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#### Question 1:

What is the difference between a theorem and an axiom?

An axiom is a basic fact that is taken for granted without proof.
Examples:
i) Halves of equals are equal.
ii) The whole is greater than each of its parts.

Theorem: A statement that requires proof is called theorem.
Examples:
i) The sum of all the angles around a point is ${360}^{\circ }$.
ii) The sum of all the angles of triangle is ${180}^{\circ }$.

#### Question 2:

Define the following terms:
(i) Line segment
(ii) Ray
(iii) Intersecting lines
(iv) Parallel lines
(v) Half line
(vi) Concurrent lines
(vii) Collinear points
(viii) Plane

(i) Line segment :A line segment is a part of line that is bounded by two distinct end-points. A line segment has a fixed length.

(ii) Ray:  A line with a start point but no end point and without a definite length is a ray.

(iii) Intersecting lines: Two lines with a common point are called intersecting lines.

(iv) Parallel lines: Two lines in a plane without a common point are parallel lines.

(v) Half line: A straight line extending from a point indefinitely in one direction only is a half line.

(vi) Concurrent lines: Three or more lines intersecting at the same point are said to be concurrent.

(vii) Collinear points: Three or more than three points are said to be collinear if there is a line, which contains all the points.

(viii) Plane: A plane is a surface such that every point of the line joining any two point on it, lies on it.

#### Question 3:

(i) six points
(ii) five lines segments
(iii) four rays
(iv) four lines
(v) four collinear points

(i) Points are A, B, C, D, P and R.

(ii)

(iii)

(iv)

(v) Collinear points are M, E, G and B.

#### Question 4:

(i) two pairs of intersecting lines and their corresponding points of intersection
(ii) three concurrent lines and their points of intersection
(iii) three rays
(iv) two line segments

(i) Two pairs of intersecting lines and their point of intersection are

(ii) Three concurrent lines are

(iii) Three rays are

(iv) Two line segments are

#### Question 5:

(i) How many lines can be drawn through a given point?
(ii) How many lines can be drawn through two given points?
(iii) At how many points can two lines at the most intersect?
(iv) If A, B and C are three collinear points, name all the line segments determined by them.

(i) Infinite lines can be drawn through a given point.

(ii) Only one line can be drawn through two given points.

(iii)  At most two lines can intersect at one point.

(iv) The line segments determined by three collinear points A, B and C are

#### Question 6:

Which of the following statements are true?
(i) A line segment has no definite length.
(ii) A ray has no end-point.
(iii) A line has a definite length.
(iv) A line $\stackrel{↔}{AB}$ is same as line $\stackrel{↔}{BA}$.
(v) A ray $\underset{AB}{\to }$ is same as ray $\underset{BA}{\to }$.
(vi) Two distinct points always determine a unique line.
(vii) Three lines are concurrent if they have a common point.
(viii) Two distinct lines cannot have more than one point in common.
(ix) Two intersecting lines cannot be both parallel to the same line.
(x) Open half-line OA is same as ray $\underset{OA}{\to }$.
(xi) Two lines may intersect at two points.
(xii) Two lines l and m are parallel only when they have no point in common.

(i) False. A line segment has a definite length.

(ii) False. A ray has one end-point.

(iii) False. A line has no definite length.

(iv) True

(v) False. $\stackrel{↔}{BA}$ and $\stackrel{↔}{AB}$ have different end-points.
(vi) True

(vii) True

(viii) True

(ix) True

(x) True

(xi) False. Two lines intersect at only one point.

(xii) True

#### Question 1:

In ancient India, the altars used for household rituals were shaped like
(a) squares and circles
(b) rectangles and squares
(c) triangles and rectangles
(d) trapeziums and pyramids

(a) squares and circles

#### Question 2:

The number of interwoven isosceles triangles in a Sriyantra is
(a) 11
(b) 9
(c) 8
(d) 7

(b) 9

#### Question 3:

Thales hailed from
(a) Babylonia
(b) Egypt
(c) Greece
(d) Rome

(c) Greece

#### Question 4:

Euclid hailed from
(a) India
(b) Greece
(c) Egypt
(d) Babylonia

(b) Greece

#### Question 5:

Pythagoras was a student of
(a) Thales
(b) Euclild
(c) Archimedes
(d) None of these

