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Name: ____________________________ Period: ________ 4.1 Tricky Triangles 1) Triangle Sum Conjecture: The sum of the measures of the angles in every triangle is ___________. 2) Determine the missing angle in each diagram a a = ______ b b= ______ 570 360 c= ______ c d 380 120 e g 1240 f 380 h i 1380 750 d= ______ 450 e= ______ h= ______ f= ______ i= ______ g= ______ j = ______ 3) Snyder Rd k m Prince Rd n City Designers planned many Tucson streets at right angles. Houghton Rd is perpendicular to both Snyder Rd and Prince Rd. Melpomene is perpendicular to Prince and Snyder Roads. a) What can you prove about Snyder Rd and Prince road according to the given information? Explain. b) The angle Prince Rd makes with Catalina Highway is a 440 angle. Find the unknown angles k= ______ m= ______ n= ______ line m 4) 2 3 4 1 line n a) Given: m || n Prove: The sum of the angles of ΞABC is 1800 5 Name: _______________________________ Period______4.2 Classifying Triangles/Exterior Angle Theorem 1. Use the diagram indicated to prove the exterior angle theorem. Your givens come from the diagram. The conclusion of your proof should say that πβ 1 = πβ 2 + πβ 3 Statements Reasons Questions 2-5: Classify each triangle by the angles, and sides. Assume that the only given information are the congruence marks, and angle indicators. Sketch an example of the type of triangle described. Mark the triangle to indicate what information is known. If no triangle can be drawn, write βnot possible.β 6) acute isosceles 7) right scalene 8) right isosceles 9) right equilateral 10) acute scalene 11) obtuse scalene 12) right obtuse 13) equilateral 12) acute equilateral Questions 13- 18 Find the measure of each indicated angle (?). 13) 14) 15) 16) 17) 18) Name: _________________________ Period: ________ For 1 & 2, shade a different triangle in each image. 1) 2) 4.3 Overlapping Triangles For 3 & 4, copy the diagram as many times as needed to shade all the different triangles in each image. 3) 4) 5) Draw your own shape that is made up of at least 4 overlapping triangles. Then recopy your design as many times as needed to shade all the different triangles in your image. Name: ___________________________ Period: _________ 4.4 Triangle Inequality Problem Set π₯+π¦ >π§ Three numbers are given as the side lengths of a triangle. Use the triangle inequality to determine whether such a triangle can exist 1) 7, 5, 4 2) 3, 6, 2 3) 5, 2, 4 4) 8, 2, 8 5) 9, 6, 5 6) 5, 8, 4 7) 4, 7, 8 8) 11, 12, 9 9) 3, 10, 8 10) 1, 13, 13 11) 2, 15, 16 12) 10, 18, 10 Two side lengths of a triangle are given, determine the range of values that are possible for the 3rd side. 13) 9, 5 14) 5, 8 15) 6, 10 16) 6, 9 17) 11, 8 18) 14, 11 Use your compass and a straight edge to draw a triangle given each set of measurements (label): 19) 7cm, 5cm, 4 cm 20) 8 cm, 8cm, 2cm 21) 3 cm, 6 cm, 2 cm 22) 6 cm, 6 cm, 6 cm Name: _________________________ Period: ________ 4.6 SSS and SAS SSS: Three sides of one triangle are congruent to the corresponding sides of another triangle SAS: Two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle Identify which property will prove these triangles congruent, SSS or SAS. If neither method works say βneitherβ 1. 2. 3. 4. 5. 6. 7. 8. Sate what additional information is required in order to know that the triangles are congruent FOR THE GIVEN REASON. Remember ORDER matters. Write the triangle congruency. Example: SAS Since one side is marked π»πΌ β πΎπΌ and from the diagram there is a pair of vertical angles so β π»πΌπ½ β β πΎπΌπ, so we need One more piece of needed information: ___________________ To prove βπ»πΌπ½ β βπΎπΌπ ππ¦ ππ΄π 9. prove by: SSS One more piece of needed information: ___________________ Ξ KMH β Ξ ________ 10. C A prove by: SAS T F One more piece of needed information: ___________________ Ξ CAT β Ξ ________ 11. prove by: SAS One more piece of needed information: ___________________ Ξ MKL β Ξ ________ 12. prove by: SSS One more piece of needed information: ___________________ Ξ XZY β Ξ ________ 4.7 Geometry Name___________________________________ ©l t2_0e1z5z QK[uztjas zSto_fpt[wHaYr^eX KLMLYC\.u [ uAhlTlN orsixgYhztbss PrQelsFewrivjeEdX. ASA and AAS Congruence Period____ State if the two triangles are congruent. If they are, state how you know. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) Worksheet by Kuta Software LLC ©a O2W0G1j5d aKZu\tcaW cS\oWfHtowXaorHeg LLjLpCU.n e ^ADlYl[ crdikgqhAtlsA arYeqsNeErGvveOdg.w i PMaa`dNeA Kwziytchu jIOnrfCignYivtZeV lGeeBoqmyeytJrSyX. 11. Use the figure to determine which (if any) triangles are congruent to one another. 12. Determine if . Explain your reasoning. 13. In the figure, is an equilateral triangle, does that mean is also equilateral? Explain your reasoning. Name: ______________________________Period: _______ 4.8 Congruence Shortcuts that Fail 1. Demonstrate (By construction) that AAA doesnβt produce two congruent triangles. 2. Using an angle of 45 degrees, and side lengths of 4cm, and 3.5cm, show that SSA will produce two different triangles which are not congruent. The diagram on the right shows the same setup except with an angle of 30 degrees and side lengths 10 and 6. Name two short cuts that donβt work: _____________ and _______________ 1. Construct one triangle that has lengths 4cm, 5cm, 7cm and another that has lengths 7cm, 7cm, and 4 cm 2. Construct a triangle that has leg lengths 5cm, 6cm, and an included angle of 50 degrees. 3. Construct a triangle that has leg lengths 4cm, 4cm and an included angle of 60 degrees. What kind of triangle is this? Name: _________________________ Period: ________ 4.9 Proving Triangles Congruent For each problem give the correct naming order of the congruent triangles. Write that name in order on the lines for the problem number (see box at bottom). Also, indicate which postulate or theorem is being used. 1. B I C A W J R H G A 7. A C B B P K οABC ο ο _______ by _________ E 3. C T G A B N οABC ο ο _______ by _________ T S A D 6. A E B C H Y οABC ο ο _______ by _________ οABC ο ο _______ by _________ 8. 9. F S E D L G A 5. H οABC ο ο _______ by _________ 10. C C C E I H S οABC ο ο _______ by _________ οGHJ ο ο _______ by _________ B A B R οABC ο ο _______ by _________ 4. 2. N A B Y C οABC ο ο _______ by _________ H οJKL ο ο _______ by _________ E D A T L K οDEF ο ο _______ by _________ 11. J A 12. O M K N S A οMNO ο ο _______ by _________ ___ ___ ___ ___ ___ _O_ ___ ___ _N_ ___ ___ ___ _S_ ___ ___ _E_ ___ _I_ ___ ___ ___ ___ ___ _T_ ___ 4 4 4 8 8 8 12 12 12 2 2 2 5 5 5 9 9 9 6 ___ ___ ___ _E_ _E_ ___ ___ ___ _O_ ___ ___ _N_ ___ _U_ ___ ___ ___ ___ _T_ ___ _E_ ___ ___ _I_ ___ . 6 6 10 10 10 1 1 1 3 3 3 7 7 7 11 11 11 (When you are done with the puzzle, there are: 3 SAS, 5 AAS, 2 ASA, and 2 SSS instances.) 13) A) Translation B) Vertical Reflection C) Rotation D) Horizontal Reflection 14) There are five different ways to find triangles are congruent: SSS, SAS, ASA, AAS and HL. For each pair of triangles, select the correct rule. Indicate if there isnβt enough information. 15) a) Mark the diagram with the given information. b) Look for any other given information that could help show that the two triangles are congruent. Do they overlap anywhere? Share any side or any angle? Mark it in the diagram. c) You should have enough information to prove the triangles are congruent. Fill in the proof. Statements Reasons 1) 1) Given 2) 2) Given 3) 3) 4) 4) d) What do you think is true about β π΄ πππ β πΆ? Explain: Name: _________________________ Period: ________ 4.10 Proving Isosceles Conjectures In this space, draw a large isosceles triangle. Use tools appropriately, do not freehand. Mark the two congruent sides of the triangle. Precision is essential. 1) Label the vertices RED with R begin the vertex angle (included by the two congruent sides). 