**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?**

**Task 1:**

- Postulate
- Theorem
- Transversal
- 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:*

*Properties of angles eg. acute, reflect etc**Properties of angles and straight lines**Properties of angles between parallel lines**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**

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.

ReplyDeleteTheorem: 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

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

ReplyDeleteTheorem: 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).

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

ReplyDeleteTheorem: 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