1. Lesson 4.4 - 4.5 Proving Triangles Congruent

2. Triangle Congruency Short-Cuts
If you can prove one of the following short cuts, you have two congruent triangles
SSS (side-side-side)
SAS (side-angle-side)
ASA (angle-side-angle)
AAS (angle-angle-side)
HL (hypotenuse-leg) right triangles only!

3. Built – In Information in Triangles

4. Identify the ‘built-in’ part

5. SAS SAS SSS Shared side Vertical angles Parallel lines -> AIA

6. SOME REASONS For Indirect Information
Def of midpoint
Def of a bisector
Vert angles are congruent
Def of perpendicular bisector
Reflexive property (shared side)
Parallel lines ….. alt int angles
Property of Perpendicular Lines

7. Side-Side-Side (SSS)
AB  DE
BC  EF
AC  DF
ABC   DEF B A C E D F

8. Side-Angle-Side (SAS)B E F A C D
AB  DE
A   D
AC  DF
ABC   DEF
included
angle

9. Angle-Side-Angle (ASA)B E F A C D
A   D
AB  DE
 B   E
ABC   DEF
included
side

10. Angle-Angle-Side (AAS)B E F A C D
A   D
 B   E
BC  EF
ABC   DEF
Non-included
side

11. Warning: No AAA Postulate
There is no such thing as an AAA postulate! E B A C F D
NOT CONGRUENT

12. Warning: No SSA Postulate There is no such thing as an SSA postulate!B F A C D
NOT CONGRUENT

13. Name That Postulate
(when possible)
SAS
ASA
SSA
SSS

14. This is called a common side. It is a side for both triangles.
We’ll use the reflexive property.

15. HL ( hypotenuse leg ) is used
only with right triangles, BUT,
not all right triangles.
ASA HL

16. Name That Postulate SAS SAS SSA SAS Vertical Angles Reflexive Property
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Vertical Angles
Reflexive Property
SSA
SAS

17. Name That Postulate
(when possible)

18. Name That Postulate
(when possible)

19. Closure Question

20. Let’s Practice B  D AC  FE A  F
Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
A  F
For AAS:

21. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
Ex 4 G I H J K
ΔGIH  ΔJIK by AAS

22. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. B A C E D
Ex 5
ΔABC  ΔEDC by ASA

23. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
Ex 6 E A C B D
ΔACB  ΔECD by SAS

24. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible.
Ex 7 J K L M
ΔJMK  ΔLKM by SAS or ASA

25. Determine if whether each pair of triangles is congruent by SSS, SAS, ASA, or AAS. If it is not possible to prove that they are congruent, write not possible. J T
Ex 8 L K V U
Not possible

26. SSS (Side-Side-Side) Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent.
If Side
Side
Side
Then
∆ABC ≅ ∆PQR

27. Example 1 SSS Congruence Postulate. Prove: ∆DEF ≅ ∆JKL
From the diagram,
SSS Congruence Postulate.
∆DEF ≅ ∆JKL

28. SAS (Side-Angle-Side) Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

29. 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, the two triangles are congruent.
If Angle
∠A ≅ ∠D
Side
Angle
∠C ≅ ∠F
Then
∆ABC ≅ ∆DEF

30. Example 2
Prove: ∆SYT ≅ ∆WYX

31. Side-Side-Side Postulate
SSS postulate: If two triangles have three congruent sides, the triangles are congruent.

32. Angle-Angle-Side Postulate
If two angles and a non included side are congruent to the two angles and a non included side of another triangle then the two triangles are congruent.

33. Angle-Side-Angle Postulate
If two angles and the side between them are congruent to the other triangle then the two angles are congruent.

34. Side-Angle-Side Postulate
If two sides and the adjacent angle between them are congruent to the other triangle then those triangles are congruent.

35. Which Congruence Postulate to Use?
1. Decide whether enough information is given in the diagram to prove that triangle PQR is congruent to triangle PQS. If so give a two-column proof and state the congruence postulate.

36. ASA
If 2 angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the 2 triangles are congruent. A Q S C R B

37. AAS
If 2 angles and a nonincluded side of one triangle are congruent to 2 angles and the corresponding nonincluded side of a second triangle, then the 2 triangles are congruent. A Q S C R B

38. AAS Proof If 2 angles are congruent, so is the 3rd Third Angle Theorem
Now QR is an included side, so ASA. A Q S C R B

39. Example
Is it possible to prove these triangles are congruent?

40. Example Is it possible to prove these triangles are congruent?
Yes - vertical angles are congruent, so you have ASA

41. Example
Is it possible to prove these triangles are congruent?

42. Example Is it possible to prove these triangles are congruent?
No. You can prove an additional side is congruent, but that only gives you SS

