5 = 180° m∠1 + m∠ 2 + m∠3 = 180° Linear pair conjecture HJJG HJJG AC and CB form transversals between parallel lines EC and AB m∠1 = m∠ 4 and m∠3 = m∠5 because AIA 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°. Measure: ∠A , ∠B , ∠E and ∠F , what can we conclude about ∠C and ∠D ? Explain! F B A C E D 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. A.Simons Page 2 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ 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. B Proof: (What definitions and coonjectures do we already know?) Given: Show: m∠A = m∠E and m∠B = m∠F m∠ C = m∠ D C A F m∠A + m∠B + m∠C = 180° and m∠E + m∠F + m∠D = 180° by the triangle sum conjecture. Since they both equal 180, m∠A + m∠B + m∠C = m∠ E + m∠F + m∠ D Now subtract equal measures m∠ A = m∠ E and m∠ B = m∠ F . m∠C = m∠D Therefore, the third angles are always congruent. A.Simons E Page 3 of 18 D Ch. 4 Note Sheet L1 Name:_____________________________ Angle between the two legs. VERTEX ANGLE One of the congruent sides. Pair of angles whose vertices are the endpoints of the base. LEG LEG Side that is not a leg. BASE BASE ANGLE BASE ANGLE Read pg 206. Do investigation 1 pg 207. Isosceles Triangle Conjecture: If a triangle is isosceles, then its base angles are congruent. Given +VBS with VB = VS , then ∠B ≅ ∠S . OR B If VB = VS , then ∠B ≅ ∠S V S **What can we say about equilateral triangles? Since they are a special case of isosceles triangle, any property that applies to isosceles triangles also applies to equilateral triangles. Do investigation 2 pg 208. Use a protractor. Converse: If a triangle has two congruent angles, then it is an isosceles triangle. S B If <B ≈ <S, then ΔVBS is isosceles OR If <B ≈ <S, then VB ≈ VS. V A.Simons Page 4 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ QUICK VOCAB REVIEW! ∠A is opposite BC and AC is opposite ∠B ∠A is between BA and AC (angles between sides) BC is between ∠B and ∠C (sides between angles) B C A INCLUDED ANGLE: angle formed between two consecutive sides of a polygon. INCLUDED SIDE: side of a polygon between two consecutive angles. Read pg 215. Do investigation 1 pg 216. Triangle Inequality Conjecture: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Construct a triangle with sides measuring 5cm, 4cm, and 2cm. Is this possible? Construct a triangle with sides measuring 5cm, 3cm, and 2cm. Is this possible? Since 2+4> 5, it makes a triangle! Since 2+5 =5, it makes a segment not a triangle! Do investigation 2 pg 217. Side-Angle Inequality Conjecture: (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. LIKEWISE! SHORTEST SIDE…SMALLEST ANGLE A.Simons LONGEST SIDE…LARGEST ANGLE If one side is the shortest side, then the angle opposite the shortest side is the smallest side. Page 5 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ Which is the largest angle? Which is the smallest angle? Why? D Which is the largest side? Why? Q 90 23 K <K largest, its opp. The longest side JI. <I smallest, its opp The shortest side JK. J < D largest, so AQ largest in ΔADQ, But m<QAU = 100 Which makes UQ the largest in ΔQAU and AQ < UQ Overall, UQ is the longest. 5.34 2.48 I 6.3 54 A 57 U Do investigation 3 pg 217-218. Triangle Exterior Angle Conjecture: 6.3 The measure of an exterior angle of a triangle is equal to the sum 57 of the measures of the remote interior angle. U B Remote Interior Angles: Interior angles of a triangle that do not share a vertex with a given exterior angle. Adjacent Interior Angles: b <A and <B a A <C and <X c Angle of a polygon that forms a linear pair with a given exterior angle of a polygon x C <X and <C D Exterior Angles: Angle that forms a linear pair with one of the interior angles of the polygon. Given ABC above. If a=50 and b=60, what is the measurement of <BCD? Explain! Given ABC above. If x = 80 and b = 30, what is the measurement of <A? Explain! 