Solution of Chapter 5. Introduction To Euclids Geometry (NCERT Mathematics Exemplar Book)

The three steps from solids to points are:

The number of dimensions, a solid has:

The number of dimensions, a surface has:

The number of dimension, a point has:

Euclid divided his famous treatise “The Elements” into:

The total number of propositions in the Elements are:

Boundaries of solids are:

Boundaries of surfaces are:

In Indus Valley Civilization (about 3000 B.C.), the bricks used for construction work were having dimensions in the ratio

A pyramid is a solid figure, the base of which is

The side faces of a pyramid are:

It is known that if x + y = 10 then x + y + z = 10 + z. The Euclid’s axiom that illustrates this statement is:

In ancient India, the shapes of altars used for house hold rituals were:

The number of interwoven isosceles triangles in Sriyantra (in the Atharvaveda) is:

Greek’s emphasized on:

In Ancient India, Altars with combination of shapes like rectangles, triangles and trapeziums were used for:

Euclid belongs to the country:

Thales belongs to the country:

Pythagoras was a student of:

Which of the following needs a proof?

Euclid stated that all right angles are equal to each other in the form of

‘Lines are parallel if they do not intersect’ is stated in the form of

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

Euclidean geometry is valid only for curved surfaces.

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

The boundaries of the solids are curves.

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

The edges of a surface are curves.

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

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

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

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.

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

The statements that are proved are called axioms.

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

“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.

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

Two distinct intersecting lines cannot be parallel to the same line.

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

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

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?

Look at the Fig. 5.3. Show that length AH > sum of lengths of AB + BC + CD.

In the Fig.5.4, we have AB = BC, BX = BY. Show that AX = CY.

In the Fig.5.5, we have X and Y are the mid-points of AC and BC and AX = CY. Show that AC = BC.

In the Fig.5.6, we have BX = 1/2 AB and BY = 1/2 BC and AB = BC. Show that BX = BY.

In the Fig.5.7, we have ∠1 =∠2, ∠2 =∠3. Show that ∠1 =∠3.

In the Fig. 5.10, we have AC = DC, CB = CE. Show that AB = DE.

In the Fig. 5.11, if OX = 1/2 XY, PX = 1/2 XZ and OX = PX, show that XY = XZ.

In the Fig.5.12:

(i) AB = BC, M is the mid-point of AB and N is the mid- point of BC. Show that AM = NC.

(ii) BM = BN, M is the mid-point of AB and N is the mid-point of BC. Show that AB = BC.

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 an 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 intersects 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.