5.1-Euclid's Definitions, Axioms and Postulates

5.1-Euclid's Definitions, Axioms and Postulates Important Formulae

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Grade 9 → Math → Euclid's Geometry → 5.1-Euclid's Definitions, Axioms and Postulates

After successful completion of this topic, you should be able to:

  • Understand Euclid's Definitions, Axioms and Postulates.

Euclid's Axioms:
(1) Things which are equal to the same thing are equal to one another.
(2) If equals are added to equals, the wholes are equal.
(3) If equals are subtracted from equals, the remainders are equal.
(4) Things which coincide with one another are equal to one another.
(5) The whole is greater than the part.
(6) Things which are double of the same things are equal to one another.
(7) Things which are halves of the same things are equal to one another

Euclid’s Five Postulates:
Postulate 1 : A straight line may be drawn from any one point to any other point.
Postulate 2 : A terminated line can be produced indefinitely.
Postulate 3 : A circle can be drawn with any centre and any radius.
Postulate 4:All right angles are equal to one another.


Photograph taken by Mark A. Wilson (Wilson44691, Department of Geology, The College of Wooster).[1], CC BY-SA 4.0, via Wikimedia Commons

Which of the following statements are true and which are false? Give reasons for your answers.
(i)  Only one line can pass through a single point.
(ii)  There are an infinite number of lines which pass through two distinct points.
(iii)  A terminated line can be produced indefinitely on both the sides.
(iv)  If two circles are equal, then their radii are equal.
(v)  In Fig. 5.9, if AB = PQ and PQ=XY, then AB = XY.

Solution:
(i)  False. This can be seen visually by the student.
(ii)  False. This contradicts Axiom 5.1.
(iii)  True.(Postulate 2)
(iv)  True. If you superimpose the region bounded by one circle on the other, then they coincide. So, their centres and boundaries coincide. Therefore, their radii will coincide.
(v)  True. The first axiom of Euclid.

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Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) Parallel lines

(ii) Perpendicular lines 

(iii) Line segment
(iv) Radius of a circle
(v) Square

Solution:

Definitions

Parallel lines: Lines in a plane that never meet and are equidistant from each other, denoted as $l_1 \parallel l_2$.

Perpendicular lines: Lines that intersect at a right angle (90°), denoted as $l_1 \perp l_2$.

Line segment: A part of a line bounded by two distinct endpoints, represented as $AB$.

Radius of a circle: The distance from the center to any point on the circle, denoted as $r$.

Square: A quadrilateral with four equal sides and four right angles, defined as $ABCD$ where $AB = BC = CD = DA$.

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Consider two ‘postulates’ given below:
(i)  Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii)  There exist at least three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.

Solution:
There are several undefined terms which the student should list. They are consistent, because they deal with two different situations —
(i) says that given two points A and B, there is a point C lying on the line in between them;
(ii) says that given A and B, you can take C not lying on the line through A and B. These ‘postulates’ do not follow from Euclid’s postulates. However, they follow from Axiom 5.1.

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If a point C lies between two points A and B such that AC = BC, then prove that:
AC= $\dfrac{1}{2}$ AB. Explain by drawing the figure.

In Question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Solution:
Make a temporary assumption that different points C and D are two mid-points of AB. Now, you show that points C and D are not two different points.

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In Fig. 5.10, if AC = BD, then prove that AB = CD.

Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)