1. Chapter 4 Congruent Triangles Identify the corresponding parts of congruent figures Prove two triangles are congruent Apply the theorems and corollaries about isosceles triangles

2. 4.1 Congruent Figures Objectives Identify the corresponding parts of congruent figures

3. What we already know… Congruent Segments –Same length Congruent Angles –Same degree measure

4. Congruent Figures Exactly the same size and shape. Don’t ASSume ! D C A B E F

5. Definition of Congruency Two figures are congruent if corresponding vertices can be matched up so that: 1. All corresponding sides are congruent 2. All corresponding angles are congruent.

6. What does corresponding mean again? Matching In the same position

7. Volunteer Draw a large scalene triangle (with a ruler) Cut out two congruent triangles that are the same Label the Vertices A, B, C and D, E, F

8. You can slide and rotate the triangles around so that they MATCH up perfectly.  ABC   DEF A B C F E D

9. matters The order in which you name the triangles matters !  ABC   DEF A B C F E D

10. Based on the definition of congruency…. Three pairs of corresponding angles Three pairs of corresponding sides 1.  A   D 3.  C   F 2.  B   E 1. AB  DE 3. CA  FD 2. BC  EF

11. It is not practical to cut out and move the triangles around

12.  ABC   XYZ Means that the letters X and A, which appear first, name corresponding vertices and that –  X   A. The letters Y and B come next, so –  Y   B and –XY  AB

13. CAUTION !! If the diagram doesn’t show the markings or You don’t have a reason –Shared sides, shared angles, vertical angles, parallel lines

14. White Boards Suppose  TIM   BER IM  ___

15. White Boards Suppose  TIM   BER IM  ER, Why ?

16. White Boards Corresponding Parts of Congruent Triangles are Congruent

17. White Boards Suppose  TIM   BER ___   R

18. White Boards Suppose  TIM   BER  M   R, Why?

19. White Boards Corresponding Parts of Congruent Triangles are Congruent

20. White Boards Suppose  TIM   BER  MTI   ____

21. White Boards Suppose  TIM   BER  MTI   RBE

22. White Boards If  ABC   XYZ m  B = 80 m  C = 50 Name four congruent angles

23. White Boards If  ABC   XYZ m  B = 80 m  C = 50  A,  C,  X,  Z

24. White Boards If  ABC   XYZ Write six congruences that must be correct

25. White Boards If  ABC   XYZ 1.  A   X 3.  C   Z 2.  B   Y 1. AB  XY 3. CA  ZX 2. BC  YZ

26. Remote time A.Always B.Sometimes C.Never D.I don’t know

27. An acute triangle is __________ congruent to an obtuse triangle. A. Always B. Sometimes C. Never D. I don’t know

28. A polygon is __________ congruent to itself. A. Always B. Sometimes C. Never D. I don’t know

29. A right triangle is ___________ congruent to another right triangle. A. Always B. Sometimes C. Never D. I don’t know

30. If  ABC   XYZ,  A is ____________ congruent to  Y. A. Always B. Sometimes C. Never D. I don’t know

31. If  ABC   XYZ,  B is ____________ congruent to  Y. A. Always B. Sometimes C. Never D. I don’t know

32. If  ABC   XYZ, AB is ____________ congruent to ZY. A. Always B. Sometimes C. Never D. I don’t know

33. 4.2 Some Ways to Prove Triangles Congruent Objectives Learn about ways to prove triangles are congruent

34. Don’t ASSume Triangles cannot be assumed to be congruent because they “look” congruent. and It’s not practical to cut them out and match them up so,

35. We must show 6 congruent pairs 3 angle pairs and 3 pairs of sides

36. WOW That’s a lot of work

37. Spaghetti Experiment Using a small amount of playdough as your “points” put together a 5 inch, 3 inch and 2.5 inch piece of spaghetti to forma triangle. Be careful, IT’S SPAGHETTI, and it will break.

38. Compare your spaghetti triangle to your neighbors Compare your spaghetti triangle to my spaghetti triangle.

39. We are lucky….. There is a shortcut –We don’t have to show ALL pairs of angles are congruent and ALL pairs of sides are congruent

40. SSS Postulate If three sides of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A B E CD F

41. Patty Paper Practice 5 inches 2.5 inches 3 inches

42. Volunteer

43. SAS Postulate If two sides and the included angle are congruent to the corresponding parts of another triangle, then the triangles are congruent. B E CD F

44. ASA Postulate If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A B CD E F

45. The order of the letters MEAN something Is SAS the same as SSA or A$$ ?

46. Construction 2 Given an angle, construct a congruent angle. Given: Construct: Steps:

47. Construction 3 Given an angle, construct the bisector of the angle Given: Construct: Steps:

48. CAUTION !! If the diagram doesn’t show the markings or You don’t have a reason –Shared sides, shared angles, vertical angles, parallel lines

49. Remote Time Can the two triangles be proved congruent? If so, what postulate can be used? A. SSS Postulate B.SAS Postulate C.ASA Postulate D.Cannot be proved congruent E.I don’t know

50. A. SSS Postulate B. SAS Postulate C. ASA Postulate D. Cannot be proved congruent E. I don’t know

59. White Board Decide Whether you can deduce by the SSS, SAS, or ASA Postulate that the two triangles are congruent. If so, write the congruence (  ABC   _ _ _ ). If not write not congruent.

