Introduction to Euclid’s Geometry
(यूक्लिड की ज्यामिति का परिचय)
Basics of Geometry
➥ Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids.
Euclid and His Contributions
➥ Euclid is known as the "Father of Geometry."
➥ He wrote a series of books called "The Elements," which organized the knowledge of geometry.
Euclid’s Axioms and Postulates
➥ Axioms (अभिगृहीत) are self-evident truths.
➥ Postulates (अभिधारण) are basic assumptions in geometry.
Euclid's Axioms
- 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 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.
- Postulate 5: If a straight line falling on two 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 that side on which the sum of angles is less than two right angles.
Fundamental Concepts in Geometry
➥ A point is an exact location in space with no dimensions.
➥ A line is a collection of points extending infinitely in both directions.
➥ A plane is a flat surface that extends infinitely in all directions.
Theorems and Proofs
➥ A theorem is a statement that can be proven.
➥ A proof is a logical argument that shows a theorem is true.
➥Concepts: A basic proposition accepted without proof, used as a starting point in mathematics or philosophy.
➥Theorem: A statement that has been proven true based on logical reasoning and established facts or assumptions.
Steps to Solve a Theorem
- Given that: This step states the initial information or conditions provided or assumed to be true.
- Prove that: The objective is to demonstrate or prove a specific statement or conclusion based on the given information.
- Construction: If applicable, this step involves creating geometric figures or setups as part of the proof process.
- Proof: The logical sequence of steps that connect the given information to the conclusion.
- Proved that: Finally, this step concludes by stating what has been proven or demonstrated through the logical process.
Exercise 5.1
1. 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.
False. An infinite number of lines can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
False. Only one line can pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
True. A line segment can be extended indefinitely on both sides.
(iv) If two circles are equal, then their radii are equal.
True. If two circles are equal, their radii must be equal.
(v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.
True. If AB = PQ and PQ = XY, then AB = XY by the transitive property of equality.
(i) Parallel lines
Parallel lines are lines in a plane that do not meet; they are always the same distance apart.
Terms to define first: Line.
(ii) Perpendicular lines
Perpendicular lines are lines that intersect at a right angle (90 degrees).
Terms to define first: Line, Right angle.
(iii) Line segment
A line segment is a part of a line that is bounded by two distinct end points.
Terms to define first: Line.
(iv) Radius of a circle
The radius of a circle is the distance from the center of the circle to any point on its circumference.
Terms to define first: Circle, Center, Circumference.
(v) Square
A square is a quadrilateral with four equal sides and four right angles.
Terms to define first: Quadrilateral, Right angle.
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
These postulates contain undefined terms like "point" and "between."
They are consistent because they do not contradict Euclid’s postulates.
They do not directly follow from Euclid’s postulates, but they are not in conflict with them.
(ii) There exist at least three points that are not on the same line.
This postulate also contains undefined terms like "point" and "line."
It is consistent and does not contradict Euclid’s postulates.
It does not directly follow from Euclid’s postulates, but it is compatible with them.
4. If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2 AB. Explain by drawing the figure.
Let A and B be two points, and let C be a point on the line segment AB such that AC = BC. This means C is the midpoint of AB.
Therefore, AC = BC. Let AC = x. Then, AB = AC + CB = x + x = 2x.
Thus, AC = 1/2 AB.
A ---- C ---- B
5. 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.
Assume there are two midpoints C and D on the line segment AB.
This implies AC = CB and AD = DB.
Since C and D are both midpoints, AC = AD and CB = DB.
Thus, C and D must coincide, meaning there is only one midpoint.
6. In Fig. 5.10, if AC = BD, then prove that AB = CD.
Given AC = BD, we need to prove AB = CD.
Consider the line segments AC and BD. Since AC = BD, we can write AB as AC + CB and CD as CB + BD.
Since AC = BD, AB = CD.
7. 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.)
Axiom 5 states that "The whole is greater than the part." This is considered a universal truth because it is a self-evident principle that holds in all contexts and is fundamental to understanding the nature of quantities and relationships.
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