1. Congruent Triangles
Chapter 5

2. Objectives Identify corresponding parts of congruent triangles.
Show triangles are congruent using the SSS, SAS, and ASA Congruence Postulates, and the AAS and HL Congruence Theorems.
Use angle bisectors and perpendicular bisectors to compute angle measures and segment lengths in situations involving triangles.
Reflect figures over lines and use reflections to discover lines of symmetry in a figure.

3. Essential Questions
How does the important property of congruence relate to triangles?
How can you identify the corresponding parts of congruent triangles?

4. Sections 5.1 Congruence and Triangles
5.2 Proving Triangles are Congruent: SSS and SAS
5.3 Proving Triangles are Congruent: ASA and AAS
5.4 Hypotenuse-Leg Congruence Theorem: HL
5.5 Using Congruent Triangles
5.6 Angle Bisectors and Perpendicular Bisectors
5.7 Reflections and Symmetry

5. Congruence and Triangles
Section 5.1

6. Objectives:
Identify congruent triangles and corresponding parts.

7. Key Vocabulary
Congruent
Congruent Figures
Corresponding Parts
CPCTC

8. Congruence Congruent Figures Not Congruent
If two geometric figures or polygons have exactly the same shape and size, they are congruent.
Congruent Figures
While positioned differently, figures 1, 2, and 3 are exactly the same shape and size.
Not Congruent
Figures 4 and 5 are exactly the same shape, but not the same size.
Figures 5 and 6 are the same size, but not exactly the same shape.

9. Two figures are congruent if they are the same size and same shape.

10. Congruent figures can be rotations of one another.

11. Congruent figures can be reflections of one another.

12. Corresponding Parts
If two polygons are congruent, then all parts of one polygon are congruent to the corresponding parts (or matching parts) of the other polygon.
Corresponding parts include corresponding angles and corresponding sides.

13. Corresponding Parts
To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS.
In a congruence statement, the order of the vertices indicates the corresponding parts.

14. Example: Congruent polygons and Corresponding Parts

15. Definition of Congruent Polygons
Two polygons are congruent if and only if their corresponding parts are congruent.
Example
Then:
Given:

16. Congruent polygons and Corresponding Parts
When you write a congruence statement such as ABC  DEF, you are also stating which parts are congruent.
Therefore, valid congruence statements for congruent polygons list corresponding vertices in the same order.
Given the valid congruence statement ∆ABC≅∆DEF
Other valid congruence statements; ∆BCA≅∆EFD or ∆CBA≅∆FED or ∆CAB≅∆FDE
Invalid congruence statements; ∆ABC≅∆FED or ∆CAB≅∆DFE

17. ∆ABC is congruent to ∆XYZ
Identifying Corresponding Congruent Parts C Z A B X Y
∆ABC is congruent to ∆XYZ

18. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent.

19. ∆ABC is congruent to ∆XYZ
Corresponding parts of these triangles are congruent.
Corresponding parts are angles and sides that “match.”

20. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent. A X

21. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent. B Y

22. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent. C Z

23. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent. AB XY

24. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent. BC YZ

25. Corresponding parts of these triangles are congruent.
∆ABC is congruent to ∆XYZ C Z A B X Y
Corresponding parts of these triangles are congruent. AC XZ

26. ∆DEF is congruent to ∆QRS

27. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent.

28. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent. D Q

29. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent. E R

30. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent. F S

31. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent. DE QR

32. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent. DF QS

33. Corresponding parts of these triangles are congruent.
∆DEF is congruent to ∆QRS F Q S D E R
Corresponding parts of these triangles are congruent. FE SR

34. Example 1 Corresponding Angles ∆JKL  ∆RST, so J  R.
Given that JKL  RST, list all
corresponding congruent parts.
SOLUTION
The order of the letters in the names of the triangles shows which parts correspond.
Corresponding Angles
∆JKL  ∆RST, so J  R.
Corresponding Sides
∆JKL  ∆RST, so JK  RS.
∆JKL  ∆RST, so K  S.
∆JKL  ∆RST, so KL  ST.
∆JKL  ∆RST, so L  T.
∆JKL  ∆RST, so JL  RT. 34

35. Example 2 a. Corresponding Angles A  F Corresponding Sides AB  FD
The two triangles are congruent. a.
Identify all corresponding
congruent parts. b.
Write a congruence statement.
SOLUTION a.
Corresponding Angles
A  F
Corresponding Sides
AB  FD
B  D
BC  DE
C  E
AC  FE
b. List the letters in the triangle names so that the corresponding angles match. One possible congruence statement is ∆ABC  ∆FDE. 35

36. Your Turn: Given STU  YXZ, list all corresponding congruent parts.
ANSWER
T  X; U  Z
ST  YX; TU  XZ;
SU  YZ; S  Y;

37. Your Turn: Which congruence statement is correct? Why? A. JKL  MNPB.
JKL  NMP C.
JKL  NPM
ANSWER
B; This statement matches up the corresponding vertices in order.

38. Practice Time! Your Turn

39. 1) Are these shapes congruent? Explain.

40. 1) Are these shapes congruent? Explain.
These shapes are congruent because they are both parallelograms of equal size.

41. 2) Are these shapes congruent? Explain.

42. These shapes are not congruent because they are different sizes.
2) Are these shapes congruent? Explain.
These shapes are not congruent because they are different sizes.

43. 3) Are these shapes congruent? Explain.

44. These shapes are congruent because they are the same size.
3) Are these shapes congruent? Explain.
These shapes are congruent because they are the same size.

