1. Congruence of Line Segments, Angles, and Triangles Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

2. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin

3. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin A line segment can be extended to any length in either direction We can choose some point of AB that is not a point of AB to form a line segment of any length AD B

4. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin Through two given points, one and only one line can be drawn, i.e., two points determine a line Through given points A and B, one and only one line can be drawn A B

5. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin Two lines cannot intersect in more than one point AMB and CMD intersect at M and cannot intersect at any other point A M B C D

6. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin One and only one circle can be drawn with any given point as center and the length of any given segment as a radius Only one circle can be drawn that has point O as its center and a radius equal in length to segment r O r

7. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin At a given point on a given line, one and only one perpendicular can be drawn to the line At point P on APB, exactly one line, PD, can be drawn perpendicular to APB and no other line through P is perpendicular to APB A P B D

8. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin From a given point not on a given line, one and only one perpendicular can be drawn to the line From point D not on AB, exactly one line DP, can be drawn perpendicular to AB and no other linefrom D is perpendicular to AB. A P B D

9. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin Distance Postulate - For any two distinct points, there is only one positive real number that is the length of the line segment joining the two points For any distinct points A and B, there is only one positive real number, represented by AB, that is the length of AB A B

10. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin The shortest distance between two points is the length of the line segment joining these two points The measure of the shortest path from A to B is the distance AB A C B

11. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin A line segment has one and only one midpoint AB has a midpoint, point P, and no other point is a midpoint of AB A P B

12. Postulates of Lines, Line Segments, and Angles ERHS Math Geometry Mr. Chin-Sung Lin An angle has one and only one bisector Angle ABC has one bisector, BD, and no other ray bisects ABC A B D C

13. Conditional Statements and Proof ERHS Math Geometry Mr. Chin-Sung Lin

14. Conditionals and Proof ERHS Math Geometry Mr. Chin-Sung Lin When the information needed for a proof is presented in a conditional statement, we use the information in the hypothesis to form a given statement, and the information in the conclusion to form a prove statement

15. Rewrite the Conditionals for Proof ERHS Math Geometry Mr. Chin-Sung Lin If a ray bisects a straight angle, it is perpendicular to the line determined by the straight angle

16. Rewrite the Conditionals for Proof ERHS Math Geometry Mr. Chin-Sung Lin If a ray bisects a straight angle, it is perpendicular to the line determined by the straight angle Given: ABC is an straight angle and BD bisects ABC Prove: BD  AC C D A B

17. Rewrite the Conditionals for Proof ERHS Math Geometry Mr. Chin-Sung Lin If a triangle is equilateral, then the measures of the sides are equal

18. Rewrite the Conditionals for Proof ERHS Math Geometry Mr. Chin-Sung Lin If a triangle is equilateral, then the measures of the sides are equal Given: ΔABC is equilateral Prove: AB = BC = CA C A B

19. Using Postulates and Definitions in Proofs ERHS Math Geometry Mr. Chin-Sung Lin

20. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: StatementsReasons C A B D R 1 2 3 4

21. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: StatementsReasons 1.DR is the bisector of ABC. 1. Given. 3 ≅ 1 and 4 ≅ 2 C A B D R 1 2 3 4

22. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: StatementsReasons 1.DR is the bisector of ABC. 1. Given. 3 ≅ 1 and 4 ≅ 2 2.1 ≅ 2 2. Definition of angle bisector. C A B D R 1 2 3 4

23. Two Column Proof Example ERHS Math Geometry Mr. Chin-Sung Lin Given: DR is the bisector of ADC, 3 ≅ 1 and 4 ≅ 2. Prove: 3 ≅ 4 Proof: StatementsReasons 1.DR is the bisector of ABC. 1. Given. 3 ≅ 1 and 4 ≅ 2 2. 1 ≅ 2 2. Definition of angle bisector. 3. 3 ≅ 4 3. Substitution postulate. C A B D R 1 2 3 4

24. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons M A B

25. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons 1.M is the midpoint of AB. 1. Given. M A B

26. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons 1.M is the midpoint of AB. 1. Given. 2.AM ≅ MB 2. Definition of midpoint. M A B

27. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons 1.M is the midpoint of AB. 1. Given. 2.AM ≅ MB 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. M A B

28. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons 1.M is the midpoint of AB. 1. Given. 2.AM ≅ MB 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. 4.AM + MB = AB 4. Partition postulate. M A B

29. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons 1.M is the midpoint of AB. 1. Given. 2.AM ≅ MB 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. 4.AM + MB = AB 4. Partition postulate. 5.2AM = AB and 2 MB = AB 5. Substitution postulate. M A B

30. Two Column Proof Exercise ERHS Math Geometry Mr. Chin-Sung Lin Given: M is the midpoint of AB. Prove: AM = ½ AB and MB = ½ AB Proof: StatementsReasons 1.M is the midpoint of AB. 1. Given. 2.AM ≅ MB 2. Definition of midpoint. 3.AM = MB 3. Definition of congruent segments. 4.AM + MB = AB 4. Partition postulate. 5.2AM = AB and 2 MB = AB 5. Substitution postulate. 6.AM = ½ AB and MB = ½ AB 6. Division postulate. M A B

31. Angles & Angle Pairs Mr. Chin-Sung Lin ERHS Math Geometry

32. Congruent Angles Mr. Chin-Sung Lin Congruent angles are angles that have the same measure DOE = ABC m DOE = m ABC ~ CB A EO D ERHS Math Geometry

33. Right Angles Mr. Chin-Sung Lin Perpendicular lines are two lines that intersect to form right angles AB  CD D O A C B ERHS Math Geometry

34. Adjacent Angles Mr. Chin-Sung Lin Adjacent angles are two angles in the same plane that have a common vertex and a common side but do not have any interior points in common AOC and COD D O A C ERHS Math Geometry

35. Vertical Angles Mr. Chin-Sung Lin Vertical angles are two angles in which the sides of one angle are opposite rays to the sides of the second angle AOC and BOD AOB and COD D O AC B ERHS Math Geometry

36. Complementary Angles Mr. Chin-Sung Lin Complementary angles are two angles the sum of whose degree measure is 90 AOB and BOC AOB and RST B CO A B T O A S R ERHS Math Geometry

37. Supplementary Angles Mr. Chin-Sung Lin Supplementary angles are two angles the sum of whose degree measure is 180 AOB and BOC AOB and RST CA B O T A B OS R ERHS Math Geometry

38. Linear Pair Mr. Chin-Sung Lin A linear pair of angles are two adjacent angles whose sum is a straight angle AOB and BOC C A B O ERHS Math Geometry

39. Theorems of Congruent Angle Pairs ERHS Math Geometry Mr. Chin-Sung Lin

40. Theorems of Angle Pairs Mr. Chin-Sung Lin Linear pair Right Angles Complementary Angles Supplementary Angles Vertical Angles ERHS Math Geometry

41. Theorem - Linear Pair Mr. Chin-Sung Lin If two angles form a linear pair, then these angles are supplementary ERHS Math Geometry

42. Theorem - Linear Pair Mr. Chin-Sung Lin If two angles form a linear pair, then these angles are supplementary Draw a diagram like the one below Given:  1 and  2 are linear pair Prove:  1 and  2 are supplementary 1 2 ERHS Math Geometry

43. Theorem - Linear Pair Mr. Chin-Sung Lin StatementsReasons 1. 1 and 2 are linear pair1. Given 2. m1 + m2 = 180 2. Definition of linear pair 3. 1 and 2 are supplementary 3. Definition of supplementary angles 1 2 ERHS Math Geometry

