1. Triangles and Angles

2. 2 Standard/Objectives: Standard 3: Students will learn and apply geometric concepts. Objectives: Classify triangles by their sides and angles. Find angle measures in triangles DEFINITION: A triangle is a figure formed by three segments joining three non- collinear points.

3. 3 Names of triangles Equilateral —3 congruent sides Isosceles Triangle—2 congruent sides Scalene— no congruent sides Triangles can be classified by the sides or by the angle

4. 4 Acute Triangle 3 acute angles

5. 5 Equiangular triangle 3 congruent angles. An equiangular triangle is also acute.

6. 6 Right Triangle 1 right angle Obtuse Triangle

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8. 8 Parts of a triangle Each of the three points joining the sides of a triangle is a vertex.(plural: vertices). A, B and C are vertices. Two sides sharing a common vertext are adjacent sides. The third is the side opposite an angle adjacent Side opposite  A

9. 9 Right Triangle Red represents the hypotenuse of a right triangle. The sides that form the right angle are the legs. hypotenuse leg

10. 10 An isosceles triangle can have 3 congruent sides in which case it is equilateral. When an isosceles triangle has only two congruent sides, then these two sides are the legs of the isosceles triangle. The third is the base. leg base Isosceles Triangles

11. 11 Identifying the parts of an isosceles triangle Explain why ∆ABC is an isosceles right triangle. In the diagram you are given that  C is a right angle. By definition, then ∆ABC is a right triangle. Because AC = 5 ft and BC = 5 ft; AC  BC. By definition, ∆ABC is also an isosceles triangle. About 7 ft. 5 ft

12. 12 Identifying the parts of an isosceles triangle Identify the legs and the hypotenuse of ∆ABC. Which side is the base of the triangle? Sides AC and BC are adjacent to the right angle, so they are the legs. Side AB is opposite the right angle, so it is t he hypotenuse. Because AC  BC, side AB is also the base. About 7 ft. 5 ft leg Hypotenuse & Base

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14. 14 Using Angle Measures of Triangles Smiley faces are interior angles and hearts represent the exterior angles Each vertex has a pair of congruent exterior angles; however it is common to show only one exterior angle at each vertex.

15. 15 Ex. 3 Finding an Angle Measure. 65  xx Exterior Angle theorem: m  1 = m  A +m  1 (2x+10)  x  + 65  = (2x + 10)  65 = x +10 55 = x

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17. 17 Finding angle measures Corollary to the triangle sum theorem The acute angles of a right triangle are complementary. m  A + m  B = 90  2x x

18. 18 Finding angle measures X + 2x = 90 3x = 90 X = 30  So m  A = 30  and the m  B=60  2x x C B A

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21. Congruence and Triangles

22. 22 Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives:  Identify congruent figures and corresponding parts  Prove that two triangles are congruent

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24. 24 Identifying congruent figures  Two geometric figures are congruent if they have exactly the same size and shape. CONGRUENT NOT CONGRUENT

25. 25 Triangles Corresponding angles  A ≅  P  B ≅  Q  C ≅  R Corresponding Sides AB ≅ PQ BC ≅ QR CA ≅ RP A B C Q P R

26. 26 ZZZZ  If Δ ABC is  to Δ XYZ, which angle is  to  C?

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28. 28 Thm 4.3 3rd angles thm  If 2  s of one Δ are  to 2  s of another Δ, then the 3rd  s are also .

29. 29 Ex: find x ) )) 22 o 87 o ) )) (4x+15) o

30. 30 Ex: continued 22+87+4x+15=1804x+15=714x=56x=14

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32. 32 Ex: ABCD is  to HGFE, find x and y. 4x-3=9 5y-12=113 4x=12 5y=125 x=3 y=25 91 ° 86 ° 113° 9 cm (5y-12)° H G F E 4x – 3 cm

33. 33 Thm 4.4 Props. of  Δs  Reflexive prop of Δ  - Every Δ is  to itself (ΔABC  ΔABC).  Symmetric prop of Δ  - If ΔABC  ΔPQR, then ΔPQR  ΔABC.  Transitive prop of Δ  - If ΔABC  ΔPQR & ΔPQR  ΔXYZ, then ΔABC  ΔXYZ. A B C P Q R X Y Z

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39. Proving Δs are  : SSS and SAS

40. 40 Standards/Benchmarks Standard 2: Students will learn and apply geometric concepts Objectives: Prove that triangles are congruent using the SSS and SAS Congruence Postulates. Use congruence postulates in real life problems such as bracing a structure.

