2.2.2 Geometric Proof
Key Objectives • Write two-column proofs. • Prove geometric theorems by using deductive reasoning. Key Terms • A theorem is any statement that you can prove. • A two-column proof is a type of proof where the statements and corresponding reasons are listed in two columns with the statements in the left column and the reasons in the right column. Theorems, Postulates, Corollaries, and Properties • Linear Pair Theorem If two angles form a linear pair, then they are supplementary. • Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. • Right Angle Congruence Theorem All right angles are congruent. • Congruent Complements Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. A proof is an argument where logic, definitions, properties, and previously proven statements are used to demonstrate that a conjecture is always true. There are several types of proofs in geometry, but a two- column proof is most commonly seen. The process of writing a proof is the same for all types of proofs. A proof is based on a conjecture. The conjecture contains a hypothesis (statements providing information that can be assumed to be true) and a conclusion (the statement that is to be proven true). In a proof, deductive reasoning is used to write a list of connected steps beginning with the hypothesis and ending with the conclusion. Each step in a proof contains two parts: a statement and a justification, or reason. “Given” is the reason for statements in the hypothesis. The reason for all other statements in the proof must be either a theorem, postulate, property, or definition. The given statements are typically listed at the start of a proof. After the given statements are made in a proof, each subsequent statement is first a conclusion based on a preceding step (or steps), and then that conclusion becomes a hypothesis for conclusions made in later steps. In other words, once a statement is justified, that statement can be used in the hypothesis for a later conclusion. The final statement in a proof is the original conjecture’s conclusion. Once the conjecture’s conclusion is stated and justified using at least one previous statement, the conjecture has been proven. If a figure is not given with the conjecture, it is often helpful to draw a figure and use markings, such as tick marks, to show the given information. Mark only the given information on the figure; do not mark the In the example below, the proof’s statements are complete and the justifications, or reasons, must be added. The information given in the problem is “ ∠ E and ∠ G are supplementary” and “ ∠ E ≅ ∠ H .” So, those two statements are the hypothesis of the conjecture to be proven. Notice that the final statement is “ ∠ H and ∠ G are supplementary.” Since this is the final statement of the proof, “ ∠ H and ∠ G are supplementary” must be the conclusion of the conjecture to be proven. information from the statement to be proven. Example 1 Writing Justifications
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