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