Introduction to Euclid's Geometry Test

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Introduction to Euclid's Geometry Test - 7

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Question 1
1 / -0

If a line AB coincides with another line PQ, which of the following relationships is true?

A
AB < PQ

B
AB > PQ

C
AB = PQ

D
Either (1) or (3)

Solution

Coinciding lines overlap each other and two lines can be overlapped if their lengths are equal.
OR
By Euclid's axiom, things that coincide with one another are equal to one another.
Question 2
1 / -0

Two points A and B lie on a plane. How many different lines can pass through A and B both?

A
0

B
Infinite

C
1

D
2

Solution

As by Euclid's axiom, for two distinct points given, there is a unique line that passes through them.

From figure also, it is clear that only one line passes through points A and B in a plane.
Question 3
1 / -0

If a = m and b = n, then which of the following relationships is true?

A
a + m = b + n

B
a + b = m – n

C
a + b = m + n

D
a – b = m + n

Solution

If a = m and b = n
a + b = m + n [If equals are added to equals, then wholes are equal.]
Question 4
1 / -0

In the given figure, AB = BD. If BP = PD, then which of the following relationships is true?

A
AP = 4PD

B
AP = 3PD

C
AP = PD

D
AD = 2BD + PD

Solution

AP = AB + BP

= BD + PB [ AB = BD]
= BP + PD + PB
= PD + PD + PD [ BP = PD]
AP = 3PD
Question 5
1 / -0

Euclid's fifth postulate gives an idea about

A
isosceles triangles

B
circles

C
parallel and intersecting lines

D
lines and points

Solution

Euclid's fifth postulate gives an idea about parallel and intersecting lines. According to the fifth postulate, 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.
Question 6
1 / -0

A plane has ________.

A
two dimensions

B
three dimensions

C
one dimension

D
no dimensions

Solution

A plane has two dimensions, i.e. length and breadth.
Question 7
1 / -0

^{_If_{and m are intersecting lines and p is any third line, then which of the following is false?}}

A
^{_Both_{and m cannot be perpendicular to p.}}

B
^{_Both_{and m cannot be parallel to p.}}

C
^{_Neither_{nor m can be both perpendicular and parallel to p at the same time.}}

D
^{_Both_{and m can be parallel, but not perpendicular to p.}}

Solution

There is no line which can be parallel and perpendicular to both and m.
Also, as both lines are intersecting, they cannot be both parallel to the same line.
Hence, (4) is the correct option.
Question 8
1 / -0

How many circles can be drawn having AB as a tangent and P as the point of contact?

A
Zero

B
Only 1

C
Only 2

D
Infinity

Solution

Since a circle can be drawn with any radius and a centre, so infinite circles can be drawn in this case.
Question 9
1 / -0

Which of the following statements is true?

A
A circle can be drawn only with a specific value of radius.

B
A circle can be drawn with any centre and any radius.

C
A circle cannot be drawn with very large value of radius.

D
A circle can be drawn with a specific centre only.

Solution

By Euclid's postulate 3, a circle can be drawn with any centre and any radius.
Question 10
1 / -0

If a = b and c = d, which of the following relationships is true?

A
a – c = b + d

B
a – c = b – d

C
a + c = b – d

D
c – a = d + b

Solution

If a = b and c = d, then a – c = b – d [By Euclid's axiom, if equals are subtracted from equals, the remainders are equal.]
Question 11
1 / -0

Which of the following statements is false?

A
The part is greater than the whole.

B
Things which are double of the same things are equal to one another.

C
Things which are halves of the same things are equal to one another.

D
If equals are subtracted from equals, the remainders are equal.

Solution

The part is not greater than the whole.
Rather the whole is greater than the part. [By Euclid's axiom]
Question 12
1 / -0

If a + b = 2c, b + c = 2d and c = d, then which of the following relationships is true?

A
a = d

B
a = 3d

C
a = c – d

D
a = 5c

Solution

Given:
a + b = 2c … (1)
b + c = 2d ... (2)
c = d … (3)
From (1) and (2),
a - c = 2c - 2d
a - d = 2d - 2d [c = d from (3)]
a - d = 0
a = d
Question 13
1 / -0

In the given figure, lines AB and CD are intersected by another line PQ. If AB and CD intersect each other on the RHS of PQ, which of the following measures of angles cannot be the value of 1 + 2?

