MATH135 Lecture Notes - Lecture 2: Angle

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Published on 29 Sep 2015
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MATH 135 - Lecture 2: Introduction to Proofs
Types of Statements:
Proposition → a statement that needs to be proven true or demonstrated false by
a valid argument
Theorem → a significant proposition
Lemma → a proposition that is often not useful on its own, but is used to prove
another theorem
Corollary → a proposition that follows almost immediately without much work
from another theorem
Axiom → a statement that is assumed to be true without proof
Common Mistakes When Dealing with Proofs:
Multiplying or dividing by a negative number when dealing with inequalities
Ex. -5x > 4 → When dividing both sides of the inequality by -5, the new
equation becomes: x < -
Note that the inequality sign is flipped.
Dividing an expression by 0
When using variables, it is not always obvious when division by 0 occurs.
Ex. a = b
a2 = ab
a2 - b2 = ab - b2
(a + b)(a - b) = b(a - b)
a + b = b
2b = b
2 = 1
Note that a = b, therefore (a - b) = 0, so you are unable to divide the
equation by 0 in the 4th step to reach the 5th step.
Taking the square root of an expression
There exists the possibility of both positive and negative roots
Stewart’s Theorem:
Let ABC be a triangle with AB = c, AC = b and BC = a. If P is a point on BC with BP = m, PC = n
and AP = d, then dad + man = bmb +
cnc.
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Document Summary

Math 135 - lecture 2: introduction to proofs. Proposition a statement that needs to be proven true or demonstrated false by a valid argument. Lemma a proposition that is often not useful on its own, but is used to prove another theorem. Corollary a proposition that follows almost immediately without much work from another theorem. Axiom a statement that is assumed to be true without proof. Multiplying or dividing by a negative number when dealing with inequalities. 5x > 4 when dividing both sides of the inequality by -5, the new equation becomes: x < - . Note that the inequality sign is flipped. When using variables, it is not always obvious when division by 0 occurs. Ex. a = b a2 = ab a2 - b2 = ab - b2 (a + b)(a - b) = b(a - b) a + b = b.

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