43. Example
Is it possible to prove these triangles are congruent? 2 1 3 4

44. Example Is it possible to prove these triangles are congruent?
Yes. The 2 pairs of parallel sides can be used to show Angle 1 =~ Angle 3 and Angle 2 =~ Angle 4. Because the included side is congruent to itself, you have ASA. 2 1 3 4

45. Included Angle
The angle between two sides
 H
 G
 I

46. Included Angle Name the included angle: YE and ES ES and YS YS and YE

47. Included Side
The side between two angles GI GH HI

48. Included Side Name the included side: Y and E E and S S and Y YEES SY

49. Side-Side-Side Congruence Postulate
SSS Post. - If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. If
then,

50. Using SSS Congruence Post.
Prove: 1) 2)
1) Given
2) SSS

51. Side-Angle-Side Congruence Postulate
SAS Post. – If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If
then,

52. Included Angle
The angle between two sides
 H
 G
 I

53. Included Angle Name the included angle: YE and ES ES and YS YS and YE

54. Included Side
The side between two angles GI GH HI

55. Included Side Name the included side: Y and E E and S S and Y YEES SY

56. Triangle congruency short-cuts

57. Given: HJ  GI, GJ  JI
Prove: ΔGHJ  ΔIHJ
HJ  GI Given
GJH & IJH are Rt <‘s
Def. ┴ lines
GJH  IJH
Rt <‘s are ≅
GJ  JI Given
HJ  HJ Reflexive Prop
ΔGHJ  ΔIHJ SAS

58. ΔABC  ΔEDC ASA Given: 1  2, A  E and AC  EC
Prove: ΔABC  ΔEDC
1  2 Given
A  E Given
AC  EC Given
ΔABC  ΔEDC ASA

59. Given: ΔABD, ΔCBD, AB  CB,
and AD  CD
Prove: ΔABD  ΔCBD
AB  CB Given
AD  CD Given
BD  BD Reflexive Prop
 ΔABD  ΔCBD SSS

60. Given: LJ bisects IJK,
ILJ   JLK
Prove: ΔILJ  ΔKLJ
LJ bisects IJK Given
IJL  IJH Definition of bisector
ILJ   JLK Given
JL  JL Reflexive Prop
ΔILJ  ΔKLJ ASA

61.  ΔTUV  ΔWXV SAS Given: TV  VW, UV VX Prove: ΔTUV  ΔWXV
TV  VW Given
UV  VX Given
TVU  WVX Vertical angles
 ΔTUV  ΔWXV SAS

62.  ΔHIJ  ΔLKJ ASA Given: Given: HJ  JL, H L Prove: ΔHIJ  ΔLKJ
HJ  JL Given
H L Given
IJH  KJL Vertical angles
 ΔHIJ  ΔLKJ ASA

63. ΔPRT  ΔSTR SAS Given: Quadrilateral PRST with PR  ST, PRT  STR
Prove: ΔPRT  ΔSTR
PR  ST Given
PRT  STR Given
RT  RT Reflexive Prop
ΔPRT  ΔSTR SAS

64. ΔPQR  ΔPSR HL Given: Quadrilateral PQRS, PQ  QR,
PS  SR, and QR  SR
Prove: ΔPQR  ΔPSR
PQ  QR Given
PQR = 90° PQ  QR
PS  SR Given
PSR = 90° PS  SR
QR  SR Given
PR  PR Reflexive Prop
ΔPQR  ΔPSR HL

65. NOT triangle congruency short cuts
Prove it!
NOT triangle congruency short cuts

66. NOT triangle congruency short-cuts
The following are NOT short cuts:
AAA (angle-angle-angle)
Triangles are similar but not necessarily congruent

67. NOT triangle congruency short-cuts
The following are NOT short cuts
SSA (side-side-angle)
SAS is a short cut but the angle is in between both sides!

68. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
Prove it!
CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

69. CPCTC
Once you have proved two triangles congruent using one of the short cuts, the rest of the parts of the triangle you haven’t proved directly are also congruent!
We say: Corresponding Parts of Congruent Triangles are Congruent or CPCTC for short

70. CPCTC example Given: ΔTUV, ΔWXV, TV  WV, TW bisects UX Prove: TU  WX
Statements: Reasons:
TV  WV Given
UV  VX Definition of bisector
TVU  WVX Vertical angles are congruent
ΔTUV  ΔWXV SAS
TU  WX CPCTC

71. Side Side Side
If 2 triangles have 3 corresponding pairs of sides that are congruent, then the triangles are congruent. X P N
3 inches
7 inches
5 inches
AC PX
AB PN
CB XN
Therefore, using SSS,
∆ABC = ∆PNX ~ = A B C
3 inches
7 inches
5 inches