41. 41 Remember?  As of yesterday, Δs could only be  if ALL sides AND angles were  NNOT ANY MORE!!!! TThere are two short cuts to add.

42. 42 Post. 19 Side-Side-Side (SSS)  post If 3 sides of one Δ are  to 3 sides of another Δ, then the Δs are .

43. 43 Meaning: If seg AB  seg ED, seg AC  seg EF & seg BC  seg DF, then ΔABC  ΔEDF. ___ A BC E DF

44. 44 Given: seg QR  seg UT, RS  TS, QS=10, US=10 Prove: ΔQRS  ΔUTS Q R S T U 10

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47. 47 Proof Statements Reasons 1. 1. given 2. QS=US 2. subst. prop. = 3. Seg QS  seg US 3. Def of  segs. 4. Δ QRS  Δ UTS 4. SSS post

48. 48 Post. 20 Side-Angle-Side post. (SAS) If 2 sides and the included  of one Δ are  to 2 sides and the included  of another Δ, then the 2 Δs are .

49. 49 If seg BC  seg YX, seg AC  seg ZX, and  C   X, then ΔABC  ΔZXY. B A C X Y Z ) (

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51. 51 Given: seg WX  seg. XY, seg VX  seg ZX, Prove: Δ VXW  Δ ZXY 12 W V X Z Y

52. 52 Proof Statements Reasons 1. seg WX  seg. XY 1. given seg. VX  seg ZX 2.  1   2 2. vert  s thm 3. Δ VXW  Δ ZXY 3. SAS post

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54. 54 Given: seg RS  seg RQ and seg ST  seg QT Prove: Δ QRT  Δ SRT. Q R S T

55. 55 Proof Statements Reasons 1. Seg RS  seg RQ1. Given seg ST  seg QT 2. Seg RT  seg RT2. Reflex prop  3. Δ QRT  Δ SRT3. SSS post

56. 56 Given: seg DR  seg AG and seg AR  seg GR Prove: Δ DRA  Δ DRG. D A R G

57. 57 Proof Statements 1.seg DR  seg AG Seg AR  seg GR 2. seg DR  Seg DR 3.  DRG &  DRA are rt.  s 4.  DRG   DRA 5. Δ DRG  Δ DRA Reasons 1.Given 2.reflex. Prop of  3.  lines form 4 rt.  s 4. Rt.  s thm 5. SAS post.

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61. Proving Triangles are Congruent: ASA and AAS

62. 62 Objectives: 1.Prove that triangles are congruent using the ASA Congruence Postulate and the AAS Congruence Theorem 2.Use congruence postulates and theorems in real-life problems.

63. 63 Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

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65. 65 Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non- included side of a second triangle, then the triangles are congruent.

66. 66 Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem Given:  A   D,  C   F, BC  EF Prove: ∆ABC  ∆DEF

67. 67 Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem You are given that two angles of ∆ABC are congruent to two angles of ∆DEF. By the Third Angles Theorem, the third angles are also congruent. That is,  B   E. Notice that BC is the side included between  B and  C, and EF is the side included between  E and  F. You can apply the ASA Congruence Postulate to conclude that ∆ABC  ∆DEF.

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69. 69 Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

70. 70 Ex. 1 Developing Proof A. In addition to the angles and segments that are marked,  EGF  JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

71. 71 Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

72. 72 Ex. 1 Developing Proof B. In addition to the congruent segments that are marked, NP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

73. 73 Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning. UZ ║WX AND UW ║WX. 1 2 3 4

74. 74 Ex. 1 Developing Proof The two pairs of parallel sides can be used to show  1   3 and  2   4. Because the included side WZ is congruent to itself, ∆WUZ  ∆ZXW by the ASA Congruence Postulate. 1 2 3 4

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76. 76 Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that  ABD and  EBC are congruent. You are given that BD  BC. Use the fact that AD ║EC to identify a pair of congruent angles.

77. 77 Proof: Statements: 1.BD  BC 2.AD ║ EC 3.  D   C 4.  ABD   EBC 5.∆ABD  ∆EBC Reasons: 1.