(a) Thales

#### Question 6:

In Indus Valley Civilisation (about 300 BC), the ratio of dimensions of the bricks used for construction was
(a) 4 : 3 : 1
(b) 4 : 2 : 1
(c) 4 : 3 : 2
(d) 4 : 4 : 1

(b) 4:2:1

#### Question 7:

Which of the following needs proof?
(a) An axiom
(b) A definition
(c) A postulate
(d) A theorem

(d) A theorem

#### Question 8:

Axioms are assumed
(a) definitions
(b) theorems
(c) universal truths in all branches of mathematics
(d) universal truths specific to geometry

(c) universal truths in all branches of mathematics

#### Question 9:

'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 theorem

(b) a definition

#### Question 10:

Euclid's statement that 'all right angles are equal to each other' was in the form of
(a) an axiom
(b) a definition
(c) a postulate
(d) a proof

(a) an axiom

#### Question 11:

Greeks emphasised
(a) inductive reasoning
(b) deductive reasoning
(c) practical use of geometry
(d) analytical geometry

(b) deductive reasoning

A solid has
(a) 0 dimension
(b) 1 dimension
(c) 2 dimensions
(d) 3 dimensions

(d) 3 dimensions

A surface has
(a) 0 dimension
(b) 1 dimension
(c) 2 dimensions
(d) 3 dimensions

(c) 2 dimensions

A point has
(a) 0 dimension
(b) 1 dimension
(c) 2 dimensions
(d) 3 dimensions

(a) 0 dimension

#### Question 15:

Boundaries of solids are
(a) lines
(b) curves
(c) surfaces
(d) points

(c) surfaces

#### Question 16:

Boundaries of surfaces are
(a) lines
(b) curves
(c) points
(d) None of these

(b) curves

#### Question 17:

The side faces of a pyramid are
(a) triangles
(b) squares
(c) trapeziums
(d) polygons

(a) triangles

#### Question 18:

The base of a pyramid is
(a) a triangle only
(b) a square only
(c) a rectangle only
(d) any polygon

(d) any polygon

#### Question 19:

The number of planes passing through three non-collinear points is
(a) 2
(b) 3
(c) 4
(d) 1

(d) 1

#### Question 20:

Into how many chapters did Euclid divide his book Elements ?
(a) 9
(b) 11
(c) 12
(d) 13

(d) 13

#### Question 21:

Which of the following is a true statement?
(a) The floor and a wall of a room are parallel planes.
(b) The ceiling and a wall of a room are parallel planes.
(c) The floor and the ceiling of a room are parallel planes.
(d) Two adjacent walls of a room are parallel planes.

(c)  The floor and the ceiling of a room are parallel planes.

#### Question 22:

Which of the following is a true statement?
(a) Only a unique line can be drawn through a given point.
(b) Infinitely many lines can be drawn through two given points.
(c) If two circles are equal, then their radii are equal.
(d) A line has a definite length.

(c) If two circles are equal, then their radii are equal.

#### Question 23:

Which of the following is a false statement?
(a) An infinite number of lines can be drawn through a given point.
(b) A unique line can be drawn through two given points.
(c) Ray .
(d) A ray has one end-point.

(c)

#### Question 24:

A point C is called the mid-point of a line segment AB if
(a) C is an interior point of AB
(b) AC = CB
(c) C is an interior point of AB, such that AC = CB
(d) AC + CB = AB

(c) C is an interior point of AB, such that AC = CB

#### Question 25:

A point C is said to lie between the points A and B if
(a) AC = CB
(b) AC + CB = AB
(c) points A, C and B are collinear
(d) None of these

(c) points A, C and B are collinear

#### Question 26:

Assertion: Every line segment has a unique mid-point.
Reason: A point C is called the mid-point of a line segment AB if C is an interior point of AB and AC = CB.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.

Assertion :-
Let us consider a line segment AB.
Assume that it has two mid-points, say C and D.

Recall that the midpoint of a line segment divides it into two equal parts and both C and D are mid-points.
That is, AC = BC  and AD = BD.
Since C is the midpoint of AB, A, C and B are collinear.
Therefore, AC + BC = AB..............(1)
Similarly, AD + DB = AB ......(2)
From (1) and (2), we get:
AC + BC = AD + DB
This is a contradiction, unless C and D coincide.
Therefore, our assumption that a line segment AB has two mid-points is incorrect.
Thus, every line segment has one and only one middle point.
Hence proved.

