2.2.2 Geometric Proof (continued)
The statements given in steps 1 and 3 are the conjecture’s hypothesis, so the corresponding reason for each is “Given information.” Notice that the step 2 statement is a conclusion based on information from step 1. Specifically, step 1 is the hypothesis for the conclusion made in step 2: IF “ ∠ E and ∠ G are supplementary” (step 1), THEN “m ∠ E + m ∠ G = 180°” (step 2). The conclusion is true because the definition of supplementary angles states that the sum of two supplementary angles is 180°. So, the reason for the statement in step 2 is the definition of supplementary angles, which can be abbreviated as “Def. of supp. ∠ s .” The statement in step 5 is a conclusion made using statements 2 and 4 as the hypothesis (noted to the left of the step 5 reason).
Example 2 Completing a Two-Column Proof A theorem is any statement that can be proven. One example of a theorem is the Linear Pair Theorem, which states that if two angles form a linear pair, then those two angles are supplementary. This conjecture is proven in the example below. The conjecture is divided into its hypothesis, provided in the “Given” statement (and figure), and its conclusion, provided in the “Prove” statement. So, the information that can be assumed to be true for a proof is provided in a “Given” statement and/or in a given figure, and the statement to be proven is provided in the “Prove” statement. The proof below is an example of a two-column proof. Notice that the statements are listed in the left column and the corresponding reasons for each statement, or justifications, are listed in the right column.
In this example, most of the proof is complete, and a few parts must be added (the second part of the statement in step 3 and the reasons for steps 1, 2, 5, and 6). The statement listed in step 1 is the “Given” statement. Therefore, the reason is “Given information.” To determine the reason for step 2, consider the statement in step 1 as the hypothesis and the statement in step 2 as the conclusion in a conditional. What reason makes the conclusion true? The definition of linear pair states that noncommon sides of the two angles in a linear pair are opposite rays that form a line. Therefore, the reason for step 2 is the definition of linear pair, abbreviated as “Def. of lin. pair.” Note that “Subst.”, which is the abbreviation for the Substitution Property of Equality, is listed as the reason in step 5.
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