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Chapter 2 Deductive Reasoning 2-1 If-Then Statements; Converses CONDITIONAL STATEMENTS are statements written in ifthen form. The clause following the “if” is called the hypothesis and the clause following “then” is called the conclusion. Examples • If it rains after school, then I will give you a ride home. •If you make an A on your test, then you will get an A on your report card. CONVERSE is formed by interchanging the hypothesis and the conclusion. Examples False Converses •If Bill lives in Texas, then he lives west of the Mississippi River. •If he lives west of the Mississippi River, then he lives in Texas Counterexample • An example that shows a statement to be false • It only takes one counterexample to disprove a statement Biconditional •A statement that contains the words “if and only if” •Segments are congruent if and only if their lengths are equal. 2-2 Properties from Algebra Addition Property • If a = b, and c = d, • then a + c = b + d Subtraction Property • If a = b, and c = d, • then a - c = b - d Multiplication Property • If a = b, • then ca = bc Division Property • If a = b, and c 0 • then a/c = b/c Substitution Property • If a = b, then either a or b may be substituted for the other in any equation (or inequality) Reflexive Property •a = a Symmetric Property • If a = b, then b = a Transitive Property • If a = b, and b = c, then a =c Distributive Property • a(b + c) = ab + ac Properties of Congruence Reflexive Property • DE DE • D D Symmetric Property • If DE FG, then FG DE • If D E, then E D Transitive Property • If DE FG, and FG JK, then DE JK • If D E, and E F, then D F 2-3 Proving Theorems Midpoint of a Segment – is the point that divides the segment into two congruent segments THEOREM 2-1 Midpoint Theorem If a point M is the midpoint of AB, then AM = ½AB and MB=½AB BISECTOR of ANGLE– is the ray that divides the angle into two adjacent angles that have equal measure. THEOREM 2-2 Angle Bisector Theorem If BX is the bisector of ABC, then: mABX = ½mABC and mXBC = ½ m ABC A• B X • C • 2-4 Special Pairs of Angles COMPLEMENTARY two angles whose measures have the sum 90º J 39º 51º K SUPPLEMENTARY two angles whose measures have the sum 180º H 133º G 47º VERTICAL ANGLES– two angles whose sides form two pairs of opposite rays. THEOREM 2-3 Vertical angles are congruent 2-5 Perpendicular Lines Perpendicular Lines– two lines that intersect to form right angles ( 90° angles) 2-4 THEOREM If two lines are perpendicular, then they form congruent adjacent angles. 2-5 THEOREM If two lines form congruent adjacent angles, then the lines are perpendicular. 2-6 THEOREM If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary 2-6 Planning a Proof Parts of a Proof 1.A diagram that illustrates the given information 2.A list, in terms of the figure, of what is given 3.A list, in terms of the figure, of what you are to prove 4.A series of statements and reasons that lead from the given information to the statement that is to be proved 2-7 THEOREM If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent. 2-8 THEOREM If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent. THE END