Introduction to Euclid's Geometry

Read the following statement:

“A square is a polygon made up of four line segments, out of which, length of three line segments are equal to the length of fourth one and all its angles are right angles”. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all angles and sides of a square are equal?

Solve the equation a – 15 = 25 and state which axiom do you use here.

Ram and Ravi have the same weight. If they each gain weight by 2 kg, how will their new weights be compared ?

Write whether the following statements are True or False? Justify your answer.

(i) Pyramid is a solid figure, the base of which is a triangle or square or some other polygon and its side faces are equilateral triangles that converges to a point at the top.

(ii) In Vedic period, squares and circular shaped altars were used for household rituals, while altars whose shapes were combination of rectangles, triangles and trapeziums were used for public worship.

(iii) In geometry, we take a point, a line and a plane as undefined terms.

(iv) If the area of a triangle equals the area of a rectangle and the area of the rectangle equals that of a square, then the area of the triangle also equals the area of the square.

(v) Euclid’s fourth axiom says that everything equals itself.

(vi) The Euclidean geometry is valid only for figures in the plane.

Read the following statement: An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all its angles are 60° each. Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all sides and all angles are equal in a equilateral triangle

Study the following statement: “Two intersecting lines cannot be perpendicular to the same line”. Check whether it is an equivalent version to the Euclid’s fifth postulate. [Hint: Identify the two intersecting lines l and m and the line n in the above statement.]

Read the following statements which are taken as axioms: (i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal. (ii) If a transversal intersect two parallel lines, then alternate interior angles are equal. Is this system of axioms consistent? Justify your answer

Read the following two statements which are taken as axioms: (i) If two lines intersect each other, then the vertically opposite angles are not equal. (ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°. Is this system of axioms consistent? Justify your answer.

Read the following axioms: (i) Things which are equal to the same thing are equal to one another. (ii) If equals are added to equals, the wholes are equal. (iii) Things which are double of the same thing are equal to one another. Check whether the given system of axioms is consistent or inconsistent.

Two salesmen make equal sales during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.

It is known that x + y = 10 and that x = z. Show that z + y = 10?

The three steps from solids to points are: (A) Solids - surfaces - lines - points (B) Solids - lines - surfaces - points (C) Lines - points - surfaces - solids (D) Lines - surfaces - points – solids

The number of dimensions, a solid has: (A) 1 (B) 2 (C) 3 (D) 0

The number of dimensions, a surface has: (A) 1 (B) 2 (C) 3 (D) 0

The number of dimension, a point has: (A) 0 (B) 1 (C) 2 (D) 3

The number of dimension, a point has: (A) 0 (B) 1 (C) 2 (D) 3

Euclid divided his famous treatise “The Elements” into: (A) 13 chapters (B) 12 chapters (C) 11 chapters (D) 9 chapters

The total number of propositions in the Elements are: (A) 465 (B) 460 (C) 13 (D) 55

Boundaries of solids are: (A) surfaces (B) curves (C) lines (D) points

Boundaries of surfaces are: (A) surfaces (B) curves (C) lines (D) points

In Indus Valley Civilization (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio (A) 1 : 3 : 4 (B) 4 : 2 : 1 (C) 4 : 4 : 1 (D) 4 : 3 : 2

A pyramid is a solid figure, the base of which is (A) only a triangle (B) only a square (C) only a rectangle (D) any polygon

The side faces of a pyramid are: (A) Triangles (B) Squares (C) Polygons (D) Trapeziums

It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is: (A) First Axiom (B) Second Axiom (C) Third Axiom (D) Fourth Axiom

In ancient India, the shapes of altars used for house hold rituals were: (A) Squares and circles (B) Triangles and rectangles (C) Trapeziums and pyramids (D) Rectangles and squares

The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is: (A) Seven (B) Eight (C) Nine (D) Eleven

Greek’s emphasized on: (A) Inductive reasoning (B) Deductive reasoning (C) Both A and B (D) Practical use of geometry

In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for: (A) Public worship (B) Household rituals (C) Both A and B (D) None of A, B, C

Euclid belongs to the country: (A) Babylonia (B) Egypt (C) Greece (D) India

Thales belongs to the country: (A) Babylonia (B) Egypt (C) Greece (D) Rome

Pythagoras was a student of: (A) Thales (B) Euclid (C) Both A and B (D) Archimedes

Which of the following needs a proof? (A) Theorem (B) Axiom (C) Definition (D) Postulate

Euclid stated that all right angles are equal to each other in the form of (A) an axiom (B) a definition (C) a postulate (D) a proof

‘Lines are parallel if they do not intersect’ is stated in the form of (A) an axiom (B) a definition (C) a postulate (D) a proof

Euclidean geometry is valid only for curved surfaces.

The boundaries of the solids are curves

The edges of a surface are curves.

The things which are double of the same thing are equal to one another.

If a quantity B is a part of another quantity A, then A can be written as the sum of B and some third quantity C.

The statements that are proved are called axioms.

“For every line l and for every point P not lying on a given line l, there exists a unique line m passing through P and parallel to l ” is known as Playfair’s axiom.

Two distinct intersecting lines cannot be parallel to the same line

Attempts to prove Euclid’s fifth postulate using the other postulates and axioms led to the discovery of several other geometries.

Does Euclid's fifth postulate imply the existence of parallel lines? Explain.

How would you rewrite Euclid's fifth postulate so that it would be easier to understand?

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.)

In the following figure, if AC = BD, then prove that AB = CD.

In the above question, point C is called a mid-point of line segment AB, prove that every line segment has one and only one mid-point.

If a point C lies between two points A and B such that AC = BC, then prove that. Explain by drawing the figure.

Consider the two 'postulates' given below:

(i) Given any two distinct points A and B, there exists a third point C, which is between A and B.

(ii) There exists 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.

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

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 the following figure, if AB = PQ and PQ = XY, then AB = XY.