A
190°

B
260°

C
279°

D
180°

Solution

Since 1 and 2 are co-interior angles and lines AB and CD intersect each other on the RHS of PQ, so these are not parallel. Also, the sum of co-interior angles is 180° only when the lines are parallel.
Question 14
1 / -0

Which of the following statements is true?

A
A terminated line can be produced finitely from both ends.

B
A terminated line can be produced indefinitely on both the sides.

C
A terminated line can be produced finitely from one end and infinitely from the other end.

D
A terminated line cannot be measured.

Solution

By Euclid's postulate 2, a terminated line can be produced indefinitely on both the sides.
Question 15
1 / -0

Which of the following statements is true?

A
A straight line lies evenly with all the points on itself.

B
A straight line does not lie evenly with all the points on itself.

C
Only end points lie on the straight line.

D
There is no difference between a plane and a surface.

Solution

According to Euclid, "A straight line lies evenly with all points on itself".
Question 16
1 / -0

There are two concentric circles. The radius of the outer circle is 12 cm. Which of the following measurements cannot be the radius of the inner circle?

A
13 cm

B
10 cm

C
11 cm

D
0.275 cm

Solution

As the radius of the outer circle (12 cm) is always greater than the inner circle, the inner concentric circle cannot have radius greater than 12 cm.
So, 13 cm cannot be the radius of the inner circle.
Question 17
1 / -0

If B is a point on line AC, then which of the following is true?

A
AB = AC

B
BC = AC

C
AB + BC = 2AC

D
AB + BC = AC

Solution

We know that lines that coincide with each other are equal to one another.
Since AB + BC coincides with AC, AB + BC = AC.
Hence, (4) is the correct option.
Question 18
1 / -0

In the given figure, AB = CD = EF.

Choose the correct relationship among the following.

A
AB + EF = CD

B
AB + EF < CD

C
AB + CD = 2EF

D
AB > CD + EF

Solution

Given:
AB = CD … (1)
CD = EF … (2)

Now, adding these two equations,
AB + CD = CD + EF
= EF + EF [CD = EF]
= 2EF
Question 19
1 / -0

Which of the following statements is true?

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

B
A plane has no dimensions.

C
The edges of a surface are planes.

D
The ends of a line have only one dimension.

Solution

i) Two distinct intersecting lines cannot be parallel to the same line. This is correct.

ii) A plane has no dimensions.
This is false as a plane has two dimensions.

iii) The edges of a surface are planes.
This is false as a plane is a flat, two-dimensional surface that extends infinitely.

iv) The ends of a line have only one dimension.
This is false as the ends of a line have no dimensions.
Question 20
1 / -0

Which of the following statements is incorrect?

A
A line is a breadthless length.

B
The ends of a line segment are points.

C
The edges of a surface are points.

D
A point has no part.

Solution

All the statements are correct, except (3). The edges of a surface are lines.
Question 21
1 / -0

l and m are two distinct non-parallel lines and t is the transversal. If the co-interior angles on the RHS of the transversal are 75° and 85° and the co-interior angles on the LHS of the transversal are 95° and 105°, then which of the following statements is true?

A
The lines and m will meet on the left hand side.

B
The lines and m will meet on the right hand side.

C
The lines and m will never meet.

D
The lines and m will coincide.

Solution

According to Euclid's fifth postulate, the side on which the sum of co-interior angles is less than 180° is the side where the lines meet.

According to the question,
75° + 85° = 160° < 180°
The lines will meet on the right hand side.
Hence, option (2) is correct.
Question 22
1 / -0

Determine whether the given statements are true (T) or false (F):

(i) The whole is always less than the part.
(ii) If x = y and if a = b, then x - a = y - b.
(iii) Two distinct intersecting lines can't be parallel to the same line.

