Introduction to Euclid`s Geometry Test

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Introduction to Euclid`s Geometry Test - 2

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Question 1
1 / -0

How many dimensions does a plane surface have?

A
One

B
Two

C
Three

D
Zero

Solution

A plane surface has two dimensions, i.e. length and breadth. Hence, (2) is the correct option.
Question 2
1 / -0

Which of the following statements is true?

A
A point has only length.

B
A point has both length and breadth.

C
A point has no dimension.

D
A point has three dimensions.

Solution

A point has no dimension (i.e. no length, no breadth and no height)
Question 3
1 / -0

How many points are common between two distinct lines?

A
0 or 1

B
2 or 3

C
4 or 5

D
6 or 7

Solution

Distinct lines are two or more lines that are not equal. So, two distinct lines can meet at 0 or 1 point.
Question 4
1 / -0

P is a point on a plane surface. How many lines can pass through P?

A
0

B
1

C
2

D
Infinite

Solution

(4) infinite lines can pass through one plane.
Question 5
1 / -0

Which of the following statements is correct?

(I) If a + b = 5c, b + c = 5d and c = d, then a = d.
(II) If a = b, c = 2d and m = 3n, then a + c - m = b + 2d + 3n.
(III) If a = 2p, b = 2p and c = 2p, then a + b = 2b + c.
(IV) If a + b = 2r and m + n = 2r, then 2a + b = m + 2n.

A
(I)

B
(II)

C
(III)

D
(IV)

Solution

(I) a + b = 5c … (1)
b + c = 5d ... (2)
c = d … (3)
From (1) and (2),
a - c = 5c - 5d
a - d = 5d - 5d [c = d from (3)]
a - d = 0
a = d (True)
Hence, statement I is correct.
(II) If a = b, c = 2d and m = 3n
Adding first two equations,
a + c = b + 2d … (1) [If equals are added to equals, then the wholes are equal.]
Now, subtracting the third equation (m = 3n) from (1),
a + c - m = b + 2d - 3n
Hence, statement II is incorrect.

(III) If a = 2p … (1)
b = 2p … (2)
c = 2p … (3)
Adding (1) and (2),
a + b = 4p … (4)

Adding (2) and (3),
b + c = 4p … (5)
a + b = b + c [Things which are equal to the same thing are equal to one another.]
Hence, statement III is incorrect.

(IV) If a + b = 2r and m + n = 2r
a + b = m + n
[By Euclid's axiom, things which are equal to the same thing are equal to one another.]
Hence, statement IV is incorrect.
Question 6
1 / -0

If p and q are two intersecting lines, also p || and q || m, then what is the relationship between and m?

A
is parallel to m.

B
and m are intersecting lines.

C
will coincide line m.

D
and m are neither parallel nor intersecting.

Solution

p and q are intersecting lines and p || .
Therefore, q intersects .
Now, q and intersect and q || m.
Hence, and m intersect. Hence, (2) is the correct option.
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

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

A
a – b = 0

B
b – d = 0

C
a + c = b + d

D
a – c = b – d

Solution

Given:
a = b
c = d
Now, we have to find a = c from the given options.
From option 1, we are unable to find a relationship between a and c.
From option 2, if this is true, then it definitely gives a = c.
So, this is the required solution.
Question 9
1 / -0

If A = 2k and B = 2k, which of the following relationships is true?

A
A = B

B
A = 2B

C
B = 2A

D
None of these

Solution

Given: A = 2k and B = 2k

A = B [ By Euclid's axiom, things which are equal to the same thing are also equal to one another]
Question 10
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 11
1 / -0

In the given figure, P is a point not lying on m. How many lines which pass through P are parallel to m?

A
Only one

B
Only two

C
Only three

D
Infinite

Solution

By play fair's axiom, for every line m and for every point not lying on m, there exists a unique line passing through that point which is parallel to m.
Question 12
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 13
1 / -0

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

A
a = b + c

B
a + b = b + c

C
a = 2c

D
2a = c

Solution

By Euclid's Axiom:
If a = k, b = k and c = k
Then a + b = 2k, b + c = 2k
Hence, a + b = b + c
Question 14
1 / -0

In the given figure, AB = CD and CD = EF.

Which of the following options is true?

A
AB + CD = EF

B
AB + CD < EF

C
AB + CD = 2EF

D
None of these

Solution

AB = CD and CD = EF,
Then AB + CD = CD + EF
= EF + EF
= 2EF
Question 15
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.