write Euclid axioms and postulates and explain them
Answers
Step-by-step explanation:
Euclidean Axioms
Here are the seven axioms given by Euclid for geometry.
Things which are equal to the same thing are equal to one another.
If equals are added to equals, the wholes are equal.
If equals are subtracted from equals, the remainders are equal.
Things which coincide with one another are equal to one another.
The whole is greater than the part.
Things which are double of the same things are equal to one another.
Things which are halves of the same things are equal to one another
Euclid’s Five Postulates
Before discussing Euclid’s Postulates let us discuss a few terms as listed by Euclid in his book 1 of the ‘Elements’. The postulated statements of these are:
Assume the three steps from solids to points as solids-surface-lines-points. In each step, one dimension is lost.
A solid has 3 dimensions, the surface has 2, the line has 1 and point is dimensionless.
A point is anything that has no part, a breadthless length is a line and the ends of a line point.
A surface is something which has length and breadth only.
It can be seen that the definition of a few terms needs extra specification. Now let us discuss these Postulates in detail.
Euclid’s Postulate 1
“A straight line can be drawn from anyone point to another point.”
This postulate states that at least one straight line passes through two distinct points but he did not mention that there cannot be more than one such line. Although throughout his work he has assumed there exists only a unique line passing through two points.
Euclid’s Geometry Postulate 1
Euclid’s Postulate 2
“A terminated line can be further produced indefinitely.”
In simple words what we call a line segment was defined as a terminated line by Euclid. Therefore this postulate means that we can extend a terminated line or a line segment in either direction to form a line. In the figure given below, the line segment AB can be extended as shown to form a line.
Euclid’s Postulate 2
Euclid’s Postulate 3
“A circle can be drawn with any centre and any radius.”
Any circle can be drawn from the end or start point of a circle and the diameter of the circle will be the length of the line segment.
Euclid’s Postulate 4
“All right angles are equal to one another.”
All the right angles (i.e. angles whose measure is 90°) are always congruent to each other i.e. they are equal irrespective of the length of the sides or their orientations.
Euclid’s Postulate 5
“If a straight line falling on two other straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on the side on which the sum of angles is less than two right angles.”