Introduction to Euclid's Geometry — Question 2
Back to all questionsWe have to define some other terms like : Ray, Straight line and a Point.
Ray - A ray is a part of a line, that has a fixed starting point but no end point. It can extend infinitely in one direction.
Straight Line - A line that connects two points in a plane and extends to infinity in both directions. It is considered one dimensional figure.
Point - A small dot made by a sharp pencil on a sheet of paper gives an idea about a point. A point has no dimension i.e, length, breadth and height, it has only a position.
(i) If the perpendicular distance between two lines is constant, then these lines are considered parallel to each other.
(ii) If the angle between two lines is equal to 90°, then these lines are considered perpendicular to each other.
(iii) A part of a line that has two endpoints and is the shortest distance between them, is a line segment.
(iv) The distance from the center to any point on the circumference of the circle is called the radius of the circle.
(v) A square is a regular quadrilateral that has four equal sides and angle between each side is a right angle.
Euclid’s Geometry — Interactive Study Guide
The Big Idea
Euclid started with things everyone agrees on (axioms and postulates) and built all of geometry by logical deduction. This is the axiomatic method — start from accepted truths, derive everything else.
Axiom vs Postulate vs Theorem
Axiom: Self-evident truth used everywhere in mathematics.
Postulate: Assumption specific to geometry (cannot be proved, must be accepted).
Theorem: A statement that has been proved using axioms, postulates, and logic.
The Famous 5th Postulate
Euclid’s 5th postulate is about parallel lines. For over 2000 years, mathematicians tried to prove it from the other 4 postulates. They failed — because it’s independent! Changing the 5th postulate leads to non-Euclidean geometry.
Quick Self-Check
- State Euclid’s first axiom. (Things equal to the same thing are equal to one another.)
- How many lines can be drawn through two distinct points? (Exactly one — Postulate 1)
- What is the modern equivalent of Euclid’s 5th postulate? (Playfair’s axiom)