3.2 Proof and Perpendicular Lines Different types of proofs Theorems about Perpendicular Lines

Theorem 3.1 If two lines intersect making congruent adjacent angles, then the lines are perpendicular.

Theorem 3.2 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Theorem 3.3 If two lines are perpendicular, then they make four right angles

Types of Proofs Two –Column proofs: Statements in one column Reason in the other column Paragraph proofs: Write as a conversation Flow Proof: Same as Two-Column proof but connected with arrows.

Proof Done in three ways: 2 columns A B C D Given AB = CD Prove AC = BD #1. AB = CD #1. Given #2. BC = BC #2. Reflexive #3. AB + BC = CD + BC #3.Add #4. AC = AB + BC #4. Segment Add BD = CD + BC #5. AC = BD #5. Substitution

Proof Done in three ways: Flow proof A B C D Given AB = CD Prove AC = BD AB = CD Given BC = BC AB + BC = CD + BC Reflexive Addition AC = AB + BC BD = CD + BC Segment Addition AC = BD Substitution

Proof Done in three ways: Paragraph Proof A B C D AB equals CD by the given. If we use the addition property of equality with BC, then BC added to AB it would equal AC; BC added to CD would be BD. AC is the sum of AB and BC by segment addition as BD is the sum of CD and BC by the same reasoning. We could substitute AC for the sum of AB and BC and BD for the sum of CD and BC. Thus proving AC = BD. QED.

Q.E.D. Q.E.D. is an initialism of the Latin phrase , which translates as "which was to be demonstrated". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation — and in the setting-out — has been exactly restated as the conclusion of the demonstration.1 The abbreviation thus signals the completion of the proof. http://en.wikipedia.org/wiki/Q.E.D.

Pick the way you want to do your proof C Given: 1 2 is complementary to Prove: 3 B D A

Pick the way you want to do your proof C Given: 1 2 is complementary to Prove: 3 B D A What is the plan?

Given: DC BD; Prove BA BD A C 1 2 B D What is the plan?

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