Reason: A point C is called the mid-point of a line segment AB if C is an interior point of AB and AC = CB.Hence, it is true

#### Question 27:

Assertion: An infinite number of lines can be drawn to pass through a given point.
Reason: A line segment has two end-points.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.

Assertion : - Infinite lines can be drawn through a given point.

Reason :-
A line segment has two end points

#### Question 28:

Assertion: 3 + 7 = 9 is a statement.
Reason: A sentence that can be judged to be true or false, but not both, is called a statement.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.

3+7=9 is obviously false and this is a statement.
Using the laws of natural numbers, this statement can be proved to be false.

Thus, 3+7=9 is a statement since this statement is false, and it is not true at the same time.

Thus, Reason and Assertion are true, and the Reason is a correct explanation of the Assertion.

#### Question 29:

Assertion: Ray $\underset{AB}{\to }$ has one end-point A.
Reason: Line segment $\overline{)AB}$ has two end-points A and B.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

​(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.

Ray AB is a segment that begins at A and passes through point B but has no end-point. In contrast, segment AB begins at A and extends to B, where it ends.

#### Question 30:

Assertion: A circle is a rectilinear figure.
Reason: A figure formed of line segments only is called a rectilinear figure.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

(d) Assertion is false and Reason is true.

A rectilinear figure is formed by line segments. For example: rectangles.
Thus, the reason is correct. A circle is not made up of line segments. Therefore, the assertion of a circle being a rectilinear figure is wrong.

#### Question 31:

Match the following columns:

 Column I Column II (a) A line segment has a (p) Infinitely many (b) A ray $\underset{BA}{\to }$ has the end point (q) definite length (c) How many lines can pass through a given point? (r) B (d) How many lines can pass through two given points? (s) Only one
(a) ........
(b) ........
(c) ........
(d) ........

 Column I Column II (a) A line segment has a (q) definite length (b) A ray $\underset{BA}{\to }$ has the end-point (r) B (c) How many lines can pass through a given point? (p) Infinitely many (d) How many lines can pass through two given points? (s) Only one

#### Question 32:

Fill in the blanks
(a) Concurrent lines ...... through a given point.
(b) Two distinct ....... in a plane cannot have more than one point in common.
(c) Two distinct points in a plane determine a ........ line.
(d) A line segment has .......... end-points.

(a) Concurrent lines pass through a given point.
(b) Two distinct lines in a plane cannot have more than one point in common.
(c) Two distinct points in a plane determine a unique line.
(d) A line segment has two end-points.

#### Question 33:

A point C lies between two points A and B, such that AC = CB. Prove that $AC=\frac{1}{2}AB$.

Given: A point C lies between two points A and B.
AC = CB

To prove: $AC=\frac{1}{2}AB$
Proof: Point C lies between two points A and B (given).
So, AC+ BC = AB.

Hence proved.

#### Question 34:

Prove that every line segment has a unique mid-point.

Let us assume that a line segment can have two mid-points, C and D, and try to prove this.
Let C and D be the two mid-points of line segment AB.
According to Euclid's fourth axiom,
AC = BC .....(1)
D is also a mid-point. So,
We have: AB = AB .....(3)
And we know that AB = AC +CB.
From equation (3), we get:
Substituting the values of BC and DB from equations (1) and (2), in equation (3), we get:
Dividing both sides by 2, we get:
This is a contradiction, unless and D coincide.
Therefore ,our assumption that a line segment AB has two mid-points is incorrect.
Thus, every line segment has a unique mid-point.
Hence proved.

#### Question 35:

In the given figure, AC = BD. Prove that AB = CD.

From the given figure, we have:
AC = AB + BC
BD =  BC + CD
It is given that AC = BD.

Therefore, AB + BC =  BC + CD..................(1)
According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal.
Subtracting BC from both sides in equation (1), we get:
AB + BC- BC = BC + CD - BC
AB = CD
Hence proved.

#### Question 36:

L, M and N are three lines in the same plane, such that L intersects M and M||N. Show that L intersects N also.

To prove: L intersects N.
Proof : Suppose L does not intersect N.
This means that lines L and N are coplanar.
Therefore, L should be parallel to N.

But this is a contradiction of the hypothesis.
Thus, our supposition is wrong and L intersects N.
Hence proved.

#### Question 37:

Find the measure of an angle that is 20° more than its complement.

Let the measure of the required angle be x$°$.
Then the measure of its complement = ${\left(90-x\right)}^{\circ }$.

