CHAPTER 2: DEDUCTIVE REASONING
Chapter 2 of Deductive Reasoning delves into the essential components of proofs, including the statement of the theorem, illustrative diagrams, given information, proving steps, and statements with reasons. Students learn to prepare for proofs by planning before writing, employing reverse reasoning when necessary, and reviewing previous theorems for methods. The chapter covers significant theorems such as the congruence of angles that are supplements or complements of congruent angles. Practice exercises reinforce the concepts introduced, aiding comprehension and skill in geometric proofs.
CHAPTER 2: DEDUCTIVE REASONING
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CHAPTER 2: DEDUCTIVE REASONING Section 2-6: Planning a Proof
PROOFS All proofs consist of the following 5 pieces: • Statement of theorem • A diagram illustrating the information • Given • Prove • Statements and Reasons (lead from given to what is to be proved)
PREPARING FOR A PROOF • Try to plan your proof before writing steps in the two-column format. • If you do not see a method that works, try reasoning back from what is to be proved. • Study proofs of previous theorems for possible methods.
1. 1 and 2 are supplementary; 3 and 4 are supplementary. m 1 + m 2 = 180; m 3 + m 4 = 180 m 1 + m 2 = m 3 + m 4 2 ≡ 4, or m 2 = m 4 m 1 = m 3, or 1 ≡ 3 Given Def. of supp. Angles Substitution Prop. Given Subtraction prop. of = PROOF Given: 1 and 2 are supplementary; 3 and 4 are supplementary; 2 ≡ 4 Prove: 1 ≡ 3 1 2 3 4
THEOREM 2-7 Theorem 2-7: If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.
THEOREM 2-8 Theorem 2-8: If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
a.Name a supplement of angle 2. Name a supplement of angle 3. What justifies your answers to a and b? Angle 1 Angle 4 Angle addition postulate EXAMPLE 1 2 3 4
If 1 and 3 are supplementary, then 1 and ______ are supplementary. If MR = NQ and MP = NP, then ___ = ___. If QRS is a right angle, then 1 and 2 are ____________ ___________ angles. MPQ PR = PQ adjacent complementary EXAMPLE Q M P 1 R 3 2 N S
Vertical angles are _________ congruent. Perpendicular lines _______ form 45 degree angles. A theorem is _______ a true statement. A postulate is ________ used as a reason in a proof. Supplements of congruent angles are ________ congruent. Two angles are ________ complements. A statement in a proof ________ has a reason. always never always sometimes always sometimes always ALWAYS, SOMETIMES, OR NEVER
CLASSWORK/HOMEWORK • Pg. 62, Classroom Exercises 1-6, 8, 10 • Pg. 63, Written Exercises 1-15
