Introduction to Euclid's Geometry

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Questions:

Generic 3

4 x 3 = 12

In the given figure, we have \(\angle ABC=\angle ACB, \angle3=\angle4\). Show that \(\angle 1=\angle2.\)

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 AC =\(1\over2\) AB. Explain by drawing the figure.

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.

Generic 4

2 x 4 = 8

In figure, C is the mid-point of AB and Dis the mid-point of AC. Prove that AD = \(1\over2\) AB.

In the given figure AB = BC and BX = BY. Show that AX = CY. State Euclid's Axiom used.

Multiple Choice Question

15 x 1 = 15

There exists a pair of straight lines that are everywhere equidistant from one another' is a direct consequence of Euclid's

(a)

first postulate

(b)

second postulate

(c)

third postulate

(d)

fifth postulate

John Playfair was a

(a)

french mathematician

(b)

Scottish mathematician

(c)

Indian mathematician

(d)

Egyptian mathematician

Which of the following statement is incorrect?

(a)

A line segment has defined length

(b)

Three line are concurrent id and only if they have a common point

(c)

two lines drawn in a plane always intersected at a point

(d)

One and only one line can be drawn passing through a given point parallel to a given line

10.

The things which are double of same thing are:

(a)

equal

(b)

halves of same thing

(c)

unequal

(d)

double of the same thing

11.

The thing which coincide with one another are:

(a)

equal

(b)

unequal

(c)

half of some thinf

(d)

triple of one another

12.

Euclid stated that things which are equal to the same thing are equal to one another in the form of:

(a)

an axiom

(b)

a definition

(c)

a postulate

(d)

a proof

13.

The things which coincide with one another are:

(a)

equal to one another

(b)

un equal

(c)

double of same thing

(d)

triple of same thing

14.

'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

15.

Two planes intersect each other to form a:

(a)

plane

(b)

Point

(c)

straight line

(d)

angle

16.

The number of line segments determined by three collinear points is:

(a)

two

(b)

Three

(c)

Only one

(d)

Four

17.

The number of lines that can pass through a given point is:

(a)

two

(b)

none

(c)

only one

(d)

infinite many

18.

How many numbers of lines do pass through two distinct points?

(a)

(b)

(c)

(d)

19.

Number of dimension(s) a surface:

(a)

(b)

(c)

(d)

20.

Pythagoras was a student of

(a)

Thales

(b)

Euclid

(c)

both (a) and (b)

(d)

Archimedes

21.

In Indus Valley Civilisation (about 300 b.C), The brick used for construction work were having dimension in the ration

(a)

1 : 3 : 4

(b)

4 : 2 : 1

(c)

4 : 4 : 1

(d)

4 : 3 : 2

1 Marks

5 x 1 = 5

22.

What is a surface?

23.

Give any one example of a geometrical line from your surroundings.

24.

What is a straight line?

25.

Express in variables the things which are double of the same thing.

26.

What does a theorem require?

Generic 2

5 x 2 = 10

27.

Prove that every line segment has one and only one mid-point.

28.

State playfair's axiom. Is it equivalent to one of the Euclid's postulate.

29.

Show that of all the line segments drawn from a given point to a line, not on it, the perpendicular line segment is the shortest.

30.

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

31.

State any two Euclid's axioms.

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Answers Key:

Generic 3

4 x 3 = 12

Given\(\angle ABC=\angle ACB\)
\(\Rightarrow \angle1+\angle4=\angle2+\angle3\)
\(\Rightarrow \angle1+\angle4-\angle4=\angle2+\angle3-\angle3\)
(As, \(\angle3=\angle4\))
\(\Rightarrow \angle1=\angle2\)

Let a line AB have two mid-points, say, C and D. Then,
AC = \(1\over2\) AB ...(1)
AD = \(1\over2\)AB ...(2)
From (1) and (2),
AC = AD.
Things which are equal to the same thing are equal to one another

AC= BC
AC + AC = BC + AC | Equals are added to equals
\(\Rightarrow \) 2AC = AB I BC + AC coincides with AB
\(\Rightarrow \) AC= \(1\over2\)2AB .
\(\Rightarrow \) AC=BC= \(1\over2\)AB.
|Things which are equal to the same thing are equal to one another.

(i) False. This can be seen visually.
(ii) False. This contradicts the Axiom.
[Given two distinct points, there is a unique line that passes through them.]
(iii) True by Euclid's Postulate
[A terminated line can be produced indefinitely.]
(iv) True. If we 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 by the first Axiom of Euclid.
[Things which are equal to the same thing are equal to one another.]

Generic 4

2 x 4 = 8

\(\because\) C is the midpoint of AB
\(\therefore \) AC = CB
AC + AC = CB + AC
| If equals are added to equals, then the wholes are equal (Euclid's Axiom (ii))]
\(\Rightarrow \) 2AC = AB I CB + AC coincides with AB
\(\Rightarrow \) \(1\over2\)(2AC) = \(1\over2\) AB
| Things which are halves of the same thing are equal (Euclid's Axiom (vii»]
\(\Rightarrow \) AC =\(1\over2\)AB
\(\Rightarrow \) \(1\over2\)AC = \(1\over2\)(\(1\over2\)AB)
| Things which are halves of the same thing are equal to one another (Euclid's Axiom (vii))]
\(1\over2\)AC = \(1\over2\)AB
AD = \(1\over4\)AB
\(\because\) D is the mid-point of AC
\(\therefore \)AD = DC =\(1\over2\)AC (as above)

We have
AB = BC
\(\Rightarrow \) AB - BX = BC - BX
|If equals are subtracted from equals, the remainders are equal (Euclid's Axiom (iii))
AB - BX = BC - BY \(|\)\( \because\) BX = BY
AB - BX coincides with AX;
BC - BY coincides with CY
[Things which coincide with one another are equal to one another (Euclid's Axiom (iv))]

Multiple Choice Question

15 x 1 = 15

(d)
fifth postulate

(b)
Scottish mathematician

10.

(a)
equal

11.

(a)
equal

12.

(a)
an axiom

13.

(a)
equal to one another

14.

(a)
an axiom

15.

16.

(b)
Three

17.

(d)
infinite many

18.

(a)
1

19.

20.

(a)
Thales

21.

(b)
4 : 2 : 1

1 Marks

5 x 1 = 5

22.

()
A surface is that which has length and breadth.

23.

()
Meeting place of two walls.

24.

()
Two planes intersect each other to form a straight line.

25.

()
Let, First thing = x
Second thing = y
then, x = 2y

26.

()
Theorem requires a proof.

Generic 2

5 x 2 = 10

27.

Let line segment \(\overline{AB}\) has 2 mid-points, say X and Y
then, \(\frac{AB}{2}=AX\ and\ \frac{AB}{2}=AY\)
\(\therefore\) AX = AY
X and Y coincides

28.

Playfair's Axiom(Statement) : For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l. it is equivalent to Euclid's fifth postulate.

29.

Let AB be perpendicular to a line l and AP is any other line segment.
In right \(\triangle ABP,\angle B>\angle P,(\therefore \angle B=90^o)\)
\(\ \Rightarrow AP>AB\ or\ AB\)

30.

Let x be the weight of Ram and Ravi each. On gaining 2 kg, weight of Ram and Ravi will be (x+2) kg each. According to Euclid's second axiom, when equals are added to equals, the wholes are equal. So, weights of Ram and Ravi are again equal.

31.

Euclid's 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.