50+60=110 The exterior angle equals the sum of the two remote interior angles. 80-30=50 the exterior angle equals the sum of the two remote interior angles. A.Simons Page 6 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ Read top of pg 221. Complete Triangle Congruence Shortcut Investigation (pg1-3). 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 D A O C C T T G O ΔCAT ≅ Δ_______ ΔCAT ≅ Δ_______ By SSS Congruence Conjecture. By SSS Congruence Conjecture. 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. A I I P Z C T Z P D E ΔPIZ ≅ Δ_______ ΔZED ≅ Δ_______ By SAS Congruence Conjecture. By SAS Congruence Conjecture. A.Simons Page 7 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ 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. D O O I C G Z P P A ΔDOG ≅ Δ_______ ΔCOP ≅ Δ_______ By ASA Congruence Conjecture. By ASA Congruence Conjecture. SAA or AAS Congruence Conjecture: If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another triangle, then the triangles are congruent. D O P I A G Z P T C ΔPIZ ≅ Δ_______ ΔCAP ≅ Δ_______ By SAS Congruence Conjecture. By SAS Congruence Conjecture. A.Simons Page 8 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ Hypotenuse Leg Congruence Conjecture: (SPECIAL CASE) If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent. O D O I C G Z P P A ΔDOG ≅ Δ_______ ΔCOP ≅ Δ_______ By HL Congruence Conjecture. By HL Congruence Conjecture. SSA or ASS? If two sides and the no-included angle of one triangle are congruent to two sides and the non-included angle of another triangle, then the triangle are NOT necessarily congruent. A C Draw a counterexample. I T Z I P Z P ΔZIP ≅ Δ_______ 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 Z P ΔZIP ≅ Δ_______ By NOT necessarily congruent. A.Simons Page 9 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ To see if you have congruent triangles, you will be checking for a SSS, SAS, ASA or AAS marked on the matching pair of triangles that you be given (as shown above). Sometimes the parts will be marked equal in the diagram. That’s the easy stuff. Other times, you will be given information that you will “translate” into equal sides and angles in order to get your congruence. You will need to deduce this information from definitions or conjectures that you already know to be true. Complete the following to help you review these statements. Remember, to mark your diagrams with the equal parts. Also never assume things are congruent! You must have a definitions or conjecture to back you up!! 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 , then VB ≅ VS ΔVBS isosceles then VB ≅ VS . B 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 . When two triangles share the exact same segment, you get a pair of equal segments. ΔBET and ΔWTE share ET ≅ TE A.Simons 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 S B If Def. Midpoint “Same Segment” V CM bisects AB , then AM ≅ MB . If B M A B Def. Median If a segment is a median, then it connects the vertex to the midpoint of the opposite side. If AM is median in ΔABC , then BM ≅ MC . Page 10 of 18 M A C Ch. 4 Note Sheet L1 Name:_____________________________ ANGLES! Isosceles Triangle Conjecture If a triangle is isosceles, then its base angles are congruent. V ΔVBS isosceles or VB ≅ VS , then ∠B ≅ ∠S . If two angles are vertical, then they are congruent. S If V E R C If ∠ VET and ∠ CER are vertical, then ∠ VET ≅ ∠ CER . B Definition of Angle Bisector If you have an angle bisector, then the ray cuts the angle into two equal angles. A Def. perpendicular lines B JJJG BD bisects ∠ABC then ∠ ABD ≅ ∠ DBC . C If two lines are perpendicular, then they intersect to form equal 90° angles. D C If B M D A CD ⊥ AB , then m∠CMA = m∠CMB = 90° . 