Originally, Geometry was a term that was used to measure earth. It is derived from the Greek word Ge – earth and Mertia – Measure. There is a world wide belief that geometry was initially used to measure the amount of tax that can be taken from the farmers who raise crops along the Nile river. Similar questions are now faced by many students while solving questions in mensuration.

Later on, a pyramid with 4 faces and square base was defined and geometry was used to measure this.

## Euclid’s Elements

This was one of the important factors that led to the advancement in geometry. Known as the Elements of Geometry, Euclid wrote these between 330 and 320 B.C. Important theorems based on plane and solid geometry were compiled together to be presented as an Axiom.

**“Axiom: A statement or proposition which is regarded as being established, accepted, or self-evidently true” **(Source: Google)

An axiom is a statement that has been widely accepted by the greatest minds in the world over centuries and proofs of which do not exist. It is completely based on logic. They are often taken as the starting point of various theorems.

**POSTULATE 1.** Any two points can be joined by a straight line.

**POSTULATE 2.** Any straight line segment can be extended indefinitely in a straight line.

**POSTULATE 3.** Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

**POSTULATE 4.** All right angles are congruent.

**POSTULATE 5.** (Parallel postulate) If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

**Euclidean geometry** is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the *Elements*. **Euclidâ€™s method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these**. Euclidean geometry is so simple at its soul that all constructions are solely done using a ruler and compass.

Until the second half of the 19th century, when *non-Euclidean geometries* attracted the attention of mathematicians, *geometry* meant Euclidean geometry. It is the most typical expression of general mathematical thinking. Rather than the memorization of simple algorithms to solve equations by rote, **it demands true insight into the subject and looks for creative ideas for applying theorems in special situations. It needs an ability to generalize from known facts, and lays the importance of proof.**

Differential Geometry is the study of geometric properties using Differential and Integral Calculus. It is a branch of mathematics dealing with geometrical forms and the intrinsic properties of curves and surfaces as related to differential calculus and mathematical analysis.

Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. ^{}Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships.

## Basic Geometry Definitions

- Line: Is determined by two distinct end points and extends indefinitely in both directions.
- Perpendicular Lines: Intersecting lines that form right angles.
- Parallel Lines: Lines that never intersect or cross each other.
- Complementary Angles: Two angles whose measures have a sum of 90 degrees.
- Supplementary Angles: Two angles whose measures have a sum of 180 degrees.
- Congruent triangles: Triangles in which corresponding parts (sides and angles) are equal in measure.
- Adjacent Angles: Adjacent Angles are two angles that share a common side. They are also supplementary angles, which means, the measure of the sums of their angles must equal 180 degrees.
- Angle-Angle (AA) Similarity: If two angles of one triangle are equal in measure to two angles of another triangle, then the two triangles are similar
- Side-side-side (SSS) Similarity: If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.
- Side-angle side SAS) Similarity: If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
- Angle-side angle Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
- Transversal: Two parallel lines intersected by a transversal form corresponding pairs of angles that are congruent. Several types of angles are formed: alternate interior, alternate exterior , corresponding and

vertical angles. - Triangle Angle Bisector: An angle bisector of a triangle divides the opposite sides into two segments whose lengths are proportional to the lengths of the other two sides.