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2-4 Deductive Reasoning 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. 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. 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. 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. 2-5 Reasoning in Algebra and Geometry 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! Writing a Two-Column Proof Given: m1 = m3 Prove: mAEC = mDEB 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 2-6 Proving Angles Congruent 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. Proving the Vertical Angles Theorem Given: 1 and 3 are vertical angles Prove: 1 3 Using the Vertical Angles Theorem • What is the value of x? Proof Using the Vertical Angles Theorem Given: 1 4 Proof: 2 3