Solution of Chapter 2. Geometry (Exemplar Mathematics Book)

Number of lines passing through five points such that no three of them are collinear is

The number of diagonals in a heptagon is

Number of line segments in Fig. 2.5 is

Measures of the two angles between hour and minute hands of a clock at 9 O’ clock are

If a bicycle wheel has 48 spokes, then the angle between a pair of two consecutive spokes is

In Fig. 2.6, ∠XYZ cannot be written as

In Fig 2.7, if point A is shifted to point B along the ray PX such that PB = 2PA, then the measure of ∠BPY is

The number of angles in Fig. 2.8 is

The number of obtuse angles in Fig. 2.9 is

The number of triangles in Fig. 2.10 is

If the sum of two angles is greater than 180°, then which of the following is not possible for the two angles?

If the sum of two angles is equal to an obtuse angle, then which of the following is not possible?

A polygon has prime number of sides. Its number of sides is equal to the sum of the two least consecutive primes. The number of diagonals of the polygon is

In Fig. 2.11, AB = BC and AD = BD = DC.

The number of isosceles triangles in the figure is

in Fig. 2.12,

∠BAC = 90° and AD ⊥ BC.

The number of right triangles in the figure is

In Fig. 2.13, PQ ⊥ RQ, PQ = 5 cm and QR = 5 cm. Then ∆ PQR is

An angle greater than 180° and less than a complete angle is called ______.

The number of diagonals in a hexagon is _____.

A pair of opposite sides of a trapezium are ________.

In Fig. 2.14, points lying in the interior of the triangle PQR are ______, that in the exterior are ______ and that on the triangle itself are ______.

In Fig. 2.15, points A, B, C, D and E are collinear such that

AB = BC = CD = DE. Then

(a) AD = AB + ____

(b) AD = AC + ____

(c) mid-point of AE is ____

(d) midpoint of CE is _____

(e) AE = _____ × AB.

In Fig. 2.16,

(a) ∠AOD is a/an _____ angle

(b) ∠COA is a/an _____ angle

(c) ∠AOE is a/an _____ angle

The number of triangles in Fig. 2.17 is ______.

Their names are _______.

Number of angles less than 180° in Fig. 2.17 is ______ and their names are __________.

The number of straight angles in Fig. 2.17 is _____.

The number of right angles in a straight angle is _____ and that in a complete angle is_____.

The number of common points in the two angles marked in Fig. 2.18 is _____.

The number of common points in the two angles marked in Fig. 2.19 is _____.

The number of common points in the two angles marked in Fig. 2.20 ______.

The number of common points in the two angles marked in Fig. 2.21 is ______.

The common part between the two angles BAC and DAB in Fig. 2.22 is _______.

A horizontal line and a vertical line always intersect at right angles.

If the arms of an angle on the paper are increased, the angle increases.

If the arms of an angle on the paper are decreased, the angle decreases.

If line PQ || line m, then line segment PQ || m

Two parallel lines meet each other at some point.

Measures of ∠ABC and ∠CBA in Fig. 2.23 are the same.

Two line segments may intersect at two points.

Many lines can pass through two given points.

Only one line can pass through a given point.

Two angles can have exactly five points in common.

Name all the line segments in Fig. 2.24.

Name the line segments shown in Fig. 2.25.

State the mid points of all the sides of Fig. 2.26.

Name the vertices and the line segments in Fig. 2.27.

Write down fifteen angles (less than 180°) involved in Fig. 2.28.

Name the following angles of Fig. 2.29, using three letters:

(a) ∠1

(b) ∠2

(c) ∠3

(d) ∠1 + ∠2

(e) ∠2 + ∠3

(f) ∠1 + ∠2 + ∠3

(g) ∠CBA – ∠1

Name the points and then the line segments in each of the following figures (Fig. 2.30):

Which points in Fig. 2.31, appear to be mid-points of the line segments? When you locate a mid-point, name the two equal line segments formed by it.

Is it possible for the same

(a) line segment to have two different lengths?

(b) angle to have two different measures?

Will the measure of ∠ABC and of ∠CBD make measure of ∠ABD in Fig. 2.32?

Will the lengths of line segment AB and line segment BC make the length of line segment AC in Fig. 2.33?

Draw two acute angles and one obtuse angle without using a protractor. Estimate the measures of the angles. Measure them with the help of a protractor and see how much accurate is your estimate.

Look at Fig. 2.34. Mark a point

(a) A which is in the interior of both ∠1 and ∠2.

(b) B which is in the interior of only ∠1.

(c) Point C in the interior of ∠1.

Now, state whether points B and C lie in the interior of ∠2 also.

Find out the incorrect statement, if any, in the following:

An angle is formed when we have

(a) two rays with a common end-point

(b) two line segments with a common end-point

(c) a ray and a line segment with a common end-point

In which of the following figures (Fig. 2.35),

(a) perpendicular bisector is shown?

(b) bisector is shown?

(c) only bisector is shown?

(d) only perpendicular is shown?

What is common in the following figures (i) and (ii) (Fig. 2.36.)?

Is Fig. 2.36 (i) that of triangle? if not, why?

If two rays intersect, will their point of intersection be the vertex ofan angle of which the rays are the two sides?

In Fig. 2.37,

(a) name any four angles that appear to be acute angles.

(b) name any two angles that appear to be obtuse angles.

In Fig. 2.38,

(a) is AC + CB = AB?

(b) is AB + AC = CB?

(c) is AB + BC = CA?

In Fig. 2.39.

(a) What is AE + EC?

(b) What is AC – EC?

(c) What is BD – BE?

(d) What is BD – DE?

Using the information given, name the right angles in each part of Fig. 2.40:

(a) BA ⊥BD

(b) RT ⊥ST

(c) AC ⊥BD

(d) RS ⊥RW

(e) AC ⊥BD

(f) AE ⊥CE

(g) AC ⊥CD

(h) OP ⊥AB

Fig. 2.40

What conclusion can be drawn from each part of Fig. 2.41, if

(a) DB is the bisector of ∠ADC?

(b) BD bisects ∠ABC?

(c) DC is the bisector of ∠ADB, CA ⊥ DA and CB ⊥ DB?

An angle is said to be trisected, if it is divided into three equal parts. If in Fig. 2.42, ∠BAC = ∠CAD = ∠DAE, how many trisectors are there for ∠BAE?

How many points are marked in Fig. 2.43?

How many line segments are there in Fig. 2.43?

In Fig. 2.44, how many points are marked? Name them.

How many line segments are there in Fig. 2.44? Name them.

In Fig. 2.45 how many points are marked? Name them.

In Fig. 2.45 how many line segments are there? Name them.

In Fig. 2.46, how many points are marked? Name them.

In Fig. 2.46 how many line segments are there? Name them.

In Fig. 2.47, O is the centre of the circle.

(a) Name all chords of the circle.

(b) Name all radii of the circle.

(c) Name a chord, which is not the diameter of the circle.

(d) Shade sectors OAC and OPB.

(e) Shade the smaller segment of the circle formed by CP.

Can we have two acute angles whose sum is

(a) an acute angle? Why or why not?

(b) a right angle? Why or why not?

(c) an obtuse angle? Why or why not?

(d) a straight angle? Why or why not?

(e) a reflex angle? Why or why not?

Can we have two obtuse angles whose sum is

(a) a reflex angle? Why or why not?

(b) a complete angle? Why or why not?

Write the name of

(a) vertices

(b) edges, and

(c) faces of the prism shown in Fig. 2.48.

How many edges, faces and vertices are there in a sphere?

Draw all the diagonals of a pentagon ABCDE and name them.