Students can go through AP Board 9th Class Maths Notes Chapter 3 The Elements of Geometry to understand and remember the concepts easily.
AP State Board Syllabus 9th Class Maths Notes Chapter 3 The Elements of Geometry
→ Geometry is structured on its building blocks namely point, line and plane.
→ In geometry there are undefined terms like point, plane and line.
→ Angles, circles and triangles are the examples for defined terms.
→ No better entrance exists than Euclid’s time honoured ‘Elements’.
→ In ‘The Elements’, Euclid developed a new system of thought which laid the foundation for the advancement of the geometry.
→ Some of the Euclid’s axioms are:
- Things which are equal to same things are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are also equal.
- Things which coincide with one another are equal to one another.
- The whole is greater than 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 postulates are
Postulate – 1 : To draw a straight line from any point to any point.
Postulate – 2 : A terminated line can be produced indefinitely.
Postulate – 3 : To describe a circle with any centre and radius.
Postulate – 4 : That all right angles are equal to one another.
Postulate – 5 : If a straight line falling on two straight lines makes the interior angles on the same side of it taken together is less than two right angles, then the two straight lines, if produced infinitely, meet on that side on which the sum of the angles is less than two right angles.
→ Equivalent versions of Euclid’s fifth postulate:
- Through a point not on a given line, exactly one parallel line may be drawn to the given line – John Play Fair (1748 – 1819).
- The sum of angles of any triangle is a constant and is equal to two right angles (Legendre).
- There exists a pair of lines everywhere equidistant from one another (Posidominus).
- If a straight line intersects any one of two parallel lines, then it will intersect the other also (Proclus).
- The statements that were proved to be true are called propositions or theorems.
- The statements neither proved nor disproved are called conjectures.
- There are non-Euclidian geometries.