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Lesson 4.1 Classifying Triangles

Lesson 4.1 Classifying Triangles. Today, you will learn to… * classify triangles by their sides and angles * find measures in triangles.  ABC. A. B. C. Equilateral Triangle. 3 congruent sides. Isosceles Triangle. 2 congruent sides. Scalene Triangle. no congruent sides.

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Lesson 4.1 Classifying Triangles

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  1. Lesson 4.1Classifying Triangles Today, you will learn to… * classify triangles by their sides and angles * find measures in triangles

  2.  ABC A B C

  3. Equilateral Triangle 3 congruent sides

  4. Isosceles Triangle 2 congruent sides

  5. Scalene Triangle no congruent sides

  6. Equiangular Triangle 3 congruent angles

  7. Acute Triangle 70° 50° 60° 3 acute angles

  8. Obtuse Triangle 95° 25° 60° 1 obtuse angle

  9. Right Triangle 30° 60° 1 right angle

  10. We classify triangles by their sides and angles. ANGLES SIDES Equiangular Acute Obtuse Right Equilateral Isosceles Scalene

  11. A CB AC AB B _____ is opposite A. _____ is opposite B. _____ is opposite C. C Identify the side opposite the given angle.

  12. Leg? hypotenuse ? leg leg

  13. Leg? base leg ? leg

  14. Theorem 4.1Triangle Sum TheoremThe sum of the measures of the interior angles of a triangle is ________ 180°

  15. 1. Find mX. Y mX = 44˚ 61º 75º X Z

  16. 1 2 If the sum of the interior angles is 180º, what do you know about 1 and 2?

  17. Corollary to the Triangle Sum TheoremThe acute angles of a right triangle are _________________. complementary

  18. 2. Find mF. D 54˚ + mF = 90˚ 54˚ mF = 36˚ F E

  19. 3. Find m1 and m2. m1 = 60˚ m2 = 120˚ 50º exterior angle 2 70º 1 adjacent angles

  20. 1 4 2 3 Theorem 4.2Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 nonadjacent interior angles. m1 + m2 = m4

  21. 1 Sum of nonadjacent interior s = ext.  4 2 3 m1 + m2 m1 + m2 + m3 = 180˚ m4 + m3 = m4 180˚ m1 + m2 = m4

  22. 4. Find mE. mE + 60˚ = 110˚ E mE = 50˚ 110˚ 60˚ D F G

  23. 5. Find x. x + 60 = 3x + 10 E x = 25 x˚ (3x + 10)˚ 60˚ D F G

  24. Lesson 4.2Congruence and Triangles Today, you will learn to… * identify congruent figures and corresponding parts * prove that 2 triangles are congruent

  25. Def. of Congruent Figures Figures are congruent if and only if all pairs of corresponding angles and sides are congruent.

  26. Ð @ XY A Ð @ B YZ Ð @ C XZ Statement of Congruence Δ ABC Δ XYZ vertices are written in corresponding order  X  Y  Z

  27. B C A 1. Mark ΔDEF to show that Δ ABC  Δ DEF. E F D

  28. B D E 35˚ 8.2 5.7 10 A C F 2. Find all missing measures. 8.2 ? B ? ? 35˚ 10 ? ? 55˚ ? 55˚ 5.7 ?

  29. K A J B 93˚ L (3y)˚ 85˚ (4x – 3) cm D H 75˚ 9 cm C 3. In the diagram, ABCD  KJHL. Find x and y. 3y = 75˚ 9 4x-3 = y = 25 x = 3

  30. A 93˚ 30˚ B E C F D (4x + 15)˚ 4.ΔABC ΔDEF. Find x. 4x + 15 = 57 x = 10.5 57

  31. D C O G A T Theorem 4.3Third Angles TheoremIf 2 angles of one triangle are congruent to 2 angles of another triangle, then… 60˚ the third angles are also congruent. ? 70˚ 60˚ 70˚ ?

  32. 5. Decide if the triangles are congruent. Justify your reasoning. H Vertical Angles Theorem E 58° G 58° J F Third Angles Theorem ΔEFG  Δ______ H G J

  33. 1 2 3 4 5 6 W X 6. M Y Z 1)WX  YZ , WX | | YZ, M is the midpoint of WY and XZ 2) 3) 4) 5) ΔWXM  ΔYZM 1) Given and 2  5 1  6 2) Alt. Int. s Theorem 3  4 3) Vertical Angles Th. 4) Def. of midpoint WM  MY and ZM  MX 5) Def. of  figures

  34. 7. Identify any figures you can prove congruent & write a congruence statement. B A C D C ACD   A B Alt. Int.  Th. Reflexive Property Third Angle Th.

  35. Theorem 4.4Properties of Congruent Triangles ABC  ABC Reflexive If ABC  XYZ, then XYZ  ABC Symmetric Transitive If ABC  XYZ and XYZ  MNO then ABC  MNO

  36. Lesson 4.3Proving Triangles are Congruent Today, you will learn to… * prove that triangles are congruent * use congruence postulates to solve problems

  37. SSS Experiment Using 3 segments, can you ONLY create 2 triangles that are congruent?

  38. B C A Y X Z Side-Side-Side Congruence Postulate If Side AB XY Side AC  XZ Side BC  YZ, then ΔABC  ΔXYZ by SSS If 3 pairs of sides are congruent, then the two triangles are congruent.

  39. 1. Does the diagram give enough info to use SSS Congruence? K L C A no J B

  40. Given: LN  NP and M is the midpoint of LP Prove:ΔNLM ΔNPM LM  MP NM  NM 2. N L P M Def of midpoint Reflexive Property NLM  NPM SSS Congruence

  41. F P 41 41 N D M E (- 5 – - 1)2 + (1 – 6)2 (6 – 2)2 + (1 – 6)2 3. Show that ΔNPM  ΔDFE by SSS if N(-5,1), P (-1,6), M (-1,1), D (6,1), F (2,6), and E (2,1). 4 NM = MP = NP = DE = EF = DF = 5 4 5

  42. SAS Experiment Using 2 congruent segments and 1 included angle, can you ONLY create 2 triangles that are congruent?

  43. B If Side AB XY Angle B  Y Side BC  YZ, then ΔABC  ΔXYZ by SAS C A Y X Z Side-Angle-Side Congruence Postulate If 2 pairs of sides and their included angle are congruent, then the two triangles are congruent.

  44. SAS? 5. 4. NO! SAS 6. 7. NO! SAS

  45. A B C D 8. Does the diagram give enough info to use SAS Congruence? A C D

  46. 9. Does the diagram give enough info to use SAS Congruence? W V no X Y Z

  47. 10. Does the diagram give enough info to use SAS Congruence? E A no B D C

  48. V X W Z VW  WY and ZW  WX Y 11. Given: W is the midpoint of VY and the midpoint of ZX Prove:ΔVWZ ΔYWX Def. of midpoint VWZ YWX Vertical Angles Th VWZ  YWX SAS Congruence

  49. Given: AB  PB , MB  AP Prove:ΔMBA ΔMBP MB  MB 12. M P A B Def of  lines ABM & PBM are right s ABM  PBM All right s are  Reflexive Property MBA  MBP SAS Congruence

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