Presentation on theme: "Conditional Statements"— Presentation transcript:
1Conditional 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.
2Conditional 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.
3Conditional 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.
4Examples 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?
5Equivalent 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.
6Definitions 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.
7Biconditional 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.
8PostulatesPostulate 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.
9PostulatesPostulate 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.
10PropertiesProperties of Equality: Addition, Subtraction, Multiplication and Division.Distributive and Substitution.ReflexiveSymmetricTransitive
12Given: Angles 1 and 2 and angles 3 and 2 are supplements 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 supplementsAngles 3 and 2 are supplementsM1 + M2 = 180M3 + M2 = 180M1+M2=M3+M2M1=M3Angle 1 is congruent to angle 3
13Given: Angles 1 and 2 and angles 3 and 2 are supplements 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 supplementsAngles 3 and 2 are supplementsM1 + M2 = 180M3 + M2 = 180M1+M2=M3+M2M1=M3Angle 1 is congruent to angle 3GivenDef. of Supp. AnglesTransitive PropertySubtraction PropertyDef of Congruent angles
14Given: Angle’s 1&2 are a linear pair Angles 1&2 are congruent Prove: g perpendicular h 1 and 2 are a linear pair1 and 2 are supplementary1 + 2 = 1801 congruent 2M1 = m2M1 + m1 = 1802(m1) =180M1=901 is a right angleG perpendicular to h
15Given: Angle’s 1&2 are a linear pair Angles 1&2 are congruent Prove: g perpendicular h 1 and 2 are a linear pair1 and 2 are supplementary1 + 2 = 1801 congruent 2M1 = m2M1 = m1 = 1802(m1) =180M1=901 is a right angleG perpendicular to hGivenLinear pair postulateDef. of supp. AnglesDef. of congruent anglesSubstitution Prop. Of EqualityCombine like TermsDivision Prop. Of EqualityDef. of right angleDef. of Perpendicular lines