1. Congruence Based on Triangles Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

2. Line Segments Associated with Triangles ERHS Math Geometry Mr. Chin-Sung Lin

3. Altitude of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin An altitude of a triangle is a line segment drawn from any vertex of the triangle, perpendicular to and ending in the line that contains the opposite side A C B C A B AC B

4. Altitude of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin If BD is the altitude of ∆ ABC then, m  BDA = 90 m  BDC = 90 C A B D

5. Altitude - Area of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin Altitudes can be used to compute the area of a triangle: A C B C A B AC B Base Altitude Base Altitude Base Area = 1/2 * Base * Altitude

6. Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Three altitudes intersect in a single point, called the orthocenter of the triangle C Orthocenter A B

7. Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Where is the orthocenter of a right triangle? Orthocenter? AC B

8. Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin The orthocenter is located at the vertex of the right angle Orthocenter A C B

9. Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Where is the orthocenter of an obtuse triangle? Orthocenter? C B A

10. Altitude - Orthocenter ERHS Math Geometry Mr. Chin-Sung Lin Orthocenter C B A The orthocenter is outside the triangle

11. Angle Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin

12. Angle Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin A line segment that bisects an angle of the triangle and terminates in the side opposite that angle A C B AC B C A B

13. Angle Bisector of a Triangle ITHS Math B Term 1 (M$4) Mr. Chin-Sung Lin If BD is the angle bisector of  ABC then,  ABD   CBD AC B D

14. Angle Bisector - Incenter ERHS Math Geometry Mr. Chin-Sung Lin The three angle bisectors of a triangle meet in one point called the incenter A B Incenter C

15. Angle Bisector - Incenter ERHS Math Geometry Mr. Chin-Sung Lin Incenter is the center of the incircle, the circle inscribed in the triangle A B Incenter C

16. Median of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin

17. Median of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin A segment from a vertex to the midpoint of the opposite side A C B AC B C A B

18. Median of a Triangle ITHS Math B Term 1 (M$4) Mr. Chin-Sung Lin If BD is the median of ∆ ABC then, AD  CD AC B D

19. Median of a Triangle - Centroid ERHS Math Geometry Mr. Chin-Sung Lin The three medians meet in the centroid or center of mass (center of gravity) A B Centroid C

20. Median of a Triangle - Centroid ERHS Math Geometry Mr. Chin-Sung Lin The centroid divides each median in a ratio of 2:1. A B Centroid C 2 1

21. Perpendicular Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin

22. Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin The perpendicular bisector of a line segment is a line, a ray, or a line segment that is perpendicular to the line segment at its midpoint AB  CD CO = OD D O A C B ~

23. Perpendicular Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin A line, a ray, or a line segment that is perpendicular to the side of a triangle at its midpoint A C B AC B C A B

24. Perpendicular Bisector of a Triangle ERHS Math Geometry Mr. Chin-Sung Lin If DE is the perpendicular bisector of the side of ∆ ABC then, AD  CD DE  AC AC B D E

25. Perpendicular Bisector - Circumcenter ERHS Math Geometry Mr. Chin-Sung Lin The three perpendicular bisectors meet in one point called the circumcenter A B Circumcenter C

26. Perpendicular Bisector - Circumcenter ERHS Math Geometry Mr. Chin-Sung Lin Circumcenter is the center of the circumcircle, the circle passing through the vertices of the triangle A B Circumcenter C

27. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin In a scalene triangle, the altitude, angle bisector, median drawn from any common vertex, and the perpendicular bisector of the opposite side are four distinct line segments A B C E DF BD: Altitude BE: Angle bisector BF: Median FG:Perpendicular Bisector G

28. Isosceles & Equilateral Triangles ERHS Math Geometry Mr. Chin-Sung Lin In isosceles & equilateral triangles, some of the altitude, angle bisector, median, and perpendicular bisector coincide C A B D BD: Altitude BD: Angle bisector BD: Median BD:Perpendicular Bisector

29. Scalene Triangle (Indirect Proof) ERHS Math Geometry Mr. Chin-Sung Lin Given: ∆ ABC is scalene, BD bisects ABC Prove: BD is not perpendicular to AC A B C D 12 34

30. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons A B C D 12 34

31. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true A B C D 12 34

32. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector A B C D 12 34

33. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector A B C D 12 34

34. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular A B C D 12 34

35. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate A B C D 12 34

36. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property A B C D 12 34

37. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate A B C D 12 34

38. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate 8. AB = CB 8. CPCTC A B C D 12 34

39. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate 8. AB = CB 8. CPCTC 9. AB ≠ CB 9. Definition of scalene triangle A B C D 12 34

40. Scalene Triangle ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. BD  AC 1. Assume the opposite is true 2. ∆ ABC is scalene, BD is angle 2. Given bisector 3. 1  2 3. Definition of angle bisector 4. 3 = 90 o, 4 = 90 o 4. Definition of perpendicular 5. 3  4 5. Substitution postulate 6. BD  BD 6. Reflexive property 7. ∆ ABD  ∆ CBD 7. ASA postulate 8. AB = CB 8. CPCTC 9. AB ≠ CB 9. Definition of scalene triangle 10. BD is not perpendicular to AC10. Contradition in statement 8 & 9, so, assumption is false. The negation of the assumption is true A B C D 12 34

41. CPCTC ERHS Math Geometry Mr. Chin-Sung Lin

42. CPCTC ERHS Math Geometry Mr. Chin-Sung Lin Corresponding Parts of Congruent Triangles are Congruent After proving that two triangles are congruent, we can conclude that their corresponding parts (angles & sides) are congruent

43. Congruent Triangles Mr. Chin-Sung Lin Given:  B   C, and AB  AC Prove: AF  AE A B C D E F ERHS Math Geometry

44. Congruent Triangles Mr. Chin-Sung Lin Given:  B   C, and AB  AC Prove: AF  AE A B C D E F ERHS Math Geometry

45. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons ERHS Math Geometry A B C D E F

46. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given ERHS Math Geometry A B C D E F

47. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given 2. A  A2. Reflexive property ERHS Math Geometry A B C D E F

48. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given 2. A  A2. Reflexive property 3.∆ ABF  ∆ ACE 3. ASA ERHS Math Geometry A B C D E F

49. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.  B   C, and AB  AC 1. Given 2. A  A2. Reflexive property 3.∆ ABF  ∆ ACE 3. ASA 4.AF  AE 4. CPCTC ERHS Math Geometry A B C D E F

50. Isosceles Triangles ERHS Math Geometry Mr. Chin-Sung Lin

51. Isosceles Triangles ERHS Math Geometry Mr. Chin-Sung Lin An isosceles triangle is a triangle that has two congruent sides AC B

52. Parts of an Isosceles Triangle ERHS Math Geometry Mr. Chin-Sung Lin Leg: the two congruent sides Base: the third side Vertex Angle: the angle formed by the two congruent side Base Angle: the angles whose vertices are the endpoints of the base AC B Base Leg Base Angle Vertex Angle

53. Base Angle Theorem (Isosceles Triangle Theorem) ERHS Math Geometry Mr. Chin-Sung Lin

54. Base Angle Theorem (Isosceles Triangle Theorem) ERHS Math Geometry Mr. Chin-Sung Lin If two sides of a triangle are congruent, then the angles opposite these sides are congruent (Base angles of an isosceles triangle are congruent)

55. Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin If two sides of a triangle are congruent, then the angles opposite these sides are congruent Draw a diagram like the one below Given: A B  CB Prove:A  C AC B

56. Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 2. 3. 4. 5. 6. AC B D

57. Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. Draw the angle bisector of 1. Any angle of measure less ABC and let D be the point than 180 has exactly one where it intersects AC bisector 2. ABD  CBD 2. Definition of angle bisector 3. A B  CB 3. Given 4. BD  BD 4. Reflexive property 5. ∆ ABD = ∆ CBD 5. SAS Postulate 6. A  C6. CPCTC AC B D

58. Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin Given: A B  CB and AD  CE Prove: ∆ ABD = ∆ CBE A C B DE

59. Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin Given: A B  CB and AD  CE Prove: ∆ ABD = ∆ CBE A C B DE

60. Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 2. 3. A C B DE

61. Base Angle Theorem - Example 1 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. A B  CB 1. Given AD  CE 2. A  C 2. Base Angle Theorem 3. ∆ ABD = ∆ CBE 3. SAS Postulate A C B DE

62. Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin Given: 1  2 and 5  6 Prove: 3  4 A C B D O 1 2 5 6 3 4

63. Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin Given: 1  2 and 5  6 Prove: 3  4 A C B D O 1 2 5 6 3 4

64. Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 2. 3. 4.5.6. A C B D O 1 2 5 6 3 4

65. Base Angle Theorem - Example 2 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 1  2 1. Given 5  6 2. A B  AB 2. Reflexive Property 3. ∆ ACB = ∆ ADB 3. ASA Postulate 4. A C  AD 4. CPCTC 5. ∆ ADC is an isosceles triangle 5. Def. of Isosceles Triangle 6. 3  4 6. Base Angle Theorem A C B D O 1 2 5 6 3 4

66. Base Angle Theorem - Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: BD  BE and AD  CE Prove: AB = CB A C B DE

67. Converse of Base Angle Theorem (Converse of Isosceles Triangle Theorem) ERHS Math Geometry Mr. Chin-Sung Lin

68. Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin If two angles of a triangle are congruent, then the sides opposite these angles are congruent

69. Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin If two angles of a triangle are congruent, then the sides opposite these angles are congruent Draw a diagram like the one below Given:A  C Prove: A B  CB AC B

70. Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 2. 3. 4. 5. 6. AC B D

71. Converse of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. Draw the angle bisector of 1. Any angle of measure less ABC and let D be the point than 180 has exactly one where it intersects AC bisector 2. ABD  CBD 2. Definition of angle bisector 3. A  C 3. Given 4. BD  BD 4. Reflexive property 5. ∆ ABD = ∆ CBD 5. AAS Postulate 6. A B  CB 6. CPCTC AC B D

72. Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin Given: A O  BO and 1  2 Prove: AC = BD A C B D O 12

73. Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin Given: A O  BO and 1  2 Prove: AC = BD A C B D O 12

74. Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 2. 3. A C B D O 12

75. Base Angle Theorem - Example 3 ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons 1. 1  2 1. Given 2. C O  DO 2. Converse of Base Angle Theorem 3. A O  BO 3. Given 4. AOC  BOD 4. Vertical Angles 5. ∆ AOC = ∆ BOD 5. SAS Postulate 6. AC  BD6. CPCTC A C B D O 12

76. Corollaries of Base Angle Theorem ERHS Math Geometry Mr. Chin-Sung Lin The median from the vertex angle of an isosceles triangle bisects the vertex angle The median from the vertex angle of an isosceles triangle is perpendicular to the base

77. Equilateral and Equiangular Triangles ERHS Math Geometry Mr. Chin-Sung Lin

78. Equilateral Triangles ERHS Math Geometry Mr. Chin-Sung Lin A equilateral triangle is a triangle that has three congruent sides AC B

79. Equilateral & Equiangular Triangles ERHS Math Geometry Mr. Chin-Sung Lin If a triangle is an equilateral triangle, then it is an equiangular triangle

80. Identify Overlapping Triangles ERHS Math Geometry Mr. Chin-Sung Lin

81. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O

82. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ ADC

83. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ BCD

84. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ DAB

85. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ CBA

86. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ DOC

87. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ AOB

88. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ AOD

89. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O ∆ BOC

90. Identify Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O Total 8 Triangles

91. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? A C B D O E

92. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ BDC A C B D O E

93. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ CEB A C B D O E

94. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ AEB A C B D O E

95. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ ADC A C B D O E

96. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ DOB A C B D O E

97. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ EOC A C B D O E

98. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ BOC A C B D O E

99. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? ∆ ABC A C B D O E

100. Identify Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin How many triangles can you identify in the following diagram? Total 8 Triangles A C B D O E

101. Shared Sides & Angles ERHS Math Geometry Mr. Chin-Sung Lin

102. Shared Side - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? A C B D O

103. Shared Side - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? A C B D O ∆ ADC & ∆ BCD DC

104. Shared Side - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? AB C O D EF

105. Shared Side - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? ∆ ACF & ∆ BDE EF AB C O D EF

106. Shared Side - 3 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? A CB D E

107. Shared Side - 3 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared side? Which line segment has been shared? ∆ AEB & ∆ ADC DE A CB D E

108. Shared Angle - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? A B C O E D

109. Shared Angle - 1 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? A B C O E D ∆ AEB & ∆ ADC  BAC

110. Shared Angle - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? A CB D E

111. Shared Angle - 2 ERHS Math Geometry Mr. Chin-Sung Lin Which two congruent-triangle candidates have a shared angle? Which angle has been shared? ∆ AEB & ∆ ADC  DAE A CB D E

