# Theorem 2.1

```Proof – A logical argument that shows a statement is true
Theorem – A statement that has been formally proven
Theorem 2.1: Segments congruence is reflexive, symmetric, and transitive.
Reflexive property of  : For any segment AB , ________________.
Examples: (a) JK  _______
(b)
XY  _______
Symmetric property of  : If AB  CD , then ___________________.
Example: (a) If GH  WO , then _________________
(b) If JK  LM , then _________________
Transitive property of  : If AB  CD and CD  EF , then _________________.
Example: (a) If JK  LM and LM  HA , then _________________
(b) If XY  SU and SU  TK , then _________________
Theorem 2.2: Angle congruence is reflexive, symmetric, and transitive.
Reflexive property of  : For any angle A, ______________.
Examples: (a) K  _______
(b)
Y  _______
Symmetric property of  : If  A   B, then _________________.
Example: (a) If G  H , then _________________
(b) If LMN  RST , then _________________
Transitive property of  : If  A   B and  B   C, then __________________.
Example: (a) If J  Z and Z  H , then _________________
(b) If YAB  LOG and LOG  UVT , then ____________________
A two-column geometric proof consists of a list of statements, and the reasons to show that
those statements are true.
1. Read the given(s) and what is to be proved. This is the beginning and end to the proof.
2. Draw a figure that illustrates what is to be proved, if one is not already given.
3. Mark the figure according to what you can deduce about it from the information given.
4. Write the steps down carefully, without skipping even the simplest one, and be sure to number each
step.
1.
E
Given: m2  m3 , mAXD  mAXC
Prove: m1  m4
D
2
3
A
1
X
4
B
C
A
2.
Given: BD bisects ABC
Prove: mABD  2 mABD
B
D
C
1. BD bisects  ABC
1. ___________________
2. __________________________
2. Definition of angle bisector
3. __________________________
3. Definition of congruent angles
4. mABD  mDBC  mABC
4. ___________________________
5. mABD  _________  mABC
5. Substitution Property of Equality
6. __________________________
6. Distributive Property (Simplify)
3. Given: AB  CD M is the midpoint of AB N is the midpoint of CD
Prove: AM  CN
Sketch:
Statements
1. ______________________
Reasons
1. ________________________________
______________________
______________________
2. AB = CD
2. ________________________________
3. AM = MB, CN = ND
3. ________________________________
4. ______________________
______________________
5. AM + MB = CN + ND
5. ________________________________
6. AM + AM = CN + CN
6. ________________________________
7. 2AM = 2CN
7. ________________________________
8. ______________________
8. Division Property of =
9. AM  CN
9. ________________________________
```