1. 2.4 Building a System of Geometry Knowledge
Obj: Identify Properties to use in Proofs

2. Algebraic Properties of Equality
Let a, b, and c be real numbers or expressions representing real numbers
Addition prop: if a=b, then a+c=b+c
Subtraction Prop if a=b, then a-c=b-c
Multiplication Prop: if a=b, then ac=bc
Division Prop: if a=b and c not equal 0, then a/c=b/c
Substitution Prop: if a=b, you may replace a with b in any true equation containing and the resulting equation will still be true.
* You can perform the same operation to both side of an equation and the equality will still hold

3. Example of use of Algebraic Properties
Solve the equation and justify each step
2x + 6 = 24

4. Equivalence Properties of Equality
Reflexive Prop: for any real number a, a=a
Symmetric Prop: For all real numbers a and b, if a=b, then b=a
Transitive Prop: For all real numbers a, b, and c, if a=b and b=c, then a=c.

5. Equivalence Properties of Congruence
Reflexive Prop: figure A figure A
Symmetric Prop: if figure A figure B, then figure B figure A
Transitive Prop: If figure A figure B, and figure B figure C, then figure A figure C

6. Two- Column Proof
A proof in which the statements are written in the left-hand column and the reasons are given in the right-hand column.

7. How to Prove a Theorem Include the following parts
Given (hypothesis of the theorem)
Prove (Conclusion of the theorem)
Picture
2 columns of statements and reasons
Conclusion

8. How long is AC? How long is BD?
What is true about AC and BD?
Why?

9. *First think through before writing!!!
Ex: Given AB = CD. Prove that AC=BD A B C D
Statement Reasons

10. Overlapping Segment Theorem
Given a segment with points A, B, C, and D arranged as shown, the following statements are true:
1) If AB=CD, then AC=BD
2) If AC=BD, then AB=CD

11. Overlapping Angle Theorem
Given <AOD with pts. B and C in its interior, the following statements are true:
1) If m<AOB=m<COD, then m<AOC=m<BOD
2) if m<AOC=m<BOD, then m<AOB=m<COD

12. Practice: Fill in the missing pieces of the proof for the 2nd part of the overlapping segment theorem
Given: ____________ Picture:
Prove: AB = CD
Statements Reasons
AC = BD 1) __________________
_________________ 2) Segment Addition Postulate
_________________ 3) Segment Addition Postulate
AB + BC = CD + BC 4) _____________________
AB = CD 5) _____________________

13. Practice Proof Given: <ABC ≅ <EFG <1 ≅ <3
Prove: <2 ≅ <4

14. Given: <ABC≅<EFG <1 ≅ <3 Prove: <2 ≅ <4
Statement Reason