112. Congruent Overlapping Triangles ERHS Math Geometry Mr. Chin-Sung Lin

113. Congruent Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A B C O E D

114. Congruent Triangles - 1 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A B C O E D ∆ AEB & ∆ ADC ∆ DOB & ∆ EOC

115. Congruent Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B D O

116. Congruent Triangles - 2 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ADC & ∆ BCD ∆ AOD & ∆ BOC A C B D O

117. Congruent Triangles - 3 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B D O E

118. Congruent Triangles - 3 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ BDO & ∆ CEO ∆ ECB & ∆ DBC A C B D O E ∆ AEB & ∆ ADC

119. Congruent Triangles - 4 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB D E

120. Congruent Triangles - 4 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ AEB & ∆ ADC ∆ ADB & ∆ AEC A CB D E

121. Congruent Triangles - 5 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD

122. Congruent Triangles - 5 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD ∆ ABD & ∆ ACF ∆ ADE & ∆ AFE ∆ ABE & ∆ ACE ∆ ABF & ∆ ACD

123. Congruent Triangles - 6 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B D O

124. Congruent Triangles - 6 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ABC & ∆ BAD ∆ AOC & ∆ BOD ∆ ACD & ∆ BDC A C B D O

125. Congruent Triangles - 7 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A D B C E F

126. Congruent Triangles - 7 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ABD & ∆ CDB ∆ ADE & ∆ CBF ∆ ABE & ∆ CDF A D B C E F

127. Congruent Triangles - 8 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EF D O G H

128. Congruent Triangles - 8 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ AGO & ∆ BHO ∆ CGE & ∆ DHF ∆ AED & ∆ BFC A C B EF D O G H

129. Congruent Triangles - 9 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EFD

130. Congruent Triangles - 9 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? ∆ ABD & ∆ ACF ∆ ADE & ∆ AFE A C B EFD

131. Congruent Triangles - 10 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD GH

132. Congruent Triangles - 10 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A CB EFD GH ∆ ABG & ∆ ACH ∆ AGE & ∆ AHE ∆ ABE & ∆ ACE ∆ ADE & ∆ AFE ∆ GDE & ∆ HFE

133. Congruent Triangles - 11 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EF D G H IJ O

134. Congruent Triangles - 11 ERHS Math Geometry Mr. Chin-Sung Lin Name the possible congruent-triangle pairs? A C B EF D G H IJ O ∆ AGO & ∆ AHO ∆ BGI & ∆ CHJ ∆ IDE & ∆ JFE ∆ AIE & ∆ AJE ∆ ADE & ∆ AFE∆ BOE & ∆ COE

135. Theorems about Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin

136. Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin The perpendicular bisector of a line segment is a line, a ray, or a line segment that is perpendicular to the line segment at its midpoint AB  CD CO = OD D O A C B ~

137. Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment Given: AB and points P and T such that PA = PB and TA = TB Prove: PT is the perpendicular bisector of AB B O P A T

138. Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment Given: Point P such that PA = PB Prove: P lies on the perpendicular bisector of AB B M P A

139. Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin If a point is on the perpendicular bisector of a line segmenton, then it is equidistant from the endpoints of the line segment Given: Point P on the perpendicular bisector of AB Prove: PA = PB B M P A

140. Theorems of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin A point is on the perpendicular bisector of a line segmenton if and only if it is equidistant from the endpoints of the line segment B M P A

141. Perpendicular Bisector Concurrence Theorems ERHS Math Geometry Mr. Chin-Sung Lin The perpendicular bisectors of the sides of a triangle are concurrent (intersect in one point) Given: MQ, the perpendicular bisector of AB NR, the perpendicular bisector of AC LS, the perpendicular bisector of BC Prove: MQ, NR, and LS intersect in P R P L S N Q M A B C

142. Perpendicular Bisector Concurrence Theorems ERHS Math Geometry Mr. Chin-Sung Lin StatementsReasons R P L S N Q M A B C

143. Construction ERHS Math Geometry Mr. Chin-Sung Lin

144. Construction of Perpendicular Bisector ERHS Math Geometry Mr. Chin-Sung Lin B M A

145. Q & A ERHS Math Geometry Mr. Chin-Sung Lin

146. The End ERHS Math Geometry Mr. Chin-Sung Lin