Geometry - Deductive Reasoning - Chapt 2

Conditional - a statement that can be written in if-then form

a.  hypothesis - the given information
b.  conclusion - what you are trying to prove

             symbolic:    p  ⇒  q

Negation - opposite of the original statement. it is or it isn't

                symbolic:   ~ p

Counterexample - an example that proves that a statement is false.

Converse - statement made by exchanging the hypothesis and conclusion.
               
Inverse - statement made by negating both the hypothesis and conclusion

Contrapositive - statement made by negating both the hypothesis and conclusion of the converse.

Equivalent statements - same truth value

Rules:
contrapositive and counterexample always the same

inverse and converse always the same.

Logic rule:  p ⇒ q, if given q ⇒ p


All cows eat grass.

Elsey is a cow then she sat grass.

biconditional - combining a true conditional with its true converse using if and only if.

State the property that justifies each statement.

1.  If m <1 = m <2 , then m <2 = m <1.

2.  If m<1 = 90 and m <2 = m<1, then m <2 = 90.

3.  If AB = RS and RS = WY, then AB = WY.

4.  If AB = CD, then 1/2AB = 1/2CD.

5.  If m <1 + m <2 = 110 and m <2 = m <3, then m <1 + m <3 = 110.

6.  RS = RS

7.  If AB = RS and TU = WY, then AB + TU = RS + WY.

Vertical angles - two non-adjacent angles formed by intersecting lines.

Vertical angles are congruent.

Complementary angles - two angles whose sum is 90°.

Theorem 2-8

Congruent Complements Theorem

If two angles are complements of the same < or congruent < then the angles are congruent.

supplementary angles - two angles whose sum is 180°.

Theorem 2-7

Congruent Supplements Theorem

If two angles are supplements of the same < or congruent < then the angles are congruent.

Linear pair

*  two adjacent angles whose non common sides are  line

*  always supplementary (180°)

Perpendicular lines: two lines that intersect to form a right angle. (90°)

Theorem 2-4

If two lines are perpendicular then they form congruent adjacent angles.

Theorem 2-5 converse to Theorem 2-4

If 2 lines form congruent adjacent angles then lines are perpendicular.


If <3 and <4 are congruent then a is perpendicular to b

Theorem 2-6

If the exterior sides of two adjacent acute angles are perpendicular then the angles are complementary.

Inductive reasoning - basing a conclusion on observations only. This doesn't have to be true.

Deductive reasoning - basing a conclusion on a fact.  If the hypothesis is true, so is the conclusion.

Law of Detachment: if p → q is true and p is true then q is true.

Law of Syllogism: if p → q and q → r are true, then p → r is true.