60. D C B A

61.  DBC   ABC SSS

62. A DC B

63. No Congruence

64. Construction 7 Given a point outside a line, construct a line parallel to the given line through the point. Given: Construct: Steps:

65. 4.3 Using Congruent Triangles Objectives Use congruent triangles to prove other things

66. Our Goal In the last section, our goal was to prove that two triangles are congruent.

67. The Reason If we can show two triangle are congruent, using the SSS, SAS, ASA postulates, then we can use the definition of Congruent Triangles to say other parts of the triangles are congruent. –Corresponding Parts of Congruent Triangles are Congruent.

68. This is an abbreviated way to refer to the definition of congruency with respect to triangles. C orresponding P arts of C ongruent T riangles are C ongruent

69. Basic Steps 1.Identify two triangles in which the two segments or angles are corresponding parts. 2.Prove that those two triangles are congruent 3.State that the two parts are congruent using the reason CPCTC.

70. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21

71. L M J K 34 21 LM = LM m  J = m  K If 2  ’s of 1  are  to 2  ’s of another , then the third  ’s are . Reflexive Property

72. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 LM = LM m  J = m  K If 2  ’s of 1  are  to 2  ’s of another , then the third  ’s are . Reflexive Property A A A A S AA

73. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21  JLM   KLM ASA A A A A S

74. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons

75. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given

76. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given 5. M is the midpoint of JK

77. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given 4. JM = KM 5. M is the midpoint of JK

78. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given 3.  JLM   KLM 4. JM = KM4. CPCTC 5. M is the midpoint of JK

79. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given 2. LM = LM2. Reflexive Property 3.  JLM   KLM 3. ASA Postulate 4. JM = KM4. CPCTC 5. M is the midpoint of JK

80. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given 2. LM = LM2. Reflexive Property 3.  JLM   KLM 3. ASA Postulate 4. JM = KM4. CPCTC 5. M is the midpoint of JK5. Definition of midpoint

81. Given: m  1 = m  2 m  3 = m  4 Prove: M is the midpoint of JK L M J K 34 21 StatementsReasons 1. m  1 = m  2 m  3 = m  4 1. Given 2. LM = LM2. Reflexive Property 3.  JLM   KLM 3. Postulate 4. JM = KM4. CPCTC 5. M is the midpoint of JK5. Definition of midpoint A A A A A A A A S S S

82. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO  3   4Definition of  bisector JK  JKReflexive Property  MKJ   OKJ SAS Postulate K O J M 1 2 3 4 A A S S S

83. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO StatementsReasons K O J M 1 2 3 4 K O J M 1 2 3 4 A A S S S

84. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO StatementsReasons 1.MK  OK; KJ bisects  MKO 1. Given 2.  3   42. Def of  bisector 3. JK  JK 3. Reflexive Property K O J M 1 2 3 4 K O J M 1 2 3 4 A A S S S

85. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO StatementsReasons 1.MK  OK; KJ bisects  MKO 1. Given 2.  3   42. Def of  bisector 3. JK  JK 3. Reflexive Property 6. JK bisects  MJO 6. K O J M 1 2 3 4 K O J M 1 2 3 4 A A S S S

86. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO StatementsReasons 1.MK  OK; KJ bisects  MKO 1. Given 2.  3   42. Def of  bisector 3. JK  JK 3. Reflexive Property 5.  1   2 5. CPCTC 6. JK bisects  MJO 6. K O J M 1 2 3 4 K O J M 1 2 3 4 A A S S S

87. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO StatementsReasons 1.MK  OK; KJ bisects  MKO 1. Given 2.  3   42. Def of  bisector 3. JK  JK 3. Reflexive Property 4.  MKJ   OKJ 4. SAS Postulate 5.  1   2 5. CPCTC 6. JK bisects  MJO 6. K O J M 1 2 3 4 K O J M 1 2 3 4 A A S S S

88. Given: MK  OK; KJ bisects  MKO; Prove: JK bisects  MJO StatementsReasons 1.MK  OK; KJ bisects  MKO 1. Given 2.  3   42. Def of  bisector 3. JK  JK 3. Reflexive Property 4.  MKJ   OKJ 4. SAS Postulate 5.  1   2 5. CPCTC 6. JK bisects  MJO6. Def of  bisector K O J M 1 2 3 4 A S A A S S S S K O J M 1 2 3 4 A A S S S

89. 4.4 The Isosceles Triangle Theorem Objectives Apply the theorems and corollaries about isosceles triangles

90. Isosceles Triangle By definition, it is a triangle with two congruent sides called legs. X Y Z Base Base Angles Legs Vertex Angle

