Axioms, Postulates and Theorems

Definition of Some Mathematical Terms (Axiom, theorem, corollary and proof)

Geometry Around Us

Our daily life is filled with geometry—the pure mathematics of points, lines, curves and surfaces. We can observe various shapes and angles in the objects that surround us. Observe, for example, this table and its rectangular surface; the boomerang and its angular shape; the bangle and its circular shape.

Euclid, an ancient Greek mathematician, observed the various types of objects around him and tried to define the most basic components of those objects. He proposed twenty-three definitions based on his studies of space and the objects visible in daily life. Let us go through this lesson to learn each of Euclid’s definitions.

Know Your Scientist

EuclidEuclid of Alexandria(325 BC−265 BC) was a great Greek mathematician. He is referred to as ‘the father of geometry’. Euclid taught at Alexandria during the reign of Ptolemy I, who ruled Egypt from 323 BC to 285 BC. Euclid wrote a series of books which are collectively known as the Elements. It is considered one of the most influential works in the history of mathematics. The Elements served as the main textbook for teaching mathematics (especially geometry) from the time of its publication up until the early 20th century. The principles in this treatise were derived by Euclid from a small set of axioms that are now referred to as Euclidean geometry.

Did You Know?

The Elements is a great collection of basic geometrical concepts and ideas. The content in it is based on the works of a few great mathematicians, including Pythagoras, Thales, Plato, Aristotle, Eudoxus and Menaechmus. This work had been the most commonly used and the premier source of geometrical concepts for over 2000 years since its publication. It is believed to be the most translated, published and studied book after the Bible in the western world.

Definitions Given by EuclidEuclid gave the definitions of a few very basic attributes of objects that are normally around us. These definitionsare listed below.

1. A point is that which has no part.

2. A line is a breadth-less length.

3. The adjacent angles equal to each other, each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than the right angle.

12. An acute angle is an angle less than the right angle.

13. A boundary points out the limit or extent of something.

14. A figure is that which is contained by any boundary or boundaries.

Definitions Given by Euclid15. A circle is a plane figure consisting of a set of points circumference of the circle, and such a straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut off by it. Also, the centre of the semicircle is the same as that of the circle.

19. Rectilinear figures are those that are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that which has its three sides equal; an isosceles triangle is that which has only two of its sides equal; and a scalene triangle is that which has its three sides unequal.

21. Of trilateral figures, a right-angled triangle is that which has a right angle; an obtuse-angled triangle is that which has an obtuse angle; and an acute-angled triangle is that which has its three angles acute.

22.Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong is that which is right-angled but not equilateral; a rhombus is that which is equilateral but not right-angled; and a rhomboid is that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let any other type of quadrilateral be called a trapezium.

23.Parallel lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Did You Know?

Although Euclid defines ‘point’, ‘line’ and ‘plane’, these geometric quantities are now considered by mathematicians as undefined. This is because these three quantities are considered the basic entities in geometry. These three entities are used for defining other geometric shapes and concepts.

Solved Examples

Easy

Example 1:

Which of the following pairs of terms are considered ‘undefined’ in geometry?

A. Point and circle

B. Point and line

C. Circle and line

D. Circle and radius

Solution:

The correct answer is B.

Point, line and plane are considered ‘undefined’ in geometry.

Example 2:

Fill in the blanks.

i) The extremities of a plane surface are known as __________.

ii) An __________ angle is greater than the right angle.

iii) The __________ of a circle is the straight line passing through the centre of the circle.

iv) A trilateral figure is contained by__________ lines.

v) The __________ is a trilateral figure whose all sides are equal.

Solution:

i) lines

ii) obtuse

iii) diameter

iv) three

v) equilateral triangle

Geometry Around UsOur daily life is filled with geometry—the pure mathematics of points, lines, curves and surfaces. We can observe various shapes and angles in the objects that surround us. Observe, for example, this table and its rectangular surface; the boomerang and its angular shape; the bangle and its circular shape.

Euclid, an ancient Greek mathematician, observed the various types of objects around him and tried to define the most basic components of those objects. He proposed twenty-three definitions based on his studies of space and the objects visible in daily life. Let us go through this lesson to learn each of Euclid’s definitions.

Know Your Scientist

EuclidEuclid of Alexandria(325 BC−265 BC) was a great Greek mathematician. He is referred to as ‘the father of geometry’. Euclid taught at Alexandria during the reign of Ptolemy I, who ruled Egypt from 323 BC to 285 BC. Euclid wrote a series of books which are collectively known as the Elements. It is considered one of the most influential works in the history of mathematics. The Elements served as the main textbook for teaching mathematics (especially geometry) from the time of its publication up until the early 20th century. The principles in this treatise were derived by Euclid from a small set of axioms that are now referred to as Euclidean geometry.

Did You Know?

The Elements is a great collection of basic geometrical concepts and ideas. The content in it is based on the works of a few great mathematicians, including Pythagoras, Thales, Plato, Aristotle, Eudoxus and Menaechmus. This work had been the most commonly used and the premier source of geometrical concepts for over 2000 years since its publication. It is believed to be the most translated, published and studied book after the Bible in the western world.

Definitions Given by EuclidEuclid gave the definitions of a few very basic attributes of objects that are normally around us. These definitionsare listed below.

1. A point is that which has no part.

2. A line is a breadth-less length.

3. The adjacent angles equal to each other, each of the equal angles is right and the straight line standing on the other is called a perpendicular to that on which it stands.

11. An obtuse angle is an angle greater than the right angle.

12. An acute angle is an angle less than the right angle.

13. A boundary points out the limit or extent of something.

14. A figure is that which is contained by any boundary or boundaries.

Definitions Given by Euclid15. A circle is a plane figure consisting of a set of points circumference of the circle, and such a straight line also bisects the circle.

18. A semicircle is the figure contained by the diameter and the circumference cut off by it. Also, the centre of the semicircle is the same as that of the circle.

19. Rectilinear figures are those that are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

20. Of trilateral figures, an equilateral triangle is that

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