Geometry 2

conditional statements

Statement, consists of 2 clauses, one of which begins with the word “if” or “when” or some equivalent word. “If a, then b”. a represents the if clause, or hypothesis, and b represents the then clause, or conclusion

Euler diagram

Circle diagrams that represent how conditional statements relate to each other, ( by Swiss mathematician, Leonhard Euler)

converse

converse of a conditional statement is found by interchanging its hypothesis and conclusion

syllogism

an argument in the form:

if a then b

if b then c

Therefore, if a then c

direct proof

if a then b

if b then c

if c then d

if d the e

if e then f

Therefore, if a then f

The startements, if a then b, if b then c, are called the premises of the argument and the statement if a then f is called the conclusion of the argument. The conclusion might be considered a theorem

Theorem

Statement that is proved by reasoning deductively from already accepted statements

indirect proof

In an indirect proof, an assumption is made at the beginning that leads to a contradiction. The contradiction indicated that the assumption is false and the desired conclusion is true.

Suppose not d is true

If not d, then e.

If e, then f,

and so on, until we come to a contradiction.

Therefore, not d is false; so, d is true

Postulate

statement that is assumed to be true without proof

Postulate 1

Two points determine a line (meaning if we draw a line 2 points are sufficient to locate its position)

Postulate 2

Three noncollinear points determine a place

Pythagorean Theorem

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

The Triangle Angle Sum Theorem

The sum of the angles of a triangle is 180 degrees

Two Circle Theorems

If the diameter of a circle is d, its circumference is pie times d

If the radius of a circle is r, its area is pie x r squared

iff

if and only if

If both a statement and its converse are true we can write “a if b” and “ a only if b”

undefined term

To avoid the problem of the dictionary maker, that is coming back to the point from which we begin- we must have a starting point: some words that we leave undefined. Among these words, the undefined terms are “point”, “line”, and “plane”