Unit 4 Congruent Triangles Homework 3 Isosceles And Equilateral Triangles

Presentation on theme: "Chapter 4 Congruent Triangles."— Presentation transcript:

1 Chapter 4Congruent Triangles

2 4. 1 Congruent Figures 4. 2 Triangle Congruence by SSS and SAS 4
4.1 Congruent Figures 4.2 Triangle Congruence by SSS and SAS 4.3 Triangle Congruence by ASA and AASStudents Will be able toRecognize Congruent figures and their Corresponding PartsProve two triangles are congruent using SSS and SASProve two triangles are congruent using ASA and AASMA.912.G.4.4 andMA.912.G.4.5 and MA.912.G.4.5MA.912.D.6.4 and MA.912.G.8.5

3 Notes forIn order to prove that two figures are congruent we need to make sure that all sides and all angles of one polygon are equal to all angles and sides of another polygon.In order to do this, we must first be able to decide which sides and angles on one polygon match with the sides and angles of another polygon… we call these matching pieces “corresponding parts”.If the polygons are congruent then the corresponding parts should be equal.

4 Notes forProving two polygons are congruent could take a lot of work. For example if we want to show that two triangles are congruent we would need to show that all 3 angles and all 3 sides of one triangle are equal to all 3 angles and all 3 sides of another triangle! This is 6 different pairs of congruent parts!!!We can use Logic and a few theorems to make some short cuts.

5 Notes forTheorem: 3rd Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle then the 3rd angles of both must be congruent.SSS Theorem: If 3 sides of one triangle are congruent to 3 sides of another triangle then the triangles are congruentSAS Theorem: If 2 sides of one triangle and the included angle of the triangle are congruent to 2 sides and the included angle of another triangle then the two triangles are congruent

6 Notes forASA Theorem: If 2 angles of one triangle and the included side of the triangle are congruent to 2 angles and the included side of another triangle then the two triangles are congruentAAS Theorem: If 2 Angles and 1 side of one triangle are congruent to 2 Angles and 1 side of another triangle then the triangles are congruentHypotenuse Leg Theorem: If two right triangles have congruent hypotenuses and another pair of equal sides then the two triangles are congruent

7 Classwork/Home Learning
Page 222 #10-19, 35, 36, 39, 40, 41Page 231 #11-14, 17, 24-26, 35-38Page 238 #13, 16-18, 25, 32-35

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9 4. 4: Corresponding Parts of Congruent Triangles are Congruent 4
4.4: Corresponding Parts of Congruent Triangles are Congruent 4.5: Isosceles and Equilateral TrianglesStudents will be able to:Use Triangle Congruence and CPCTS to prove that parts of two triangles are CongruentUse and apply the properties of isosceles and equilateral TrianglesMA.912.G.4.4 andMA.912.G.4.5 and MA.912.G.4.5MA.912.D.6.4 and MA.912.G.8.5

10 Notes for 4.4 and 4.5Once you know that two triangles are congruent based on SSS, SAS, ASA, AAS and HL you can now make conclusions about specific corresponding parts of triangles.If you know that two shapes are exactly the same size and exactly the same shape (ie: They are congruent) then it makes sense that specific angles and specific sides that are corresponding should be the same too… this is what CPCTC means.

11 Notes for 4.4 and 4.5With Isosceles and Equilateral Triangles we know even more information because we know that sides are across from equal anglesThis means that in an Isosceles triangle we have 2 equal sides and the two angles across from them are also equal.In an equilateral triangle, all sides and all angels are equal and all angles measure 60 degrees.

12 Classwork/HomeworkPage 247 #6, 11-13, 23-26Page 254 #6-9, 16-19, 37-40

13 4. 6: Congruence in Right Triangles 4
4.6: Congruence in Right Triangles 4.7: Congruence in Overlapping TrianglesStudents will be able toProve right triangles are congruent using the Hypotenuse Leg TheoremIdentify congruent overlapping triangles and use congruent triangle theorems to prove triangles are congruent.MA.912.G.4.4 andMA.912.G.4.5 and MA.912.G.4.5MA.912.D.6.4 and MA.912.G.8.5

14 Notes for 4.6 and 4.7Hypotenuse Leg Theorem: If two right triangles have congruent hypotenuses and another pair of equal sides then the two triangles are congruentWhen figures are overlapped it may be useful to separate the figures and identify the shared parts

15 Classwork / Home Learning
Page 262# 15, 29-31Page 268# 8-13, 17, 29-32

16 Office AidI am at a meeting in the main office… here’s your list of things to do:Come and see me in the office first!!!!At the back of the room there is a hole puncher and papers – please hole punchClean up my classroomThe baskets in the back have papers and folders please put the papers into the correct student folders. If the student has no folder, just leave out.