Therefore, the measure of the required angle is ${55}^{\circ }$
.

#### Question 38:

Find the measure of an angle that is 20° less than its supplement.

Let the measure of the angle be x$°$.
Then, the measure of its supplement will be (180 - x)$°$.

Therefore, the measure of the required angle is ${80}^{\circ }$
.

#### Question 39:

Find the measure of an angle if five times its complement is 12° less than twice its supplement.

Let the measure of the angle be x$°$.
Then, its complement = ${\left(90-x\right)}^{\circ }$
And its supplement = ${\left(180-x\right)}^{\circ }$
$⇒5\left(90-x\right)=2\left(180-x\right)-12\phantom{\rule{0ex}{0ex}}⇒450-5x=360-2x-12\phantom{\rule{0ex}{0ex}}⇒450-348=3x\phantom{\rule{0ex}{0ex}}⇒3x=102\phantom{\rule{0ex}{0ex}}⇒x=34$
Therefore, the measure of the required angle is ${34}^{\circ }$.

#### Question 1:

Which of the following needs a proof?
(a) Postulate
(b) Axiom
(c) Definition
(d) Theorem

(d) Theorem

#### Question 2:

How many planes pass through three non-collinear points?
(a) 3
(b) 1
(c) 2
(d) Infinitely many

(b) 1

#### Question 3:

How many lines can be drawn through
(a) a given point?
(b) two given points?

a)  An infinite number of lines can be drawn through a given point.

b) Only one line can be drawn through two given points.

#### Question 4:

A, B and C are three collinear points. How many line segments can be determined by them? Name these line segments.

If A, B and C are three collinear points, then three line segments can be determined by them. These are .

#### Question 5:

In the given figure, AC = BD. Prove that AB = CD.
Figure

From the given figure, we have:
AC  = AB + BC
BD  =  BC + CD
It is given that AC = BD.
So, AB + BC =  BC + CD..................(1)
According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal.
Subtracting BC from both sides in equation (1), we get:
AB + BC- BC = BC + CD - BC
AB = CD

Hence proved.

#### Question 6:

Show that every line segment has one and only one middle point.

Let us consider a line segment AB.
Assume that it has two mid-points, say C and D.

Recall that the midpoint of a line segment divides it into two equal parts and both C and D are mid-points.
That is, AC = BC  and AD = BD.
Since C is the midpoint of AB, A, C and B are collinear.
Therefore, AC + BC = AB..............(1)
Similarly, AD + DB = AB ......(2)
From (1) and (2), we get:
AC + BC = AD + DB
This is a contradiction, unless C and D coincide.
Therefore, our assumption that a line segment AB has two mid-points is incorrect.
Thus, every line segment has one and only one middle point.
Hence proved.

#### Question 7:

Define the following terms:
(a) Parallel lines
(b) Intersecting lines
(c) Concurrent lines

(a) Parallel Lines: Two lines are said to be parallel if

(i) they never meet or never intersect each other even if they are extended to infinity
(ii) they are coplanar

(b)Intersecting linesLines that have one and only one point in common are known as intersecting lines. For intersection, the following conditions have to be met:
(i) A
minimum of two lines are required for intersection.
(ii) The common point where all the intersecting lines meet is called the point of intersection.
(iii)
All the intersecting lines form angles at the point of intersection.

In the figure, there are two lines AB and CD, which intersect at O.

(c) Concurrent lines: A set of lines or curves are said to be concurrent if they intersect at the same point.

In the  above  figure, the three lines are concurrent because they intersect at point P, which is
the point of concurrence.

#### Question 8:

If L, M and N are three straight lines, such that L || M and L || N, prove that M || N.

Given:LM and N are three straight lines, such that L || M and L || N.
To Prove :M
|| N

Proof: Suppose M is not parallel to N. Then, M and N intersect at a point, say P. Now, P is not on L.
Since, P is on M and M || L
Through a point P, outside L , there are two lines M and N both parallel to L
This is absurd.
Therefore, our supposition is wrong.
Hence, M  || N.

#### Question 9:

Which of the following is a true statement?
(a) A line has a definite length.
(b) A ray has two end-points.
(c) A point always determines a unique line.
(d) Three lines are concurrent when they have only one point in common.

(d) Three lines are concurrent when they have only one point in common is a true statement.