45. Name all corresponding parts.
4) ∆BAD is congruent to ∆THE
Name all corresponding parts. D E A B T H

46. 4) ∆BAD is congruent to ∆THE
Name all corresponding parts. D E A B T H
ANGLES
SIDES B T BA TH A H AD HE D E DB ET

47. 5) ∆FGH is congruent to ∆JKL
Name all corresponding parts. F J G H K L

48. 5) ∆FGH is congruent to ∆JKL
Name all corresponding parts. F J G H K L
ANGLES
SIDES F J FG JK H L GH KL G K HF LJ

49. 6) ∆QRS is congruent to ∆BRX
Name all corresponding parts. S R B Q X

50. 6) ∆QRS is congruent to ∆BRX
Name all corresponding parts. S R B Q X
ANGLES
SIDES Q B QR BR S X QS BX R R SR XR

51. 7) ∆EFG is congruent to ∆FGH
Name all corresponding parts. E H G F

52. 7) ∆EFG is congruent to ∆FGH
Name all corresponding parts. E H G F
ANGLES
SIDES E H EF HF F F EG HG G G GF GF

53. Corresponding Parts of Congruent Triangles are Congruent
Informal Geometry
4/28/2017
CPCTC
Stands for
Corresponding Parts of Congruent Triangles are Congruent

54. Definition CPCTC
The bi-conditional phrase “if and only if” in the congruent polygon definition means that both the conditional and its converse are true.
Therefore, definition of CPCTC is;
If the corresponding parts of two triangles are congruent, then the two triangles are congruent.
AND
If two triangles are congruent, then the corresponding parts of the two triangles are congruent.

55. O because ________. CPCTC CPCTC Practice
If CAT  DOG, then A  ___
because ________. O
CPCTC O D G C A T
Add markings!

56. Q CPCTC B CPCTC CPCTC Practice If FJH  QRS, then ___
and F  ___ because _______. Q
CPCTC
If XYZ  ABC, then ___
and Y  ___ because _______. B
CPCTC

57. Example 3: Naming Congruent Corresponding Parts
Given: ∆PQR  ∆STW
Identify all pairs of corresponding congruent parts.
Angles: P  S, Q  T, R  W
Sides: PQ  ST, QR  TW, PR  SW

58. Your Turn
If polygon LMNP  polygon EFGH, identify all pairs of corresponding congruent parts.
Angles: L  E, M  F, N  G, P  H
Sides: LM  EF, MN  FG, NP  GH, LP  EH

59. Example 4 D  G DE  GE DEF  GEF DF  GF DFE  GFE EF  EF
Use the two triangles at the right. a.
Identify all corresponding
congruent parts. b.
Determine whether the triangles
are congruent. If they are congruent, write a congruence statement. E D F G
SOLUTION a.
Corresponding Angles
D  G
Corresponding Sides
DE  GE
DEF  GEF
DF  GF
DFE  GFE
EF  EF 59

60. Example 4 E D F G b.
All three sets of corresponding angles are congruent and all three sets of corresponding sides are congruent, so the two triangles are congruent. A congruence statement is DEF  GEF. 60

61. Example 5
In the figure, HG || LK. Determine whether the triangles are congruent. If so, write a congruence statement.
SOLUTION
Start by labeling any information you can conclude from the figure. You can list the following angles congruent.
HJG  KJL
Vertical angles are congruent.
H  K
Alternate Interior Angles Theorem
G  L
Alternate Interior Angles Theorem
The congruent sides are marked on the diagram, so
HJ  KJ, HG  KL, and JG  JL. Since all corresponding parts are congruent, HJG  KJL. 61

62. Your Turn:
In the figure, XY || ZW. Determine whether the two triangles are congruent. If they are, write a congruence statement.
ANSWER
yes; Sample answer: XVY  ZVW

63. Example 6 In the diagram, PQR  XYZ. Find the length of XZ.
Find mQ. a. b.
SOLUTION a.
Because XZ  PR, you know that XZ = PR = 10. b.
Because Q  Y, you know that mQ = mY = 95°. 63

64. Your Turn: Given ∆ABC  ∆DEF, find the length of DF and mB. 3; 28°
ANSWER
3; 28°

65. Example 7A: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find the value of x.
BCA and BCD are rt. s.
Def. of  lines.
BCA  BCD
Rt.   Thm.
mBCA = mBCD
Def. of  s
Substitute values for mBCA and mBCD.
(2x – 16)° = 90°
2x = 106
Add 16 to both sides.
x = 53
Divide both sides by 2.

66. Example 7B: Using Corresponding Parts of Congruent Triangles
Given: ∆ABC  ∆DBC.
Find mDBC.
∆ Sum Thm.
mABC + mBCA + mA = 180°
Substitute values for mBCA and mA.
mABC = 180
mABC = 180
Simplify.
Subtract from both sides.
mABC = 40.7
DBC  ABC
Corr. s of  ∆s are  .
mDBC = mABC
Def. of  s.
mDBC  40.7°
Trans. Prop. of =

67. Your Turn Given: ∆ABC  ∆DEF Find the value of x. AB  DE
Corr. sides of  ∆s are .
AB = DE
Def. of  parts.
Substitute values for AB and DE.
2x – 2 = 6
2x = 8
Add 2 to both sides.
x = 4
Divide both sides by 2.

68. Your Turn Given: ∆ABC  ∆DEF Find mF. ∆ Sum Thm.
mEFD + mDEF + mFDE = 180°
ABC  DEF
Corr. s of  ∆ are .
mABC = mDEF
Def. of  s.
mDEF = 53°
Transitive Prop. of =.
Substitute values for mDEF and mFDE.
mEFD = 180
mF = 180
Simplify.
mF = 37°
Subtract 143 from both sides.