44. Theorem - Right Angles Mr. Chin-Sung Lin If two angles are right angles, then these angles are congruent ERHS Math Geometry

45. Theorem - Right Angles Mr. Chin-Sung Lin If two angles are right angles, then these angles are congruent Draw a diagram like the one below Given:  1 and  2 are right angles Prove:  1 =  2 21 ERHS Math Geometry ~

46. Theorem - Right Angles Mr. Chin-Sung Lin StatementsReasons 1. 1 and 2 are right angles1. Given 2. m1 = 90; m2 = 90 2. Definition of right angle 3. m1 = m2 3. Substitution postulate 4. 1 = 2 4. Definition of congruent angles 21 ERHS Math Geometry ~

47. Theorem - Complementary Angles Mr. Chin-Sung Lin If two angles are complementary to the same angle, then these angles are congruent ERHS Math Geometry

48. Theorem - Complementary Angles Mr. Chin-Sung Lin If two angles are complementary to the same angle, then these angles are congruent Draw a diagram like the one below Given:  1 and  2 are complementary  3 and  2 are complementary Prove:  1 =  3 2 3 1 ERHS Math Geometry ~

49. Theorem - Complementary Angles Mr. Chin-Sung Lin StatementsReasons 1. 1 and 2 are complementary1. Given 3 and 2 are complementary 2. m1 + m2 = 90 2. Definition of complementary m3 + m2 = 90 angles 3. m1 + m2 = m3 + m2 3. Substitution postulate 4. m2 = m24. Reflexive property 5. m1 = m35. Subtraction postulate 6. 1 = 3 6. Definition of congruent angles 2 3 1 ERHS Math Geometry ~

50. Theorem - Supplementary Angles Mr. Chin-Sung Lin If two angles are supplementary to the same angle, then these angles are congruent ERHS Math Geometry

51. Theorem - Supplementary Angles Mr. Chin-Sung Lin If two angles are supplementary to the same angle, then these angles are congruent Draw a diagram like the one below Given:  1 and  2 are supplementary  3 and  2 are supplementary Prove:  1 =  3 1 3 2 ERHS Math Geometry ~

52. Theorem - Supplementary Angles Mr. Chin-Sung Lin StatementsReasons 1. 1 and 2 are supplementary1. Given 3 and 2 are supplementary 2. m1 + m2 = 180 2. Definition of supplementary m3 + m2 = 180 angles 3. m1 + m2 = m3 + m2 3. Substitution postulate 4. m2 = m24. Reflexive property 5. m1 = m35. Subtraction postulate 6. 1 = 3 6. Definition of congruent angles 1 3 2 ERHS Math Geometry ~

53. Theorem - Vertical Angles Mr. Chin-Sung Lin If two angles are vertical angles, then these angles are congruent ERHS Math Geometry

54. Theorem - Vertical Angles Mr. Chin-Sung Lin If two angles are vertical angles, then these angles are congruent Draw a diagram like the one below Given:  1 and  3 are vertical angles Prove:  1 =  3 1 3 2 4 ERHS Math Geometry ~

55. Theorem - Vertical Angles Mr. Chin-Sung Lin StatementsReasons 1. 1 and 3 are vertical angles1. Given 2. 1 and 2 are linear pair 2. Definition of vertical angle 3 and 2 are linear pair 3. m1 + m2 = 180 3. Definition of linear pair m3 + m2 = 180 4. m1 + m2 = m3 + m2 4. Substitution postulate 5. m2 = m25. Reflexive property 6. m1 = m36. Subtraction postulate 7. 1 = 3 7. Definition of congruent angles 1 3 2 4 ERHS Math Geometry ~

56. Theorems of Angle Pairs Review Mr. Chin-Sung Lin Linear pair Right Angles Complementary Angles Supplementary Angles Vertical Angles ERHS Math Geometry