#### Question 10:

Which statement is true?
(a) A line segment $\overline{)AB}$, when extended in both directions, is called ray $\underset{AB}{\to }$.
(b)
(c) Ray $\underset{AB}{\to }$ has one end-point A.
(d) Ray $\underset{AB}{\to }$ has two end-points A and B.

(c) Ray $\underset{AB}{\to }$ has one end-point A.

#### Question 11:

Which statement is false?
(a) Two circles are equal only when their radii are equal.
(b) A figure formed by line segments is called a rectilinear figure.
(c) Only one line can pass through a single point.
(d) A terminated line can be produced indefinitely on both sides.

(c) Only one line can pass through a single point.
Infinitely many lines can pass through a single point.

#### Question 12:

From the given figure, name the following:
(a) Three lines
(b) One rectilinear figure
(c) Four concurrent points

(a) $Line\stackrel{↔}{RS}$ and $Line\stackrel{↔}{AB}$
(b) $CEFG$
(c) No point is concurrent.

#### Question 13:

The given question is followed by two statements I and II. The answer is
(a) if the question can be answered by using only one statement and not the other.
(b) if the question can be answered by using either of the two statements alone.
(c) if the question can only be answered by using both the statements.
(d) if the question cannot be answered even by using both the statements.
A point C is the mid-point of the line segment AB if
I. AC = CB
II. C is the interior point of AB.
III. AC = CB and C is the interior point of AB.
The given statement is true only when
(a) I holds
(b) II holds
(c) III holds
(d) none holds

(c) III holds
A point C is the mid-point of the line segment AB if
AC = CB and C is the interior point of AB.

#### Question 14:

Is D the mid-point of the line segment AB? It is given that
I. AE = CB
II. DE = CD

AE = CB..............(Given).............I
DE = CD..............(Given)............II
Subtracting II from I we get ,
AE-DE =  CB-CD......(additive property of equality)
Therefore, the distance between AD and DB is same.
So, D is the mid-point of the line segment AB.

#### Question 15:

How many lines can be drawn using four distinct points in a plane, when
(i) all the points are collinear?
(ii) when no three of the four lines are collinear?

(i)  When the four points are collinear, only one line can be drawn through them.

(ii) When no three of the four lines are collinear, six lines can be drawn.

#### Question 16:

Prove that two distinct lines cannot have more than one point in common.

Proof:
Let us consider that two lines intersect at two distinct points P and Q.
Thus, we see that the two lines l and m pass through two distinct points P and Q.
But this assumption clashes with the axiom, which states that “given two distinct points, there is a unique line that passes through them.”
Thus, our assumption that two lines can pass through two distinct points is  wrong.
So, two distinct lines cannot have more than one point in common.

Hence proved.

#### Question 17:

Let us define a statement as a sentence that can be judged to be true or false.
Which of the following is not a statement?
(a) 3 + 5 = 7.
(b) Kunal is a tall boy.
(c) The sum of the angles of a triangle is 90°.
(d) The angles opposite to equal sides of a triangle are equal.

(b) Kunal is a tall boy.
It cannot be proved in isolation whether ​Kunal is tall. Kunal can only be a tall boy in comparison with other boys or some other individual for this to be a statement.

#### Question 18:

State Euclid's axioms.

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 equals.
iii) If equals are subtracted from equals, the remainders are equal.
iv) Things which coincide with one another are equal to one another.
v) The whole is greater than a part.
vi) Things which are double of the same thing are equal to one another.
vii) Things which are halves of the same thing are equal to one another.

#### Question 19:

Match the following columns.

 Column I Column II (a) How many lines can be drawn through one given point? (p) One only (b) How many lines can be drawn through two given points? (q) Infinitely many (c) How many end-points does a line $\stackrel{↔}{AB}$ have? (r) Two only (d) How many end-points does a line segment $\overline{)AB}$ have? (s) None
(a) .......,
(b) .......,
(c) .......,
(d) .......,

 Column I Column II (a) How many lines can be drawn through one given point? (q) Infinitely many (b) How many lines can be drawn through two given points? (p) One only (c) How many end-points does a line $\stackrel{↔}{AB}$ have? (s) None (d) How many end-points does a line segment $\overline{)AB}$ have? (r) Two only

#### Question 20:

(i)
Assertion: A circle is not a rectilinear figure.
Reason: A figure formed by straight lines only is called a rectilinear figure.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.
(ii)
Assertion: All right angles are equal to one another.
Reason: A unique line passes through a single point.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.