A
(i) - F, (ii) - T, (iii) - F

B
(i) - F, (ii) - F, (iii) - T

C
(i) - F, (ii) - T, (iii) - T

D
(i) - F, (ii) - F, (iii) - F

Solution

(i) False
This statement is false and does not follow the Euclid's axiom that the whole is greater than the part
(ii) True
This statement is true and follows the Euclid's axiom: ''If equals are subtracted from equals, the remainders are equal.''
(iii) True
This statement is true and follows the Euclid's axiom: ''Two distinct intersecting lines cannot be parallel to the same line.''

Question 23

1 / -0

Match the following:

Column 1	Column 2
(i) It has no length.	(a) Straight line
(ii) It is a breadth less length.	(b) Surface
(iii) It has only length and breath.	(c) Line
(iv) It lies evenly with the points on itself.	(d) Point

(i) - (a); (ii) - (c); (iii) - (b); (iv) - (d)

(i) - (b); (ii) - (c); (iii) - (a); (iv) - (d)

(i) - (d); (ii) - (c); (iii) - (b); (iv) - (a)

(i) - (c); (ii) - (d); (iii) - (b); (iv) - (a)

Solution

According to definitions given by Euclid,
A point has no length.
A line is a breadth less length.
A surface has only length and breath.
Straight line is a line which lies evenly with the points on itself.

Question 24
1 / -0

Directions: Read the given statement and choose the correct option accordingly.

Statement (I): P is a point not lying on m. There is only one line parallel to m that can pass through point P.
Statement (II): Infinite number of planes can pass through three non-collinear points.

A
Only statement I is true.

B
Only statement II is true.

C
Both statements I and II are true.

D
Neither of the statements is true.

Solution

Statement I:
If P is a point not lying on m, then there is only one line parallel to m that can pass through P. So, statement I is true.

Statement II:

Let A, B and C be three non-collinear points.
Then, only one plane can pass through three non-collinear points.
So, statement (II) is false.
Question 25
1 / -0

Fill in the blanks:

(i) A P is that which has length and breadth only.

(ii) The edge of a surface is a Q .

(iii) A R is a surface which lies evenly with straight lines on itself.

(iv) Boundaries of a surface can be S or straight lines.

A
P - surface, Q - line, R - plane surface, S - planes

B
P - plane surface, Q - line, R - surface, S - surfaces

C
P - surface, Q - line, R - plane surface, S - curves

D
P - curve, Q - line, R - plane surface, S - curves

Solution

(i) A surface is that which has length and breadth only.
(ii) The edge of a surface is a line.
(iii) A plane surface is a surface which lies evenly with straight lines on itself.
(iv) Boundaries of a surface can be curves or straight lines.

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Introduction to Euclid's Geometry Test - 7

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Introduction to Euclid's Geometry Test - 7

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Question 1

AB < PQ

AB > PQ

AB = PQ

Either (1) or (3)

Solution

Question 2

0

Infinite

1

2

Solution

Question 3

a + m = b + n

a + b = m – n

a + b = m + n

a – b = m + n

Solution

Question 4

AP = 4PD

AP = 3PD

AP = PD

AD = 2BD + PD

Solution

Question 5

isosceles triangles

circles

parallel and intersecting lines

lines and points

Solution

Question 6

two dimensions

three dimensions

one dimension

no dimensions

Solution

Question 7

Both and m cannot be perpendicular to p.

Both and m cannot be parallel to p.

Neither nor m can be both perpendicular and parallel to p at the same time.

Both and m can be parallel, but not perpendicular to p.

Solution

Question 8

Zero

Only 1

Only 2

Infinity

Solution

Question 9

A circle can be drawn only with a specific value of radius.

A circle can be drawn with any centre and any radius.

A circle cannot be drawn with very large value of radius.

A circle can be drawn with a specific centre only.

Solution

Question 10

a – c = b + d

a – c = b – d

a + c = b – d

c – a = d + b

Solution

Question 11

The part is greater than the whole.

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.

If equals are subtracted from equals, the remainders are equal.

Solution

Question 12

a = d

a = 3d

a = c – d

a = 5c

Solution

^{_Both_{and m cannot be perpendicular to p.}}

^{_Both_{and m cannot be parallel to p.}}

^{_Neither_{nor m can be both perpendicular and parallel to p at the same time.}}

^{_Both_{and m can be parallel, but not perpendicular to p.}}