57. Exercise: Theorems of Congruent Angle Pairs ERHS Math Geometry Mr. Chin-Sung Lin

58. Theorem - Complementary Angles Mr. Chin-Sung Lin If two angles are congruent, then their complements are congruent ERHS Math Geometry

59. Theorem - Supplementary Angles Mr. Chin-Sung Lin If two angles are congruent, then their supplements are congruent ERHS Math Geometry

60. Theorem - Right Angles Mr. Chin-Sung Lin If two lines intersect to form congruent angles, then they are perpendicular ERHS Math Geometry

61. Congruent Polygons & Congruent Triangles Mr. Chin-Sung Lin ERHS Math Geometry

62. Congruent Polygons Mr. Chin-Sung Lin Polygons are congruent if and only if there is a one-to- one correspondence between their vertices such that all corresponding sides and corresponding angles are congruent ERHS Math Geometry

63. Congruent Polygons Mr. Chin-Sung Lin Corresponding parts of congruent polygons are congruent ABCD ≅ WXYZ ERHS Math Geometry AB ≅ WX BC ≅ XY CD ≅ YZ DA ≅ ZW  A =  W  B =  X  C =  Y  D =  Z

64. Congruent Polygons Mr. Chin-Sung Lin The polygons will have the same shape and size, but one may be a rotated, or be the mirror image of the other ERHS Math Geometry

65. Congruent Triangles Mr. Chin-Sung Lin AC B Z Y X Two triangles are congruent if the vertices of one triangle can be matched with the vertices of the other triangle such that corresponding angles are congruent and the corresponding sides are congruent ERHS Math Geometry

66. Congruent Triangles Mr. Chin-Sung Lin AC B Z Y X Corresponding parts of congruent triangles are congruent Corresponding parts of congruent triangles are equal in measure ERHS Math Geometry

67. Congruent Triangles Mr. Chin-Sung Lin Corresponding Angles  A ≅  Xm  A = m  X  B ≅  Ym  B = m  Y  C ≅  Zm  C = m  Z AC B Z Y X Corresponding Sides AB ≅ XYAB = XY BC ≅ YZBC = YZ CA ≅ ZXCA = ZX ∆ ABC ≅ ∆ XYZ ERHS Math Geometry

68. Congruent Triangles Mr. Chin-Sung Lin AC B Z Y X ∆ ABC ≅ ∆ XYZ ∆ BAC ≅ ∆ YXZ ∆ CAB ≅ ∆ ZXY Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

69. Congruent Triangles – Exercise Mr. Chin-Sung Lin If ∆ OPQ  ∆ DEF ∆ POQ  ∆ FED  ∆ OQP  ∆ EFD  ∆ QOP  Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

70. Congruent Triangles – Exercise Mr. Chin-Sung Lin If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ OQP  ∆ EFD  ∆ QOP  Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

71. Congruent Triangles – Exercise Mr. Chin-Sung Lin If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ EFD  ∆ QOP  Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

72. Congruent Triangles – Exercise Mr. Chin-Sung Lin If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ DFE ∆ EFD  ∆ QOP  Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

73. Congruent Triangles – Exercise Mr. Chin-Sung Lin If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ DFE ∆ EFD  ∆ PQO ∆ QOP  Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

74. Congruent Triangles – Exercise Mr. Chin-Sung Lin If ∆ OPQ  ∆ DEF ∆ POQ  ∆ EDF ∆ FED  ∆ QPO ∆ OQP  ∆ DFE ∆ EFD  ∆ PQO ∆ QOP  ∆ FDE Congruence can be represented by more than one way, as long as the corresponding vertices in the same order ERHS Math Geometry

75. Equivalence Relation of Congruence Mr. Chin-Sung Lin ERHS Math Geometry

76. Reflexive Property Mr. Chin-Sung Lin AC B ∆ ABC ≅ ∆ ABC Any geometric figure is congruent to itself ERHS Math Geometry