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 Vertical Angle Conjecture Corresponding Angles Conjecture E If two parallel lines are cut by a transversal, then corresponding angles are congruent. T I A RA is an altitude of ΔTRI then m∠ RAT = m∠ RAI = 90° . If B A HJJG HJJG If AB & DC , then ∠ EFA ≅ ∠ FGD . F C G D Alternate Interior Angles Conjecture “Same Angle” When two triangles share the exact same angle, you get a pair of equal angles. If two parallel lines are cut by a transversal, then alternate interior angles are congruent. I A R G T N ΔTIA and ΔNIR share ∠TIA ≅ ∠RIN . HJJG HJJG If LK & NJ , then ∠KHM ≅ ∠HMN Def. perpendicular bisector ANGLES AND SIDES! Complete the Triangle Congruence Shortcut A.Simons J . N M B M CD is the perp. bisector of AB , then AM ≅ MB and ∠ CMA = ∠ CMB = 90° . If K L C A line (or part of a line) that passes through the midpoint of a segment and is perpendicular to the segment. H A Page 11 of 18 D Ch. 4 Note Sheet L1 Name:_____________________________ Once you can PROVE that two triangles are congruent, by using the conjectures above, then ANY of the corresponding parts will be equal! Congruent Triangles: If two triangles are congruent, then all of their corresponding parts (sides and angles) are congruent. Also called Corresponding Parts of Congruent Triangles are Congruent Corresponding triangles must match corresponding vertices! (Shows how they match-up!) So if you know that following: ΔCAT ≅ ΔDOG then you can say any of the ∠C ≅ ∠D ∠A ≅ ∠O ∠T ≅ ∠G CA ≅ DO AT ≅ OG CT ≅ DG CPCTC How to prove that parts are equal! (Show Deductively) P 1 ∠PAC ≅ ∠TAC and PA ≅ AT . ∠PCA ≅ ∠TCA ? Know: Is The triangles share a side, so So… CA = CA A T of course. C ΔPAC ≅ ΔTAC by SAS Congruence and ∠ PCA ≅ ∠ TCA because CPCTC or Corresponding Parts of Congruent Triangles are Congruent. 2 Given: Z is midpoint of Prove: PI ≅ DE ? DI and EP . I P Z DI and EP was given, so IZ = ZD and PZ = ZE by Definition of Midpoint D ∠PZI ≅ ∠EZD by the Vertical Angles Conjecture (or vertical angles are congruent.) Therefore, Δ ZIP ≅ Δ ZDE by SAS Congruence and PI ≅ DE because CPCTC or Corresponding Parts of Congruent Triangles are Congruent. Z is midpoint of A.Simons E Page 12 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ TRICKY! Know: Is R RE & TC and TR & CE . RE ≅ EC ? E What is congruent? T C RE & TC given so ∠RET ≅ ∠ETC and TR & CE given so ∠RTE ≅ ∠CET because parallel lines make equal alternate interior angles. TE = TE the triangles share a side, ΔTRE ≅ ΔECT by ASA Congruence. So any of the corresponding sides are congruent. RE and EC are NOT corresponding sides. Note: you could say any of the following though: So, probably not equal. RE ≅ TC , RT ≅ EC or ∠R ≅ ∠C . TRICKIER! Start by un-overlapping possible pairs of congruent triangles: LE = EP and LV = OP Prove: ∠ ALP ≅ ∠ APL A Given: V O E L P LE = EP so ∠ELP ≅ ∠EPL If isosceles, then base angles are equal. LP = PL Same segment LV = OP Given information ∠LOP ≅ ∠PVL by SAS Congruence ∠ALP ≅ ∠APL CPCTC Try Proofs in Worksheet Packet! A.Simons Page 13 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ 4.7 Page 238 Example B Flowchart Proof Given: Prove: EC ≅ AC and ER ≅ AR ∠A ≅ ∠E E R C EC ≅ AC Given ER ≅ AR Given A ΔREC ≅ ΔRAC ∠A ≅ ∠E SSS Cong. Conj. CPCTC RC ≅ RC Same segment 4.7 Page 239 Top. Explain why the angle bisector construction works. Flowchart Proof Given: Prove: ∠ABC with BA ≅ BC and CD ≅ AD BD is the angle bisector of ∠ABC . BA ≅ BC Given CD ≅ AD Given ΔBAD ≅ ΔBCD ∠1 ≅ ∠2 SSS Cong. Conj. CPCTC BD ≅ BD Same segment BD is the angle bisector of ∠ABC . Def. of angle bisector A.Simons Page 14 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ NEED EQUAL SEGMENTS TO GET… B Def. Isosceles Triangle Def. Median If a triangle’s legs are congruent, then it is isosceles. If a segment connects the vertex to the midpoint of the opposite side, A then it is a median. V VB ≅ VS then ΔVBS isosceles. S then AM is median in ΔABC . B Def. Midpoint