Draw a triangle with three acute angles. Construct angle

TABLE 1.6 Further Algebraic Properties of Equality a, b, and c are real numbers Reflexive Property: a = a . Symmetric Property: If a = b , then b = a . Distributive Property: a b + c = a ⭈ b + a ⭈ c . Substitution Property: If a = b , then a replaces b in any equation. Transitive Property: If a = b and b = c , then a = c . TABLE 1.5 Properties of Equality a, b, and c are real numbers Addition Property of Equality: If a = b , then a + c = b + c . Subtraction Property of Equality: If a = b , then a - c = b - c . Multiplication Property of Equality: If a = b , then a c = b c . Division Property of Equality: If a = b and c Z 0, then = . b c a c Reminder Additional properties and techniques of algebra are found in Appendix A. To deal with this question, you must ask “What” it is that is known Given and “Why” the conclusion Prove should follow from this information. Completing the proof often requires deducing several related conclusions and thus several intermediate “whys”. In correctly piecing together a proof, you will usually scratch out several con- clusions and reorder them. Each conclusion must be justified by citing the Given hy- pothesis, a previously stated definition or postulate, or a theorem previously proved. Selected properties from algebra are often used as reasons to justify statements. For instance, we use the Addition Property of Equality to justify adding the same number to each side of an equation. Reasons found in a proof often include the properties found in Tables 1.5 and 1.6. As we discover in Example 1, some properties can be used interchangably. EXAMPLE 1 Which property of equality justifies each conclusion? a If 2x - 3 = 7, then 2x = 10. b If 2x = 10, then x = 5. Solution a Addition Property of Equality; added 3 to each side of the equation. b Multiplication Property of Equality; multiplied each side of the equation by . OR Division Property of Equality; divided each side of the equation by 2. 쮿 1 2 Before considering geometric proof, we study algebraic proof, in which each state- ment in a sequence of steps is supported by the reason why we can make that statement claim. The first claim in the proof is the Given problem; and the sequence of steps must conclude with a final statement representing the claim to be proved called the Prove statement .