**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**

Definition of words:

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

Theorem:A rule in algebra or other branches of mathematics expressed by symbols or formulae.

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

Converse:is an if-then statement in which the hypothesis and the conclusion are switched.

Postulate : postulate or axiom in a ,mathematical statement, serves as a starting point from which other statements are logically derived. Like

ReplyDelete___/a__/b___

/ /

The starting line is a thus from there we know that b is also the same as a as it has the same angle.

Theorem: Like Axiom , is statement that has been proven on the basis of previously established statements.

Example: /w___y/

w + y = 180

Transversal: A line that cross through other lines

Converse: The hypothesis of the question , as in the predicted answer.