72. Side Angle Side
If two sides and the INCLUDED ANGLE in one triangle are congruent to two sides and INCLUDED ANGLE in another triangle, then the triangles are congruent. X P N
3 inches
5 inches
60°
CA XP
CB XN
C  X
Therefore, by SAS,
∆ABC ∆PNX = ~ A B C
3 inches
5 inches
60°

73. Angle Side Angle
If two angles and the INCLUDED SIDE of one triangle are congruent to two angles and the INCLUDED SIDE of another triangle, the two triangles are congruent. X
CA XP
A P
C X
Therefore, by ASA,
∆ABC ∆PNX = ~
60° A
3 inches
3 inches
70°
70° P N
60° C B

74. Side Angle Angle
Triangle congruence can be proved if two angles and a NON-included side of one triangle are congruent to the corresponding angles and NON-included side of another triangle, then the triangles are congruent.
60°
70°
60°
70°
5 m
5 m
These two triangles are congruent by SAA

75. Remembering our shortcuts
SSS
ASA
SAS
SAA

76. Corresponding parts
When you use a shortcut (SSS, AAS, SAS, ASA, HL) to show that 2 triangles are ,
that means that ALL the corresponding parts are congruent.
EX: If a triangle is congruent by ASA (for instance), then all the other corresponding parts are . A C B G E F
That means that EG  CB FE
What is AC congruent to?

77. Corresponding parts of congruent triangles are congruent.

78. Corresponding Parts of Congruent Triangles are Congruent.
CPCTC
If you can prove congruence using a shortcut, then you KNOW that the remaining corresponding parts are congruent.
You can only use CPCTC in a proof AFTER you have proved congruence.

79. For example: Prove: AB  DE A Statements Reasons B C AC  DF Given
C  F Given
CB  FE Given
ΔABC  ΔDEF SAS
AB  DE CPCTC D F E

81. Using SAS Congruence Prove: Δ VWZ ≅ Δ XWY Δ VWZ ≅ Δ XWY SAS PROOF
Given
Δ VWZ ≅ Δ XWY
Vertical Angles
SAS

82. Proof 1) MB is perpendicular bisector of AP
Given: MB is perpendicular bisector of AP
Prove:
1) MB is perpendicular bisector of AP
2) <ABM and <PBM are right <‘s 3) 4) 5) 6)
1) Given
2) Def of Perpendiculars
3) Def of Bisector
4) Def of Right <‘s
5) Reflexive Property
6) SAS

83. Proof 1) O is the midpoint of MQ and NP 2) 3) 4) 1) Given
Given: O is the midpoint of MQ and NP
Prove:
1) O is the midpoint of MQ and NP 2) 3) 4)
1) Given
2) Def of midpoint
3) Vertical Angles
4) SAS

84. Proof
Given:
Prove: 1) 2) 3)
1) Given
2) Reflexive Property
3) SSS

85. Proof 1) 2) 3) 4) 1) Given 2) Alt. Int. <‘s Thm
Prove: 1) 2) 3) 4)
1) Given
2) Alt. Int. <‘s Thm
3) Reflexive Property
4) SAS

86. Checkpoint
Decide if enough information is given to prove the triangles are congruent. If so, state the congruence postulate you would use.

87. Congruent Triangles in the Coordinate Plane
Use the SSS Congruence Postulate to show that
∆ABC ≅ ∆DEF
Which other postulate could you use to prove the triangles are congruent?

88. Congruent Triangles

89. Standardized Test Practice
EXAMPLE 2
Standardized Test Practice
SOLUTION
By counting, PQ = 4 and QR = 3. Use the Distance Formula to find PR. d = y 2 – 1 ( ) x +

90. Write a proof.
GIVEN
KL NL, KM NM
PROVE
KLM NLM
Proof
It is given that
KL NL and KM NM
By the Reflexive Property,
LM LM.
So, by the SSS Congruence Postulate,
KLM NLM

91. GUIDED PRACTICE
for Example 1
Decide whether the congruence statement is true. Explain your reasoning.
DFG HJK
SOLUTION
Three sides of one triangle are congruent to three sides of second triangle then the two triangle are congruent.
Side DG HK, Side DF JH,and Side FG JK.
So by the SSS Congruence postulate, DFG HJK.
Yes. The statement is true.

92. Included Angle
The angle between two sides
 H
 G
 I

93. Included Angle Name the included angle: YE and ES ES and YS YS and YE

94. In the diagram at the right, what postulate or theorem can you use to prove that
RST VUT
Given
S U
Given
RS UV
Δ RST ≅ Δ VUT
SAA
Vertical angles
RTS UTV

95. Now For The Fun Part…
Proofs!

96. Given: JO  SH; O is the midpoint of SH Prove:  SOJ  HOJ

97. Given: BC bisects AD
A   D
Prove: AB  DC
A C E
B D

98. WORK