1. G-09 Congruent Triangles and their parts
“I can name corresponding sides and angles of two triangles.”

2. Reflexive Property
AB = AB

3. Symmetric Property Transitive Property
If A = B, then B = A
If A = B and B = C, then A = C
Transitive Property

4. Addition, Subtraction, Multiplication, Division Property (=)

5. Distributive Property
If A(B + C), then AB + AC Or
If (B + C)A, then BA + CA
If A = B, then A can be substituted for any B in the expression
Substitution

6. Angle/Segment Addition Postulate

7. Definition of Congruence
If AB = CD, then AB  CD
Congruent segments are segments that have the same length.
Congruent angles are angles that have the same measure.

8. Definition of Vertical Angles
Vertical angles are two nonadjacent angles formed by two intersecting lines.
Vertical Angles are congruent
1 and 2 are vertical angles

9. Definition of Perpendicular Lines
Perpendicular lines intersect to form 90 angles.
Perpendicular lines are form congruent angles

10. Definition of Complementary/Supplementary Angles
Complementary Angles: 2 angles that add up to be 90°
Supplementary Angles: 2 angles that add up to be 90°

11. Definition of Midpoint/Bisector
The midpoint M of AB is the pt that bisects, or divides, the segment into 2 congruent segments.
(segments) If M is the midpt of AB, then AM = MB
An angle bisector is a ray that divides an angle into two congruent angles.
JK bisects LJM; thus LJK  KJM.

12. Definition of Right Angles
All right angles are congruent
If A and B are right angles, then A  B

13. Third Angle Theorem

14. Definition of Congruent Triangles
If two or more triangles have corresponding angles and sides that are congruent, then those triangles are congruent.

16. In a congruence statement, the order of the vertices indicates the corresponding parts.
When you write a statement such as ABC  DEF, you are also stating which parts are congruent.
Helpful Hint

17. Example 1 A. Given: ∆PQR  ∆STU
Identify all pairs of corresponding congruent parts.
Angles:
Sides:

18. Example 1 B. Given: ∆ABC  ∆DEF
Identify all pairs of corresponding congruent parts.
Angles:
Sides:

19. Example 1 C. Given: ∆JKM  ∆LKM
Identify all pairs of corresponding congruent parts.
Angles:
Sides:

20. Example 2
A. Given: polygon ABCD  EFGH

21. Example 2
B. Given: polygon ABCD  EFGH

22. Example 2
C. Given: polygon DEFGH  IJKLM

23. Example 3a:
Given: K is the midpt. of JL,
Prove:
Statement
Reason

24. Statement
Reason
K is the midpt. of
Given
Definition of Midpoint
Reflexive Property
are right angles
Definition of Perpendicular lines
Right angles are congruent
Third Angle Thm.
Definition of Congruent Triangles

25. Given: YWX and YWZ are right angles.
Example 3b
Given: YWX and YWZ are right angles.
YW bisects XYZ. W is the midpoint of XZ. XY  YZ.
Prove: ∆XYW  ∆ZYW

26. Example 3b: Statement Reason YWX and YWZ are right angles. Given
YWX  YWZ
YW bisects XYZ
XYW  ZYW
W is mdpt. of XZ
XW  ZW
YW  YW
X  Z
XY  YZ
∆XYW  ∆ZYW

27. Given: AD bisects BE. BE bisects AD. AB  DE, A  D
Example 3c
Given: AD bisects BE.
BE bisects AD.
AB  DE, A  D
Prove: ∆ABC  ∆DEC

28. Example 3c: Statement Reason A  D Given BCA  DCE ABC  DEC
AB  DE
AD bisects BE, BE bisects AD
BC  EC, AC  DC
∆ABC  ∆DEC

29. Example 3d Given: PR and QT bisect each other. PQS  RTS, QP  RT
Prove: ∆QPS  ∆TRS

30. Example 3d: Statement Reason QP  RT Given PQS  RTS
PR and QT bisect each other
QS  TS, PS  RS
QSP  TSR
QSP  TRS
∆QPS  ∆TRS

31. Example 3e
Use the diagram to prove the following.
Given: MK bisects JL. JL bisects MK. JK  ML. JK || ML.
Prove: ∆JKN  ∆LMN

32. Example 3e: Statement Reason JK  ML Given JK || ML JKN  NML
JL and MK bisect each other.
JN  LN, MN  KN
Vert. s Thm.
Third s Thm.
∆JKN ∆LMN
Def. of  ∆s

33. Example 4a
Given: ∆ABC  ∆DBC.
Find the value of x.
Find mDBC.

34. Example 4b
Given: ∆ABC  ∆DEF
1. Find the value of x.
2. Find mF.

35. Example 4c
Given: ∆ABD  ∆CBD
1. Find the value of x.
2. Find AD.

36. Example 4d
Given: ∆RSU  ∆TSU
1. Find the value of x.
2. Find UT.