77. Symmetric Property Mr. Chin-Sung Lin AC B If ∆ ABC ≅ ∆ XYZ then ∆ XYZ ≅ ∆ ABC A congruence may be expressed in either order ERHS Math Geometry Z Y X

78. Transitive Property Mr. Chin-Sung Lin AC B If ∆ ABC ≅ ∆ RST and ∆ RST ≅ ∆ XYZ then ∆ ABC ≅ ∆ XYZ Two geometric figures congruent to the same geometric figure are congruent to each other ERHS Math Geometry Z Y X RT S

79. Postulates that Prove Triangles Congruent Mr. Chin-Sung Lin ERHS Math Geometry

80. Postulates that Prove Triangles Congruent Mr. Chin-Sung Lin Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) ERHS Math Geometry

81. Side-Side-Side Congruence (SSS) Mr. Chin-Sung Lin If the three sides of one triangle are congruent, respectively, to the three sides of a second triangle, then two triangles are congruent ERHS Math Geometry

82. Side-Angle-Side Congruence (SAS) Mr. Chin-Sung Lin If two sides and the included angle of one triangle are congruent, respectively, to the ones of another triangle, then two triangles are congruent ERHS Math Geometry

83. Angle-Side-Angle Congruence (ASA) Mr. Chin-Sung Lin If two angles and the included side of one triangle are congruent, respectively, to the ones of another triangle, then two triangles are congruent ERHS Math Geometry

84. Angle-Angle-Side Congruence (AAS) Mr. Chin-Sung Lin If two of corresponding angles and a not-included side are congruent, respectively, to the ones of another triangle, then the triangles are congruent ERHS Math Geometry

85. Side-Side-Angle Case (SSA) Mr. Chin-Sung Lin ERHS Math Geometry

86. Side-Side-Angle Case (SSA) Mr. Chin-Sung Lin The condition does not guarantee congruence, because it is possible to have two incongruent triangles. This is known as the ambiguous case ERHS Math Geometry

87. Angle-Angle-Angle Case (AAA) Mr. Chin-Sung Lin The AAA says nothing about the size of the two triangles and hence shows only similarity and not congruence ERHS Math Geometry

88. Postulates that Prove Triangles Congruent Mr. Chin-Sung Lin Side-Side-Side Congruence (SSS) Side-Angle-Side Congruence (SAS) Angle-Side-Angle Congruence (ASA) Angle-Angle-Side Congruence (AAS) Side-Side-Angle Congruence (SSA) Angle-Angle-Angle Congruence (AAA) ERHS Math Geometry

89. Identify the Postulate Mr. Chin-Sung Lin ERHS Math Geometry

90. Postulate for Proving Congruence Mr. Chin-Sung Lin Given  A ≅  X,  B ≅  Y and AB ≅ XY Prove ∆ ABC ≅ ∆ XYZ A C B X Z Y ERHS Math Geometry

91. Postulate for Proving Congruence Mr. Chin-Sung Lin Given  A ≅  X,  B ≅  Y and AB ≅ XY Prove ∆ ABC ≅ ∆ XYZ A C B X Z Y ASA ERHS Math Geometry

92. Postulate for Proving Congruence Mr. Chin-Sung Lin Given O is the midpoint of AX and BY Prove ∆ ABO ≅ ∆ XYO A B X O Y ERHS Math Geometry

93. Postulate for Proving Congruence Mr. Chin-Sung Lin Given O is the midpoint of AX and BY Prove ∆ ABO ≅ ∆ XYO A B X O Y SAS ERHS Math Geometry

94. Postulate for Proving Congruence Mr. Chin-Sung Lin Given CA is an angle bisector of  DCB, and  B ≅  D Prove ∆ ACD = ∆ ACB A B C D ERHS Math Geometry

95. Postulate for Proving Congruence Mr. Chin-Sung Lin Given CA is an angle bisector of  DCB, and  B ≅  D Prove ∆ ACD = ∆ ACB A B C D AAS ERHS Math Geometry

