Introduction to Euclid's Geometry Test

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Introduction to Euclid's Geometry Test - 3

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Question 1
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 2
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Fill in the blanks:

(i) A line has _______ dimension(s).
(ii) All ________ angles are equal to one another.
(iii) In a test, Awaab's marks are equal to Delilah's marks. After giving grace marks, marks of both become double. Now Awaab's marks are still equal to Delilah's. This is as per the Euclid's __________ axiom.
(iv) According to the Euclid's axiom, things which _______ one another are equal to one another.

A
(i) one (ii) right (iii) second (iv) coincide with

B
(i) two (ii) acute (iii) third (iv) are opposite to

C
(i) one (ii) obtuse (iii) third (iv) are diagonally opposite to

D
(i) two (ii) acute (iii) third (iv) coincide with

Solution

(i) A line has one dimension because only one coordinate is needed to specify a point on it.
(ii) All right angles are equal to one another because all measure 90°.
(iii) According to the Euclid's second axiom , if equals are added to equals, the wholes are equal.
(iv) According to the Euclid's fourth axiom, things which coincide with one another are equal to one another.

Question 3

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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 4
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Which statement is correct according to Postulate 5?

A
If a straight line falling on two straight lines makes the interior angles on the same side of it taken together more 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.

B
If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than one right angle, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than one right angle.

C
If a straight line falling on two straight lines makes the interior angles on the opposite side of it taken together with two angles (each 30 degrees), then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

D
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.

Solution

According to Euclid's 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 5
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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
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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 7
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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 8
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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.

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Introduction to Euclid's Geometry Test - 3

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Introduction to Euclid's Geometry Test - 3

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Question 1

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

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

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

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

Solution

Question 2

(i) one (ii) right (iii) second (iv) coincide with

(i) two (ii) acute (iii) third (iv) are opposite to

(i) one (ii) obtuse (iii) third (iv) are diagonally opposite to

(i) two (ii) acute (iii) third (iv) coincide with

Solution

Question 3

(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

Question 4

If a straight line falling on two straight lines makes the interior angles on the same side of it taken together more 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.

If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than one right angle, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than one right angle.

If a straight line falling on two straight lines makes the interior angles on the opposite side of it taken together with two angles (each 30 degrees), then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

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.

Solution

Question 5

(I)

(II)

(III)

(IV)

Solution

Question 6

AB < PQ

AB > PQ

AB = PQ

Either (1) or (3)

Solution

Question 7

a + m = b + n

a + b = m – n

a + b = m + n

a – b = m + n

Solution

Question 8

isosceles triangles

circles

parallel and intersecting lines

lines and points

Solution

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