91. Experiment - Goal Discover Properties of an Isosceles Triangle

92. Supplies Blank sheet of paper Ruler Pencil Scissors

93. Procedure 1.Fold a sheet of paper in half.

94. Procedure 2. Draw a line with the ruler going from the folded edge (very important) to the corner of the non folded edge. Folded edge

95. Procedure 3. Cut on the red line Cut here

96. Procedure 4. Open and lay flat. You will have a triangle

97. Procedure 5. Label the triangle P Q S R

98. Procedure 6.Since  PRQ fits exactly over  PSQ (because that’s the way we cut it),  PRQ   PSQ P Q S R

99. Procedure 7. What conclusions can you make? Be careful not to ASSume anything. P Q S R

100. Conclusions P Q S R 1.  PRS   PSR 2.PQ bisects  RPS 3.PQ bisects RS 4.PQ  RS at Q 5.PR  PS

101. These conclusions are actually Theorems and corollaries

102. Theorem The base angles of an isosceles triangle are congruent. A B C

103. Corollary An equilateral triangle is also equiangular.

104. Corollary An equilateral triangle has angles that measure 60.

105. Corollary The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

106. Theorem If two angles of a triangle are congruent, then it is isosceles. A B C

107. Corollary An equiangular triangle is also equilateral.

108. White Board Practice Find the value of x 30º xº

109. x = 75º

110. White Board Practice Find the value of x 2x - 4 x + 5 2x + 2

111. x = 9

112. White Board Practice Find the value of x 56 º62 º x 41 42

113. x = 42

114. 4.5 Other Methods of Proving Triangles Congruent Objectives Learn two new ways to prove triangles are congruent

115. Proving Triangles  We can already prove triangles are congruent by the ASA, SSS and SAS. There are two other ways to prove them congruent…

116. AAS Theorem If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A B CD E F

117. The Right Triangle leg hypotenuse right angle A B C acute angles

118. HL Theorem If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. A B CD E F

119. Five Ways to Prove   ’s All Triangles: ASA SSS SAS AAS Right Triangles Only: HL

120. White Board Practice State which of the congruence methods can be used to prove the triangles congruent. You may choose more than one answer. SSS Postulate SAS Postulate ASA Postulate AAS Theorem HL Theorem

133. 4.6 Using More than One Pair of Congruent Triangles Objectives Construct a proof using more than one pair of congruent triangles.

134. Sometimes two triangles that you want to prove congruent have common parts with two other triangles that you can easily prove congruent.

135. More Than One Pair of  ’s  Given: X is the midpt of AF & CD Prove: X is the midpt of BE A B C D E F X

136. Lecture 7 (4-7) Objectives Define altitudes, medians and perpendicular bisectors.

137. Median of a Triangle A segment connecting a vertex to the midpoint of the opposite side. midpoint vertex

138. Median of a Triangle Each triangle has three Medians midpoint vertex

139. Median of a Triangle Each triangle has three Medians midpoint vertex

140. Median of a Triangle Notice that the three medians will meet at one point. If they do not meet, then you are not drawing the segments well.

141. Altitude of a Triangle A segment drawn from a vertex perpendicular to the opposite side. vertex perpendicular

142. Altitude of a Triangle Each Triangle has three altitudes vertex perpendicular

143. Altitude of a Triangle Each triangle has three altitudes vertex perpendicular

144. Altitude of a Triangle Notice that the three altitudes will meet at one point. If they do not meet, then you are not drawing the segments well.

145. Special Cases - Altitudes Obtuse Triangles: Two of the altitudes are drawn outside the triangle. Extend the sides of the triangle

146. Special Cases - Altitudes Right Triangles: Two of the altitudes are already drawn for you.

147. Perpendicular Bisector A segment (line or ray) that is perpendicular to and passes through the midpoint of another segment. Must put the perpendicular and congruent markings !

148. Angle Bisector A ray that cuts an angle into two congruent angles.

149. Theorem If a point lies on the perpendicular bisector of a segment of a segment, then the point is equidistant from the endpoints.

150. Theorem If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

151. Remember When you measure distance from a point to a line, you have to make a perpendicular line.

152. Theorem If a point lies on the bisector of an angle then the point is equidistant from the sides of the angle.

153. Construction 10 Given a triangle, circumscribe a circle about the triangle. Given: Construct: Steps:

154. Construction 11 Given a triangle, inscribe a circle within the triangle. Given: Construct: Steps:

155. Remote Time A.Always B.Sometimes C.Never D.I don’t know

156. An altitude is _____________ perpendicular to the opposite side. A. Always B. Sometimes C. Never D. I don’t know

157. A median is ___________ perpendicular to the opposite side. A. Always B. Sometimes C. Never D. I don’t know

158. An altitude is ______________ a perpendicular bisector. A. Always B. Sometimes C. Never D. I don’t know

159. An angle bisector is _______________ perpendicular to the opposite side. A. Always B. Sometimes C. Never D. I don’t know

160. A perpendicular bisector of a segment is ___________ equidistant from the endpoints of the segment. A. Always B. Sometimes C. Never D. I don’t know