78. 78 Proof: Statements: 1.BD  BC 2.AD ║ EC 3.  D   C 4.  ABD   EBC 5.∆ABD  ∆EBC Reasons: 1. Given

79. 79 Proof: Statements: 1.BD  BC 2.AD ║ EC 3.  D   C 4.  ABD   EBC 5.∆ABD  ∆EBC Reasons: 1.Given 2.Given

80. 80 Proof: Statements: 1.BD  BC 2.AD ║ EC 3.  D   C 4.  ABD   EBC 5.∆ABD  ∆EBC Reasons: 1.Given 2.Given 3.Alternate Interior Angles

81. 81 Proof: Statements: 1.BD  BC 2.AD ║ EC 3.  D   C 4.  ABD   EBC 5.∆ABD  ∆EBC Reasons: 1.Given 2.Given 3.Alternate Interior Angles 4.Vertical Angles Theorem

82. 82 Proof: Statements: 1.BD  BC 2.AD ║ EC 3.  D   C 4.  ABD   EBC 5.∆ABD  ∆EBC Reasons: 1.Given 2.Given 3.Alternate Interior Angles 4.Vertical Angles Theorem 5.ASA Congruence Theorem

83. 83 Note: You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that  D   C and  A   E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.

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87. Using Congruent Triangles

88. 88 Objectives: Use congruent triangles to plan and write proofs. Use congruent triangles to prove constructions are valid.

89. 89 Planning a proof Knowing that all pairs of corresponding parts of congruent triangles are congruent can help you reach conclusions about congruent figures.

90. 90 Planning a proof For example, suppose you want to prove that  PQS ≅  RQS in the diagram shown at the right. One way to do this is to show that ∆PQS ≅ ∆RQS by the SSS Congruence Postulate. Then you can use the fact that corresponding parts of congruent triangles are congruent to conclude that  PQS ≅  RQS.

91. 91 Ex. 1: Planning & Writing a Proof Given: AB ║ CD, BC ║ DA Prove: AB ≅CD Plan for proof: Show that ∆ABD ≅ ∆CDB. Then use the fact that corresponding parts of congruent triangles are congruent.

92. 92 Ex. 1: Planning & Writing a Proof Solution: First copy the diagram and mark it with the given information. Then mark any additional information you can deduce. Because AB and CD are parallel segments intersected by a transversal, and BC and DA are parallel segments intersected by a transversal, you can deduce that two pairs of alternate interior angles are congruent.

93. 93 Ex. 1: Paragraph Proof Because AD ║CD, it follows from the Alternate Interior Angles Theorem that  ABD ≅  CDB. For the same reason,  ADB ≅  CBD because BC ║DA. By the Reflexive property of Congruence, BD ≅ BD. You can use the ASA Congruence Postulate to conclude that ∆ABD ≅ ∆CDB. Finally because corresponding parts of congruent triangles are congruent, it follows that AB ≅ CD.

94. 94 Ex. 2: Planning & Writing a Proof Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Plan for proof: Prove that ∆MAS ≅ ∆TAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that  M ≅  T. Because these angles are formed by two segments intersected by a transversal, you can conclude that MS ║ TR.

95. 95 Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3.  MAS ≅  TAR 4.∆MAS ≅ ∆TAR 5.  M ≅  T 6.MS ║ TR Reasons: 1.Given

96. 96 Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3.  MAS ≅  TAR 4.∆MAS ≅ ∆TAR 5.  M ≅  T 6.MS ║ TR Reasons: 1.Given 2.Definition of a midpoint

97. 97 Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3.  MAS ≅  TAR 4.∆MAS ≅ ∆TAR 5.  M ≅  T 6.MS ║ TR Reasons: 1.Given 2.Definition of a midpoint 3.Vertical Angles Theorem

98. 98 Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3.  MAS ≅  TAR 4.∆MAS ≅ ∆TAR 5.  M ≅  T 6.MS ║ TR Reasons: 1.Given 2.Definition of a midpoint 3.Vertical Angles Theorem 4.SAS Congruence Postulate

99. 99 Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3.  MAS ≅  TAR 4.∆MAS ≅ ∆TAR 5.  M ≅  T 6.MS ║ TR Reasons: 1.Given 2.Definition of a midpoint 3.Vertical Angles Theorem 4.SAS Congruence Postulate 5.Corres. parts of ≅ ∆’s are ≅

100. 100 Given: A is the midpoint of MT, A is the midpoint of SR. Prove: MS ║TR. Statements: 1.A is the midpoint of MT, A is the midpoint of SR. 2.MA ≅ TA, SA ≅ RA 3.  MAS ≅  TAR 4.∆MAS ≅ ∆TAR 5.  M ≅  T 6.MS ║ TR Reasons: 1.Given 2.Definition of a midpoint 3.Vertical Angles Theorem 4.SAS Congruence Postulate 5.Corres. parts of ≅ ∆’s are ≅ 6.Alternate Interior Angles Converse.

