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Published byMisael Mayhew Modified about 1 year ago

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Conditional Statements Logical statements that have two parts, a hypothesis and a conclusion. The “if” part contains the hypothesis and the “then” part the conclusion. Ex: If it is raining then there are clouds in the sky. Make the following statement a conditional statement: Two angles are supplementary if they are a linear pair.

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Conditional Statements The negation of a statement is the opposite of the original. Ex: We are in Geometry class. We are not in Geometry class. Conditional Statements may be true or false. They needed to be proven one way or the other. You must show examples of it being true or use a counterexample to prove it false.

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Conditional Statements The converse of a conditional statement switches the hypothesis with the conclusion. The inverse negates both the hypothesis and the conclusion. The contrapositive is the converse with both the conclusion and hypothesis negated.

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Examples Conditional- If angle A=99 degrees, then angle A is obtuse. Converse- If angle A is obtuse, then angle A=99 degrees. Inverse- If angle A is not equal to 99 degrees, then angle A is not obtuse. Contrapositive- If angle A is not obtuse, then angle A is not equal to 99 degrees. Which are true and which are false?

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Equivalent Statements A conditional statement and its contrapositive are either both true or both false. The converse and inverse of the conditional are either both true or both false. These are examples of equivalent statements.

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Definitions and Perpendicular Lines Definitions can be written as conditional statements or as the converse of the statement if both are true. The definition of perpendicular lines can be written in either of these forms. If two lines intersect to form a right angle, then they are perpendicular. If two lines are perpendicular, then they intersect to form a right angle.

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Biconditional Statements When both the conditional statement and its converse are true, you can write them as a single biconditional statement using “if and only if.” Any valid definition can be written as a biconditional statement. Two lines are perpendicular if and only if they intersect to form a right angle.

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Postulates Postulate 5 – Through any two points there exists exactly one line. Postulate 6 – A line contains at least two points. Postulate 7 – If two lines intersect, then their intersection is exactly one point. Postulate 8 – Through any three noncollinear points there exists exactly one plane.

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Postulates Postulate 9 – A plane contains at least three noncollinear points. Postulate 10 – If two points lie in a plane, then the line containing them lies in the plane. Postulate 11 – If two planes intersect, then their intersection is a line.

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Properties Properties of Equality: Addition, Subtraction, Multiplication and Division. Distributive and Substitution. Reflexive Symmetric Transitive

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Angle Pair Relationship Theorems Congruent Supplements Congruent Complements Right Angles Congruence

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Given: Angles 1 and 2 and angles 3 and 2 are supplements. Prove: Angle 1 is congruent to angle 3. Angles 1 and 2 are supplements Angles 3 and 2 are supplements M1 + M2 = 180 M3 + M2 = 180 M1+M2=M3+M2 M1=M3 Angle 1 is congruent to angle 3

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Given: Angles 1 and 2 and angles 3 and 2 are supplements. Prove: Angle 1 is congruent to angle 3. Angles 1 and 2 are supplements Angles 3 and 2 are supplements M1 + M2 = 180 M3 + M2 = 180 M1+M2=M3+M2 M1=M3 Angle 1 is congruent to angle 3 Given Def. of Supp. Angles Transitive Property Subtraction Property Def of Congruent angles

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Given: Angle’s 1&2 are a linear pair Angles 1&2 are congruent Prove: g perpendicular h 1 and 2 are a linear pair 1 and 2 are supplementary = congruent 2 M1 = m2 M1 + m1 = 180 2(m1) =180 M1=90 1 is a right angle G perpendicular to h g 2 h 1

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Given: Angle’s 1&2 are a linear pair Angles 1&2 are congruent Prove: g perpendicular h 1 and 2 are a linear pair 1 and 2 are supplementary = congruent 2 M1 = m2 M1 = m1 = 180 2(m1) =180 M1=90 1 is a right angle G perpendicular to h Given Linear pair postulate Def. of supp. Angles Given Def. of congruent angles Substitution Prop. Of Equality Combine like Terms Division Prop. Of Equality Def. of right angle Def. of Perpendicular lines g 2 g 1

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