96. Postulate for Proving Congruence Mr. Chin-Sung Lin Given ∆ ABC is an isosceles triangle and BD is the median Prove ∆ ABD ≅ ∆ CBD A C B D ERHS Math Geometry

97. Postulate for Proving Congruence Mr. Chin-Sung Lin Given ∆ ABC is an isosceles triangle and BD is the median Prove ∆ ABD ≅ ∆ CBD SSS A C B D ERHS Math Geometry

98. Postulate for Proving Congruence Mr. Chin-Sung Lin Given DE ≅ AE, BE ≅ CE, and  1 ≅  2 Prove ∆ DBC ≅ ∆ ACB A C B D E 12 ERHS Math Geometry

99. Postulate for Proving Congruence Mr. Chin-Sung Lin A C B D E 12 SAS ERHS Math Geometry Given DE ≅ AE, BE ≅ CE, and  1 ≅  2 Prove ∆ DBC ≅ ∆ ACB

100. Identify Congruent Triangles & the Postulate Mr. Chin-Sung Lin ERHS Math Geometry

101. Congruent Triangles Mr. Chin-Sung Lin Given: AB  XY, BC  YZ, and  B   Y Prove: A B C X Y Z ERHS Math Geometry

102. Congruent Triangles Mr. Chin-Sung Lin Given: AB  XY, BC  YZ, and  B   Y Prove: ∆ ABC  ∆ XYZ ERHS Math Geometry A B C X Y Z

103. Congruent Triangles Mr. Chin-Sung Lin A B C X Y Z SAS ERHS Math Geometry Given: AB  XY, BC  YZ, and  B   Y Prove: ∆ ABC  ∆ XYZ

104. Congruent Triangles Mr. Chin-Sung Lin Given: AB  AC, and BD  CD Prove: A B C D ERHS Math Geometry

105. Congruent Triangles Mr. Chin-Sung Lin Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD A B C D ERHS Math Geometry

106. Congruent Triangles Mr. Chin-Sung Lin Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD SSS A B C D ERHS Math Geometry

107. Congruent Triangles Mr. Chin-Sung Lin Given: AO  XO, and BO  YO Prove: A B X O Y ERHS Math Geometry

108. Congruent Triangles Mr. Chin-Sung Lin Given: AO  XO, and BO  YO Prove: ∆ AOB  ∆ XOY A B X O Y ERHS Math Geometry

109. Congruent Triangles Mr. Chin-Sung Lin Given: AO  XO, and BO  YO Prove: ∆ AOB  ∆ XOY A B X O Y SAS ERHS Math Geometry

110. Congruent Triangles Mr. Chin-Sung Lin Given:  D   B, and  DAC   BAC Prove: A B C D ERHS Math Geometry

111. Congruent Triangles Mr. Chin-Sung Lin Given:  D   B, and  DAC   BAC Prove: ∆ ABC  ∆ ADC A B C D ERHS Math Geometry

112. Congruent Triangles Mr. Chin-Sung Lin Given:  D   B, and  DAC   BAC Prove: ∆ ABC  ∆ ADC A B C D AAS ERHS Math Geometry

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

114. Congruent Triangles Mr. Chin-Sung Lin Given:  B   C, and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F ERHS Math Geometry

115. Congruent Triangles Mr. Chin-Sung Lin Given:  B   C, and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F ASA ERHS Math Geometry

116. Two-Column Proof of Congruent Triangles ERHS Math Geometry Mr. Chin-Sung Lin

117. Prove Congruent Triangles Mr. Chin-Sung Lin Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD A B C D ERHS Math Geometry

118. Prove Congruent Triangles Mr. Chin-Sung Lin Given: AB  AC, and BD  CD Prove: ∆ ABD  ∆ ACD A B C D ERHS Math Geometry

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

120. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1. AB  AC, and BD  CD 1. Given ERHS Math Geometry A B C D

121. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1. AB  AC, and BD  CD 1. Given 2. AD  AD 2. Reflexive property ERHS Math Geometry A B C D

122. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1. AB  AC, and BD  CD 1. Given 2. AD  AD 2. Reflexive property 3. ∆ ABD  ∆ ACD 3. SSS ERHS Math Geometry A B C D

123. Congruent Triangles Mr. Chin-Sung Lin Given:  B   C, and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F ERHS Math Geometry

124. Congruent Triangles Mr. Chin-Sung Lin Given:  B   C, and AB  AC Prove: ∆ ABF  ∆ ACE A B C D E F ERHS Math Geometry

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

126. 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

127. 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

128. 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

129. Prove Congruent Triangles Mr. Chin-Sung Lin Given: ∆ ABC, AD is the bisector of BC, and AD  BC Prove: ∆ ABD  ∆ ACD A B C D ERHS Math Geometry

130. Prove Congruent Triangles Mr. Chin-Sung Lin A B C D ERHS Math Geometry Given: ∆ ABC, AD is the bisector of BC, and AD  BC Prove: ∆ ABD  ∆ ACD

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

132. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC ERHS Math Geometry A B C D

133. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD 2. Reflexive property ERHS Math Geometry A B C D

134. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD 2. Reflexive property 3.D is the midpoint of BC3. Definition of segment bisector ERHS Math Geometry A B C D

135. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD 2. Reflexive property 3.D is the midpoint of BC3. Definition of segment bisector 4.BD  DC 4. Definition of midpoint ERHS Math Geometry A B C D

136. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD 2. Reflexive property 3.D is the midpoint of BC3. Definition of segment bisector 4.BD  DC 4. Definition of midpoint 5.ADB and ADC are right 5. Definition of perpendicular angles lines ERHS Math Geometry A B C D

137. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD 2. Reflexive property 3.D is the midpoint of BC3. Definition of segment bisector 4.BD  DC 4. Definition of midpoint 5.ADB and ADC are right 5. Definition of perpendicular angles lines 6.ADB  ADC6. Right angles are congruent ERHS Math Geometry A B C D

138. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.∆ ABC, AD is the bisector of BC, 1. Given and AD  BC 2. AD  AD 2. Reflexive property 3.D is the midpoint of BC3. Definition of segment bisector 4.BD  DC 4. Definition of midpoint 5.ADB and ADC are right 5. Definition of perpendicular angles lines 6.ADB  ADC6. Right angles are congruent 7. ∆ ABD  ∆ ACD 7. SAS ERHS Math Geometry A B C D

139. Congruent Triangles Mr. Chin-Sung Lin Given: O is the midpoint of AX, and B  Y Prove: ∆ AOB  ∆ XOY A B X O Y ERHS Math Geometry

140. Congruent Triangles Mr. Chin-Sung Lin Given: O is the midpoint of AX, and B  Y Prove: ∆ AOB  ∆ XOY ERHS Math Geometry A B X O Y

141. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons ERHS Math Geometry A B X O Y

142. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.O is the midpoint of AX, and 1. Given B  Y ERHS Math Geometry A B X O Y

143. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.O is the midpoint of AX, and 1. Given B  Y 2. AO  XO 2. Definition of midpoint ERHS Math Geometry A B X O Y

144. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.O is the midpoint of AX, and 1. Given B  Y 2. AO  XO 2. Definition of midpoint 3.AOB  XOY 3. Vertical angle theorem ERHS Math Geometry A B X O Y

145. Prove Congruent Triangles Mr. Chin-Sung Lin StatementsReasons 1.O is the midpoint of AX, and 1. Given B  Y 2. AO  XO 2. Definition of midpoint 3.AOB  XOY 3. Vertical angle theorem 4. ∆ AOB  ∆ XOY 4. AAS ERHS Math Geometry A B X O Y

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

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