Geometry: Chapter 2 By: Antonio Nassivera, Dalton Hogan and Tom Kiernan.

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Presentation transcript:

Geometry: Chapter 2 By: Antonio Nassivera, Dalton Hogan and Tom Kiernan

Conditional Statements  A conditional statement is an if-then statement about two similar things.  Ex. If it snows, then it is cold out.  P->Q  P= The hypothesis  Q= The conclusion

Hypothesis  The hypothesis is the part after the word if.  Conditional: If it snows, then it is cold out.  The hypothesis of that statement would be the “it snows” portion.  The hypothesis is the P part of the conditional statement “P->Q”.

Conclusion  The conclusion portion of an if-then statement or conditional statement is the part following the word then.  Using the same conditional as before the hypothesis was be the “it is cold out” portion.  The conclusion is the Q part of the conditional statement that follows after then.

Converse  A converse switches the hypothesis and conclusion order so it is Q->P not P->Q like in a conditional statement.  So if we were to use the same conditional as before it would be: If it is cold out, then it is snowing.

Inverse  An inverse, unlike a converse, negates the hypothesis and conclusion of the conditional statement instead of switching their order.  An inverse can be looked at as ~P->~Q  Using the same conditional as before the inverse of it would be: If it does not snow, then it is not cold out.

Contrapositive  A contrapositive is a statement that switches and negates the hypothesis and conclusion.  In other words it would be like making a conditional statement be the converse and the inverse at the same time.  It would look like ~Q->~P  Using the same conditional as the previous ones it would be: If it is not snowing, then it is not cold out.

Biconditional  A biconditional is a statement that connects the conditional and its converse with if and only if.  Symbol: P Q  Ex. Conditional: If I eat, then I am hungry. Converse: If I am hungry, then I eat. Biconditional: I eat if and only if I am hungry.

Law of Detachment  If a conditional is true and it’s hypothesis is true then its conclusion is true.  If P->Q and P are true statements then Q is a true statement.  Ex. If it is an A day, then I have gym. Today is an A day. Therefore I have gym.

Law of Syllogism  If two conditionals are true then they can be combined using the hypothesis from the first conditional and the conclusion from the second conditional.  If P->Q and Q->R are true then P->R is true.  If it is an A day, then I have gym. If I have gym, then I can play sports. If it is an A day, then I can play sports.

Properties  Addition property: If a=b then a+c=b+c  Subtraction property: If a=b then a–c=b-c  Multiplication property: If a=b then ac=bc  Division property: if a=b, and c does not equal 0, then a/c=b/c

More Properties  Reflexive property: a=a or a is congruent to a  Symmetric property: If a=b then b=a. Or the same but congruent, not equal  Transitive property: If a=b and b=c then a=c.  or the same but congreunt not equal.  Substitution property: If a=b then b can be replace a in any equation.  Distributive property: a(b+c)=ab+ac

Angle Postulates/Theorems  Vertical angles are congruent.  All right angles are congruent.  If two angles are congruent and supplementary then each angle is a right angle.