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102. 102 Ex. 3: Using more than one pair of triangles. Given:  1 ≅  2,  3≅  4. Prove ∆BCE≅∆DCE Plan for proof: The only information you have about ∆BCE and ∆DCE is that  1 ≅  2 and that CE ≅CE. Notice, however, that sides BC and DC are also sides of ∆ABC and ∆ADC. If you can prove that ∆ABC≅∆ADC, you can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ∆BCE and ∆DCE. 2 1 4 3

103. 103 Given:  1 ≅  2,  3 ≅  4. Prove ∆BCE ≅ ∆DCE Statements: 1.  1≅  2,  3≅  4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE Reasons: 1.Given 4 3 2 1

104. 104 Given:  1 ≅  2,  3 ≅  4. Prove ∆BCE ≅ ∆DCE Statements: 1.  1≅  2,  3≅  4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE Reasons: 1.Given 2.Reflexive property of Congruence 4 3 2 1

105. 105 Given:  1 ≅  2,  3 ≅  4. Prove ∆BCE ≅ ∆DCE Statements: 1.  1≅  2,  3≅  4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE Reasons: 1.Given 2.Reflexive property of Congruence 3.ASA Congruence Postulate 4 3 2 1

106. 106 Given:  1 ≅  2,  3 ≅  4. Prove ∆BCE ≅ ∆DCE Statements: 1.  1≅  2,  3≅  4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE Reasons: 1.Given 2.Reflexive property of Congruence 3.ASA Congruence Postulate 4.Corres. parts of ≅ ∆’s are ≅ 4 3 2 1

107. 107 Given:  1 ≅  2,  3 ≅  4. Prove ∆BCE ≅ ∆DCE Statements: 1.  1≅  2,  3≅  4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE Reasons: 1.Given 2.Reflexive property of Congruence 3.ASA Congruence Postulate 4.Corres. parts of ≅ ∆’s are ≅ 5.Reflexive Property of Congruence 4 3 2 1

108. 108 Given:  1 ≅  2,  3 ≅  4. Prove ∆BCE ≅ ∆DCE Statements: 1.  1≅  2,  3≅  4 2.AC ≅ AC 3.∆ABC ≅ ∆ADC 4.BC ≅ DC 5.CE ≅ CE 6.∆BCE≅∆DCE Reasons: 1.Given 2.Reflexive property of Congruence 3.ASA Congruence Postulate 4.Corres. parts of ≅ ∆’s are ≅ 5.Reflexive Property of Congruence 6.SAS Congruence Postulate 4 3 2 1

109. 109 Ex. 4: Proving constructions are valid In Lesson 3.5 – you learned to copy an angle using a compass and a straight edge. The construction is summarized on pg. 159 and on pg. 231. Using the construction summarized above, you can copy  CAB to form  FDE. Write a proof to verify the construction is valid.

110. 110 Plan for proof Show that ∆CAB ≅ ∆FDE. Then use the fact that corresponding parts of congruent triangles are congruent to conclude that  CAB ≅  FDE. By construction, you can assume the following statements: –AB ≅ DE Same compass setting is used –AC ≅ DF Same compass setting is used –BC ≅ EF Same compass setting is used

111. 111 Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove  CAB ≅  FDE Statements: 1.AB ≅ DE 2.AC ≅ DF 3.BC ≅ EF 4.∆CAB ≅ ∆FDE 5.  CAB ≅  FDE Reasons: 1.Given 4 3 2 1

112. 112 Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove  CAB ≅  FDE Statements: 1.AB ≅ DE 2.AC ≅ DF 3.BC ≅ EF 4.∆CAB ≅ ∆FDE 5.  CAB ≅  FDE Reasons: 1.Given 2.Given 4 3 2 1