Def. segment bisector C If a line (or part of a line) passes through the midpoint of the segment, then it is a bisector. M S If a point divides the segment into two equal segments, then it is a midpoint. M If SM ≅ MG , then M is G the midpoint of SG . NEED EQUAL ANGLES TO GET…. Converse of the Isosceles Triangle Conj. If a triangle has two congruent angles, then it is an isosceles triangle. V S B A D B ∠ABD ≅ ∠DBC , JJJG then BD bisects ∠ ABC . C Def. Altitude C If two lines intersect to form equal adjacent angles, then they are perpendicular. B M D A Corresponding Angles Conjecture E If two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel. A ∠EFA ≅ ∠FGD , HJJG HJJG then AB & DC . B F C If G D Alternate Interior Angles Conjecture If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel. R m∠RAT = m∠RAI = 90 A °, then RA is an altitude of ΔTRI A.Simons A m∠CMA = m∠CMB , then CD ⊥ AB . If If a segment goes from a vertex perpendicular to the line that contains the opposite side, then it is an altitude. B If Definition of Angle Bisector If AM ≅ MB , then CM bisects AB . If Def. perpendicular lines If a ray cuts the angle into two equal angles, then you have an angle bisector. C If BM ≅ MC , If If ∠ B ≅ ∠ S , then ΔVBS isosceles. M T I . If ∠ KHM ≅ ∠ HMN HJJG HJJG then LK & NJ . H K L , J N M Page 15 of 18 Ch. 4 Note Sheet L1 Need Equal Segments and Angles to get…. Name:_____________________________ Def. perpendicular bisector If a line (or part of a line) that passes through the midpoint of a segment and is perpendicular to the segment, C then it is the perpendicular bisector. AM ≅ MB and ∠CMA = 90° , then CD is the perp. bisector of AB . If B M A D SUMMARY Basic Procedure for Proofs “parts” refers to sides and/or angles. 1. Get equal parts by using given info. and known definitions and conjectures. 2. State the triangles are congruent by SSS, SAS, ASA or AAS. 3. Use CPCTC to get more equal parts. 4. Connect that info. to what you were trying to prove . Hints [if you get stuck]: Mark the diagram with what you have stated as congruent in your proof. ( If given M is the midpoint of midpoint before marking the diagram!) AB , convert it to AM = MB by def. of Look at the diagram to find equal parts. Brainstorm and then apply previous conjectures and definitions. Work (or think) backwards! Draw overlapping triangles separately. Re-draw figures without all of the “extra segments” in there. Draw an auxillary line. Break a problem into smaller parts. A.Simons Page 16 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ Investigate. What can we conclude about Isocceles Triangles and the measurement of it’s altitude, median and angle bisector from one base angle? Isosceles Triangle with vertex angle A, Isosceles Triangle with vertex angle B. Median BM AM = 1.64 cm MC = 1.64 cm A Altitude BE m∠ABE = 90 . D Median BM Altitude BL Angle Bisector BS AB = 5.43 cm AC = 5.43 cm CB = 6.96 cm A VB = VS B C C Angle Bisector BD m∠ABD = 27D m∠DBC = 27D B A L A E,M,D C S M B B C 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. Equilateral/Equiangular Triangle Conjecture: Every equilateral triangle is equiangular. Conversely, every equiangular triangle is equilateral. A.Simons Page 17 of 18 Ch. 4 Note Sheet L1 Name:_____________________________ Look at the following isosceles triangles, what can we conclude about segments from the base angles? A A N M P A S T L C C C B B B Medians MC = BN Angle Bisectors PC = BS Altitudes BT = LC Medians to the Congruent Sides Theorem: In an isosceles triangle, the medians to the congruent sides are congruent. Angle Bisectors to the Congruent Sides Theorem: In an isosceles triangle, the angle bisectors to the congruent sides are congruent. Altitudes to the Congruent Sides Theorem: In an isosceles triangle, the altitudes to the congruent sides are congruent. A.Simons Page 18 of 18