## Tuesday, September 27, 2011

### Geometry Part 1

What is Geometry?
Geometry is a subject in mathematics that focuses on the study of shapes, sizes, relative configurations, and spatial properties. Derived from the Greek word meaning “earth measurement”, geometry is one of the oldest sciences. It was first formally organized by the Greek mathematician Euclid around 300 BC when he arranged 465 geometric propositions into 13 books, titled ‘Elements’.

What are Angle Properties, Postulates, and Theorems?
In order to study geometry in a logical way, it will be important to understand key mathematical properties and to know how to apply useful postulates and theorems.

Define the following (& # 4, 6 & 8 post as a comment)
1. Postulate
2. Theorem
3. Transversal
4. Converse
Nb: suggest you do a personal note or concept map to summarise the various types of geometrical properties.
The syllabus requires you to know the following:
1. Properties of angles eg. acute, reflect etc
2. Properties of angles and straight lines
3. Properties of angles between parallel lines
4. Properties of Triangle

courtesy of Lincoln Chu S1-02 2010

courtesy of Goh Jia Sheng S1-02 2010

Lets look at some of these Postulates
A. Corresponding Angles Postulate
If a transversal intersects two parallel lines, the pairs of corresponding angles are congruent. Converse also true: If a transversal intersects two lines and the corresponding angles are congruent, then the lines are parallel.
The figure above yields four pairs of corresponding angles.

### B. Parallel Postulate

Given a line and a point not on that line, there exists a unique line through the point parallel to the given line. The parallel postulate is what sets Euclidean geometry apart from non-Euclidean geometry.
There are an infinite number of lines that pass through point E, but only the red line runs parallel to line CD. Any other line through E will eventually intersect line CD.

## Angle Theorems

### C. Alternate Exterior Angles Theorem

If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.
Converse also true: If a transversal intersects two lines and the alternate exterior angles are congruent, then the lines are parallel.
The alternate exterior angles have the same degree measures because the lines are parallel to each other.

### D. Alternate Interior Angles Theorem

If a transversal intersects two parallel lines, then the alternate interior angles are congruent.
Converse also true: If a transversal intersects two lines and the alternate interior angles are congruent, then the lines are parallel.
The alternate interior angles have the same degree measures because the lines are parallel to each other.

E. Same-Side Interior Angles Theorem
If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.
The sum of the degree measures of the same-side interior angles is 180°.

### F. Vertical Angles Theorem

If two angles are vertical angles, then they have equal measures.
The vertical angles have equal degree measures. There are two pairs of vertical angles.
sources:
http://www.wyzant.com
http://www.mathsteacher.com.au/year9/ch13_geometry/05_deductive/geometry.htm

1. Postulates : Postulates are statements that are assumed to be true without proof. Postulates serve two purposes - to explain undefined terms, and to serve as a starting point for proving other statements.

Theorem: Theorems are statements that can be deduced and proved from definitions, postulates, and previously proved theorems.

Transversal: A line that cuts across two or more (usually parallel) lines.

Converse : a reversed conditional; if a conditional is p--->q, than its converse is q--->p

2. Postulate: An assumption used as a basis for mathematical reasoning(assumed without proof).

Theorem: A proposition that has been or is to be proved on the basis of explicit assumptions. e.g formulae???(not sure)

Transversal: somehing that intersect other lines

Converse: basically a reversed equation. For example, (1+2=x)'s converse is (x=2+1).

3. Postulate: Suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief

Theorem: a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths.
• a rule in algebra or other branches of mathematics expressed by symbols or formulae.

Transversal: (of a line) intersecting a system of lines.

Converse: having characteristics that are the reverse of something else already mentioned