1. 2-4 Deductive Reasoning

2. Law of Detachment
Deductive reasoning (sometimes called logical reasoning) is the process of reasoning logically from given statements or facts to a conclusion.
Law of Detachment: If the hypothesis of a true conditional is true, then the conclusion is true.
If p  q is true and p is true, then q is true.

3.  What can you conclude from the given true statements?
If a student gets an A on the Final, then the student will pass the course. Felicia got an A on her History Final.
If a ray divides an angle into two congruent angles, then the ray is an angle bisector. RS divides ARB so that ARS  SRB.
If two angles are adjacent, then they share a common vertex. 1 and 2 share a common vertex.

4. Law of Syllogism
Another law of deductive reasoning is the Law of Syllogism.
The Law of Syllogism allows you to state a conclusion from two true conditional statement when the conclusion of one statement is the hypothesis of the other statement.
If p  q is true and q  r is true, then p  r is true.
Example: If it is July, then you are on summer vacation. If you are on summer vacation, then you work at Smoothie King.
Conclusion: If it is July, then you work at Smoothie King.

5. What can you conclude from the given information?
If a figure is a square, then the figure is a rectangle. If a figure is a rectangle, then the figure has four sides.
If you do gymnastics, then you are flexible. If you do ballet, then you are flexible.
If a whole number ends in 0, then it is divisible by 10. If a whole number is divisible by 10, then it is divisible by 5.

6. 2-5 Reasoning in Algebra and Geometry

7. Proofs A proof is a convincing argument that uses deductive reasoning.
A proof logically shows why a conjecture is true.
A two-column proof lists each statement on the left and the justification (or reason) for each statement on the right.
Each statement MUST follow logically from the steps before it!

8. Writing a Two-Column Proof
Given: m1 = m3 Prove: mAEC = mDEB

9. Writing a Two-Column Proof
Given: AB  CD Prove: AC  BD AB  CD AB + BC = CD + BC AB + BC = AC CD + BC = BD AC = BD AC  BD

10. 2-6 Proving Angles Congruent

11. Theorems A theorem is a conjecture or statement that you prove true.
Vertical Angles Theorem: Vertical angles are congruent.
Congruent Supplements Theorem: If two angles are supplements of the same angle (or congruent angles), then the two angles are congruent.
Congruent Complements Theorem: If two angles are complements of the same angle (or congruent angles), then the two angles are congruent.
Theorem 2-4: All right angles are congruent.
Theorem 2-5: If two angles are congruent and supplementary, then each is a right angle.

12. Proving the Vertical Angles Theorem
Given: 1 and 3 are vertical angles Prove: 1  3

13. Using the Vertical Angles Theorem
What is the value of x?

14. Proof Using the Vertical Angles Theorem
Given: 1  4 Proof: 2  3