113. 113 Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove  CAB ≅  FDE Statements: 1.AB ≅ DE 2.AC ≅ DF 3.BC ≅ EF 4.∆CAB ≅ ∆FDE 5.  CAB ≅  FDE Reasons: 1.Given 2.Given 3.Given 4 3 2 1

114. 114 Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove  CAB ≅  FDE Statements: 1.AB ≅ DE 2.AC ≅ DF 3.BC ≅ EF 4.∆CAB ≅ ∆FDE 5.  CAB ≅  FDE Reasons: 1.Given 2.Given 3.Given 4.SSS Congruence Post 4 3 2 1

115. 115 Given: AB ≅ DE, AC ≅ DF, BC ≅ EF Prove  CAB ≅  FDE Statements: 1.AB ≅ DE 2.AC ≅ DF 3.BC ≅ EF 4.∆CAB ≅ ∆FDE 5.  CAB ≅  FDE Reasons: 1.Given 2.Given 3.Given 4.SSS Congruence Post 5.Corres. parts of ≅ ∆’s are ≅. 4 3 2 1

116. 116 Given: QS  RP, PT ≅ RT Prove PS ≅ RS Statements: 1.QS  RP 2.PT ≅ RT Reasons: 1.Given 2.Given 4 3 2 1

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122. Isosceles, Equilateral and Right  s Pg 236

123. 123 Standards/Objectives: Standard 2: Students will learn and apply geometric concepts Objectives: Use properties of Isosceles and equilateral triangles. Use properties of right triangles.

124. 124 Isosceles triangle’s special parts  A is the vertex angle (opposite the base)  B and  C are base angles (adjacent to the base) A B C Leg Base

125. 125 Thm 4.6 Base  s thm If 2 sides of a  are , the the  s opposite them are .( the base  s of an isosceles  are  ) A BC If seg AB  seg AC, then  B  C ) (

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127. 127 Thm 4.7 Converse of Base  s thm If 2  s of a  are  the sides opposite them are . ) ( A B C If  B   C, then seg AB  seg AC

128. 128 Corollary to the base  s thm If a triangle is equilateral, then it is equiangular. A B C If seg AB  seg BC  seg CA, then  A  B   C

129. 129 Corollary to converse of the base angles thm If a triangle is equiangular, then it is also equilateral. ) ) ( A B C If  A   B   C, then seg AB  seg BC  seg CA

130. 130 Example: find x and y X=60 Y=30 X Y 120

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133. 133 Thm 4.8 Hypotenuse-Leg (HL)  thm If the hypotenuse and a leg of one right  are  to the hypotenuse and leg of another right , then the  s are . _ _ _ _ A BC X Y Z If seg AC  seg XZ and seg BC  seg YZ, then  ABC   XYZ

134. 134 Given: D is the midpt of seg CE,  BCD and  FED are rt  s and seg BD  seg FD. Prove:  BCD   FED B C D F E

135. 135 Proof Statements 1.D is the midpt of seg CE,  BCD and <FED are rt  s and seg BD  to seg FD 2.Seg CD  seg ED 3.  BCD   FED Reasons 1.Given 2.Def of a midpt 3.HL thm

136. 136 Are the 2 triangles  ) ( ( ) ( ( Yes, ASA or AAS

137. 137 Find x and y. 75 x x y 2x + 75=180 2x=105 x=52.5 y=75 90 x y 60 x=60 y=30

138. 138 Find x. ) ) ( )) (( 56ft 8xft 56=8x 7=x

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142. Triangles and Coordinate Proof

143. 143 Objectives: 1. Place geometric figures in a coordinate plane. 2. Write a coordinate proof.

144. 144 Placing Figures in a Coordinate Plane So far, you have studied two-column proofs, paragraph proofs, and flow proofs. A COORDINATE PROOF involves placing geometric figures in a coordinate plane. Then you can use the Distance Formula (no, you never get away from using this) and the Midpoint Formula, as well as postulate and theorems to prove statements about figures.

145. 145 Ex. 1: Placing a Rectangle in a Coordinate Plane Place a 2-unit by 6-unit rectangle in a coordinate plane. SOLUTION: Choose a placement that makes finding distance easy (along the origin) as seen to the right.

