2.2.3 Flowchart and Paragraph Proofs

Key Objectives • Write flowchart and paragraph proofs. • Prove geometric theorems by using deductive reasoning. Key Terms • A flowchart proof uses boxes and arrows to show the structure of a proof. • In a paragraph proof , the statements and reasons of the proof are presented as sentences in a paragraph. Theorems, Postulates, Corollaries, and Properties • Common Segments Theorem Given collinear points A , B , C , and D arranged on a segment such that A and D are the endpoints, B is between A and C , and C is between B and D . If AB ≅ CD , then AC ≅ BD . • Vertical Angles Theorem Vertical angles are congruent. • Theorem If two congruent angles are supplementary, then each angle is a right angle. Example 1 Writing a Flowchart Proof A flowchart proof uses boxes and arrows to show the structure of a proof. The steps in a flowchart proof move from left to right or from top to bottom, shown by arrows connecting each box. The boxes used in a flowchart proof contain the statements and the reason for each statement is listed under the corresponding box.

In this example, the conjecture, figure, and two- column proof are given and a flowchart proof must be written, based on the two-column proof. The statement and reason from step 1 in the two- column proof should be listed in and under either the top box or the box farthest to the left. Here, the flowchart proof is using a top/down layout, so the given statement is shown in the top box with the corresponding reason underneath the box. The statements and reasons from steps 2, 3, and 4 are each listed in and under a box. The arrows are used to show the progression of the steps. Notice that the information from steps 1 and 3 are shown in the figure at the bottom.

Example 2 Reading a Paragraph Proof In a paragraph proof, the statements and reasons of the proof are presented as sentences in a paragraph. The sentences are written in a logical order, typically beginning with a sentence (or sentences) summarizing the information given in the “Given” statement(s) and in the figure, and ending with a sentence that summarizes the conjecture’s conclusion.

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