Unformatted text preview: Answers: Chapter 4 Triangle Congruence Lesson 4-8 lsosceles and Equilateral Triangles; 3. 118° 4. 49° 5 2f 6 90° 7 Y=5 8 X : 4 9 20 10. 50 13. 69° 14. 33° 15. 130° or 172‘” D 16. 31 17. Z = 92 18. 3’ = 48 19. 26 20. 20 21. It is given that AABC is isosc., E e E, P is the mdpt. of E, and Q is the mdpt. of lit—C. By the Mdpt. Formula, the coords. of P are (a, b) and the coords. of Q are (3a, b). By the Dist. Formula, PC = Q5 = «982 + b2 , so PC 2 QB by the def_ of z 5395,. Solutions: Chapter 4 Triangle Congruence Lesson 4-8 lsosceles and Equilateral Triangles; 3. Think: Use lsosc. A Thm., A A Sum Thm., and Vert. A Thm. mAB = mAA = 31° mAA + mAB + mAABC = 180 31+ 31+ mAABC: 130 mAABC: 118° mAECD: mAABC =118° 4. Think: Use Isosc. A Thm. and A A Sum Thrn. mAJ = mAK mAJ+ mAK+ mAL = 180 2mAK+ 82 = 130 2mAK= 98 mAK= 49° 5. Think: Use lsosc. A Thrn. mAX: mAY 5t— 13 = 31+ 3 2f=16 t=8 rnAX= 5t— 13 = 27° 6. Think: Use Isosc. A Thm. and A A Sum Thm. rnAB = mAC = 2x mAA + mAB+ mAC: 180 4x+ 2X+ 2X: 180 BX: 180 x: 22.5 mAA = 4x: 90° 7. Think: Use Equilat. A Thm. and A A Sum Thm. AREASEAT mAR+ mAS+ mAT: 180 12y+12y+ 12y: 180 36y: 180 y: 5 8. Think: Use Equilat. A Thm. and A A Sum Thm. AL E AM "=“ AN mAL + mAM+ mAN: 180 3(10x+ 20) = 180 30x: 120 X = 4 Solutions: Chapter 4 Triangle Congruence Lesson 4-8 lsosceles and Equilateral Triangles; 9. Think: Use Equiang. A T_hm- _ _ ABE 303 AC BC: AC 6y + 2 =— y + 23 7y: 21 y: 3 BC: 6y+ 2 = 6(3) + 2 = 20 10. Think: Use Equiang. A Thm. i-i—JE W; H—K HJ = JK 7t+ 15 =10t 15: Br 1: 5 JK: 101‘ = 10(5) = 50 13. Think: use lsosc. A Thm.. A 4 Sum Thm., and Vert. 4%. Thrn. sz = mAACB mAA + mAB+ mAACB = 180 96 + 2mAACB = 180 mAACB = 42° [TIADCE = mAACB = 42° sz = mAE mAD + mAE+ mADCE= 180 2mAE+ 42 = 180 mAE = 69° 14. Think: Use lsosc. A Thm. and A 4 Sum Thm. mAU = méS = 57° mASHU + mAS + sz = 180 mASHT+ mATFi'U + 57 + 57 = 180 2mATRU = 66 m4 THU = 33° 15. mAD=mAE x2=3x+10 x2— 3x— 10 =0 (x— 5)(x+2}=0 x=50r—2 sz+m4E+m1F=180 x2+3x+10+m2F=180 m2F=180— 500r180— 8 =130°0r172° Solutions: Chapter 4 Triangle Congruence Lesson 4-8 lsosceles and Equilateral Triangles; 16. Think: Use lsosc. A Thm. and A A Sum Thm. mAA = mAB=(6y+1)° mAA + mAB+ mAC: 180 2(6y+ 1) + 21y+ 8 = 180 33y: 165 Y= 5° mAA: 6y+1= 31° 17. Think: Use Equilat. A Thm. and A A Sum Thm. AFE AG 2 AH mAF+ mAG + mAH = 180 3(§+14)=180 2:92 18. Think: Use Equilat. A Thm. and A A Sum Thm. AL 5 AM 5 AN mAL + mAi'i/ir + mAN: 180 3(1.5y— 12): 180 y— 8 = 40 y = 48 19. Think: use Equiang. A Thm. 8—0 a 0—D 2 @ BC = CD 3 5 —x 2=—x6 2+ 4+ 6x+8=5x+24 x=16 3 BC:— 2 2X+ = g(1:3) + 2 = 2s 2 20. Think: use Equiang. A Thrn. We Y—ZE X—Z XY: X2 5 2x=—x— 2 5 1 5—2x X=10 XZ=XY =2x =2(10)=20 Solutions: Chapter 4 Triangle Congruence Lesson 4-8 lsosceles and Equilateral Triangles; 21. Proof: It is given that AABC is isoscLE "=“ ET P is mdpt. of E, and Q is mdpt. of AC. By Mdpt. Formula, coords. of P are (a, b), and coords. of O are (33, b). By Dist. Formula, PC: 08: 9a2+b2, so We Q—Bby def. ofs. ...
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