146. 146 Ex. 1: Placing a Rectangle in a Coordinate Plane One vertex is at the origin, and three of the vertices have at least one coordinate that is 0.

147. 147 Ex. 1: Placing a Rectangle in a Coordinate Plane One side is centered at the origin, and the x- coordinates are opposites.

148. 148 Note: Once a figure has been placed in a coordinate plane, you can use the Distance Formula or the Midpoint Formula to measure distances or locate points

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151. 151 Ex. 2: Using the Distance Formula A right triangle has legs of 5 units and 12 units. Place the triangle in a coordinate plane. Label the coordinates of the vertices and find the length of the hypotenuse.

152. 152 Ex. 2: Using the Distance Formula One possible placement is shown. Notice that one leg is vertical and the other leg is horizontal, which assures that the legs meet as right angles. Points on the same vertical segment have the same x-coordinate, and points on the same horizontal segment have the same y-coordinate.

153. 153 Ex. 2: Using the Distance Formula You can use the Distance Formula to find the length of the hypotenuse. d = √(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = √(12-0) 2 + (5-0) 2 = √169 = 13

154. 154 Ex. 3 Using the Midpoint Formula In the diagram, ∆MLN ≅ ∆KLN). Find the coordinates of point L. Solution: Because the triangles are congruent, it follows that ML ≅ KL. So, point L must be the midpoint of MK. This means you can use the Midpoint Formula to find the coordinates of point L.

155. 155 Ex. 3 Using the Midpoint Formula L (x, y) = x 1 + x 2, y 1 +y 2 2 2 Midpoint Formula =160+0, 0+160 2 2 Substitute values = (80, 80) Simplify.

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158. 158 Writing Coordinate Proofs Once a figure is placed in a coordinate plane, you may be able to prove statements about the figure.

159. 159 Ex. 4: Writing a Plan for a Coordinate Proof Write a plan to prove that SQ bisects  PSR. Given: Coordinates of vertices of ∆PQS and ∆RQS. Prove SQ bisects  PSR. Plan for proof: Use the Distance Formula to find the side lengths of ∆PQS and ∆RQS. Then use the SSS Congruence Postulate to show that ∆PQS ≅ ∆RQS. Finally, use the fact that corresponding parts of congruent triangles are congruent (CPCTC) to conclude that  PSQ ≅  RSQ, which implies that SQ bisects  PSR.

160. 160 Ex. 4: Writing a Plan for a Coordinate Proof Given: Coordinates of vertices of ∆PQS and ∆RQS. Prove SQ bisects  PSR.

161. 161 NOTE: The coordinate proof in Example 4 applies to a specific triangle. When you want to prove a statement about a more general set of figures, it is helpful to use variables as coordinates. For instance, you can use variable coordinates to duplicate the proof in Example 4. Once this is done, you can conclude that SQ bisects  PSR for any triangle whose coordinates fit the given pattern.

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163. 163 No coordinates – just variables

164. 164 Ex. 5: Using Variables as Coordinates Right ∆QBC has leg lengths of h units and k units. You can find the coordinates of points B and C by considering how the triangle is placed in a coordinate plane. Point B is h units horizontally from the origin (0, 0), so its coordinates are (h, 0). Point C is h units horizontally from the origin and k units vertically from the origin, so its coordinates are (h, k). You can use the Distance Formula to find the length of the hypotenuse QC. k units h units C (h, k) B (h, 0)Q (0, 0) hypotenuse

165. 165 Ex. 5: Using Variables as Coordinates OC = √(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 = √(h-0) 2 + (k - 0) 2 = √h 2 + k 2 k units h units C (h, k) B (h, 0)Q (0, 0) hypotenuse

166. 166 Ex. 5 Writing a Coordinate Proof Given: Coordinates of figure OTUV Prove ∆OUT  ∆UVO Coordinate proof: Segments OV and UT have the same length. OV = √(h-0) 2 + (0 - 0) 2 =h UT = √(m+h-m) 2 + (k - k) 2 =h

167. 167 Ex. 5 Writing a Coordinate Proof Horizontal segments UT and OV each have a slope of 0, which implies they are parallel. Segment OU intersects UT and OV to form congruent alternate interior angles  TUO and  VOU. Because OU  OU, you can apply the SAS Congruence Postulate to conclude that ∆OUT  ∆UVO.

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