2) Carefully construct the angle bisector of β R. Label point Q on Μ Μ Μ Μ π·πΈ where the angle bisector ray Μ Μ Μ Μ . Mark the congruent angles. intersects π·πΈ 3) Complete the proof to show that Ξ ERQ β Ξ DRQ Statement 1. ___________________________ 2. β ERQ β β DRQ 3. ___________________________ 4. Ξ ERQ β Ξ DRQ Reason 1. Given 2. ___________________________ 3. ___________________________ 4. SAS 4) Using CPCTC, what other parts of the triangle are congruent? ________ β ________ and ________ β ________ 5) On your diagram, measure πΈπ πππ π·π, what can you say about point Q? 6) On your diagram, measure β πΈππ πππ β π·ππ , what can you say about these angles? 7) Describe a series of transformations that would map Ξ ERQ onto Ξ DRQ. Be specific. The isosceles triangle theorems are frequently abbreviated as 1. ΞXYZ is an isosceles triangle with perimeter 48 cm If XY = 18 Find XW _______ Ξ MPQ is an isosceles triangle. πβ πππ = 45° ππππ π‘βπ πβ πππ 2. M N P 3. Q 5 π₯ 2 2x β5 Find AB 2x (3y β 5)0 Solve for y 4. 400 5. (5x + 15)0 solve for x Name: _________________________ Period: ________ 4.11 CPCTC worksheet MARK THE DIAGRAMS WITH THE GIVEN INFORMATION! #1: οHEY is congruent to οMAN by ______. What other parts of the triangles are congruent by CPCTC? ______ ο ______ ______ ο ______ H ______ ο ______ M A Y E N #2: οCAT ο ______, by _____ C R THEREFORE: ______ ο ______, by CPCTC ______ ο ______, by CPCTC ______ ο ______, by CPCTC A T P #3: C Given: AC ο AR and ο1 ο ο2 Prove: ο3 ο ο4 R 3 A 1 L Proof: 1. 2. 3. 4. 5. AC ο AR ____________ οCAL ο οRAS οLCA ο οSRA ο3 ο ο4 1. 2. 3. 4. 5. _______________ Given ________________ ________________ ________________ 4 2 S MARK THE DIAGRAMS WITH THE GIVEN INFORMATION! #4: L Given: Prove: M οNLM ο οLNO and οOLN ο οMNL οM ο οO O N Proof: 1. 2. 3. 4. 5. οNLM ο οLNO 1. 2. 3. 4. 5. _________________ _________________ οLMN ο ο______ _________________ _________________ Given Reflexive Property of ο _________________ _________________ C #5 Given: AC ο BC and AX ο BX Prove: ο1 ο ο2 Proof: 1. __________________________ 2. __________________________ 3. οAXC ο _______ 4. ________________ 1. 2. 3. 4. 1 2 3 4 A X B Given Reflexive Prop. of Congruence ____________ ____________ #6 W Z Given: ο1 ο ο2 and ο3 ο ο4 Prove: XY ο ZW Proof: 1. __________________________ 2. XZ ο XZ 3. οXWZ ο _______ 4. ________________ 2 4 1 X 1. 2. 3. 4. 3 Given ________________ ________________ ________________ Y Name: ______________________________Period: __________ 2 3. 4. 4.12 Using Proof Blocks 5. 6. 7. 8. Name: ______________________________ Period: __________ 1. Given: π΅πΆ β π·πΈ & β π΅ β β πΈ Prove: π΄πΆ β π΄π· 2. Given: β π· β β π, β πΈ β β π, πΈπ· β ππ Prove: π·πΉ β ππ 3. Given that β πΊ β β πΎ, and the information in the diagram, prove π»πΌ β π½πΏ 4.13Proof blocks & CPCTC 4. Using the information in the diagram prove that β π β β π. Μ 5. Given that πΊπ» β₯ π½πΌ, I is the midpoint of π»πΎ and Μ Μ Μ Μ πΊπ» β π½πΌ Prove: β πΊ β β π½ 6. Given that ππ β₯ ππ, and the information in the diagram to prove that ππ is the angle bisector of β πππ Name: _________________________________Period: _________ 1) Classify βABC by its angles and its side lengths ___________ 4.14 Quiz 4 Review _________________ 2) Classify each triangle by its side length βABD ______________ and βADC ___________________ 3) While surveying a triangular plot of land, a surveyor finds πβ π = 430 . The measure of β π ππ is twice that of β π ππ. What is the πβ π ? Given βXYZ β βJKL, identify the congruent corresponding parts Μ β _____ 4) π½πΏ 5) β π β ____ 6) β πΏ β ____ Μ Μ Μ Μ Μ Μ Μ Μ πππ ππ 8) Given: T is the midpoint of both ππ Prove: βPTS β βRTQ 9 & 10: Find the Measures of each angle Statements 7) Μ Μ Μ Μ ππ β ___ Reasons 11. Find the measure of β π΄ππ 13. 12. Find the measure of β CBD 14. 15. Look at the diagrams below and determine which (if any) triangle congruence theorems you can use to prove two triangles are congruent. Circle the ones that work and label them with SSS, SAS, AAS, ASA or HL.