AMSCO'S Geometry. New York - Rye High School

Deductive Reasoning 101ABCIn the logic-based proof above, notice that the Law of Detachment is citedas a reason for reaching our conclusion. In a typical geometry proof, however,the laws of logic are used to deduce the conclusion but the laws are not listedamong the reasons.Let us restate this proof in the format used often in Euclidean geometry.Wewrite the information known to be true as the “given” statements and the conclusionto be proved as the “prove”. Then we construct a two-column proof.Inthe left column, we write statements that we know to be true, and in the right column,we write the reasons why each statement is true.Given: In ABC, AB ' BC.Prove: ABC is a right angle.Proof: Statements Reasons1. AB ' BC1. Given.2. ABC is a right angle. 2. If two lines are perpendicular, thenthey intersect to form right angles.ABNotice how the Law of Detachment was used in this geometry proof. Bycombining statement 1 with reason 2, the conclusion is written as statement 2,just as p and p → r led us to the conclusion r using logic. In reason 2, we usedthe conditional form of the definition of perpendicular lines.Often in a proof, we find one conclusion in order to use that statement tofind another conclusion. For instance, in the following proof, we will begin byproving that ABC is a right angle and then use that fact with the definition ofa right triangle to prove that ABC is a right triangle.Given: In ABC, AB ' BC.Prove: ABC is a right triangle.CProof: Statements Reasons1. AB ' BC1. Given.2. ABC is a right angle. 2. If two lines are perpendicular, thenthey intersect to form right angles.3. ABC is a right triangle. 3. If a triangle has a right angle, then itis a right triangle.Note that in this proof, we used the conditional form of the definition of perpendicularlines in reason 2 and the converse form of the definition of a righttriangle in reason 3.The proof can also be written in paragraph form, also called a paragraphproof. Each statement must be justified by stating a definition or another