Understanding Elementary Shapes

Algebraic Expressions

An expression, its terms and factors of the terms can be represented by a tree diagram to make it easily comprehensible to you......

Data Handling

The word data means information in the form of numerical figures or a set of given facts. E.g. The percentage of marks scored by 10 students.......

The Triangles and its Properties

A closed figure formed by joining three non-collinear points is called a triangle. The three sides and three angles of a triangle are collectively known as elements of the triangle......

Visualising Solid Shapes

A solid is any enclosed three-dimensional shape, i.e., it has 3 dimensions- length, width and height, whereas there are some common (flat) shapes which can be easily drawn on paper. They have only.....

Fractions

Fractions having the same denominator are called like fractions, whereas fractions having different denominator are called unlike fractions......

Perimeter and Area

Mensuration : The process, art or the act of measuring is called mensuration. Anything that can be measured is said to be mensurable.......

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Understanding Elementary Shapes

1.0Measuring Line Segments

When we measure a line segment, we measure its length or distance from one end point to the other. When we measure the length, we must know the units of measurement. Today the metric system is used at most universally. The standard unit in this system is metre.

Conversion of Units of Length

Conversion	Equivalent
10 millimetres (mm)	= 1 centimetre (cm)
10 centimetres (cm)	= 1 decimetre (dm)
10 decimetres (dm)	= 1 metre (m) = 1000 mm
10 metres (m)	= 1 decametre (dam)
10 decametres (dam)	= 1 hectometre (hm)
10 hectometres (hm)	= 1 kilometre (km) = 1000 m

However most often we use the following units. 10 millimetres = 1 centimetres 100 centimetres $= 1$ metres 1000 metres $= 1$ kilometre $1 cm = 0.3937$ inch. By using a ruler: The ruler has centimetre and millimetre marks on one edge and other edge is divided into inches.

One small division $=$ One-tenth of one $cm = 0.1 cm = 1 mm$ Two small division $=$ Two-tenths of one centimetre $= 0.2 cm = 2 mm$

There is a global standard, the international system of units (SI), the modern form of the metric system.

The distance between the endpoints of a line segment is called its length.

2.0Comparison of Line Segments

Comparison by Observation

The simplest way of comparing two line segments is to observe their lengths. Here we can easily observe that line segment 'b' is placed directly below line segment 'a'. Line segment 'a' extends further to the right.

But if the lengths are almost equal then comparison by observation is not easy.

Comprision by Observation if length are not Equal

Comparison by Divider

Let us compare the line segments $\overline{AB}$ and $\overline{CD}$ , using a divider.

Place one point of the divider on A and open the other leg of the divider until the other point coincides with B . This measures the length of $\overline{AB}$ . Now take the divider as it is and place one point of the divider at C and the other point along $\overline{CD}$ . We will observe: (i) If the other point touches $\overline{CD}$ exactly at D , then $\overline{AB} = \overline{CD}$ . (ii) If the other point of the divider is beyond the point D on $\overline{CD}$ , then $\overline{AB} > \overline{CD}$ . (iii) If the other point is between $C$ and $D$ on $\overline{CD}$ , then $\overline{AB} < \overline{CD}$ .

Comparison by Tracing

We can also compare two-line segments, say $\overline{AB}$ and $\overline{CD}$ , by tracing one of them and overlapping the traced line on the other, with one endpoint coinciding. We can easily make out which line is longer, which is shorter, or whether they are both equal. $\overline{AB}$ is placed on $\overline{CD}$ , with the endpoints C and A coinciding. Since the other two endpoints $B$ and $D$ do not coincide, we can say that

(i) $\overline{AB}$ is not equal to $\overline{CD}$ . (ii) $\overline{AB}$ is shorter than $\overline{CD}$ as the endpoint B of $\overline{AB}$ falls short of D , the endpoint of $\overline{CD}$ .

$500 cm < 1 km$

3.0Measure of Angles

The magnitude or measure of the angle is the measure of rotation. Suppose a ray OP starts rotating around 0 , from the fixed position OA to different position $OP_{1}, OP_{2}, OP_{3}$ etc. then measure of the angle will equal the measure of this rotation.

Measuring Angles Using Protractor

The word angle comes from the Latin word angulus, meaning "a corner".

The protractor is an instrument used to measure angles and draw angles of required magnitude. Suppose you have to measure the angle BAC. Place the protractor such that its centre falls on the vertex A of the angle and its horizontal edge (zero line) on the arm AC. Now look at the protractor to find out which line of division on the rim falls on the arm AB .

Read the degree measure from the protractor, use the anticlockwise, i.e. the inner scale. Thus, by measurement $∠ BAC = 6 0^{\circ}$ .

Degree Measure of An Angle

$OQ$ rotates from position $OP$ . When it has made one complete rotation, it reaches $OP$ again. We say that the angle thus formed is 360 degrees. It is written as $36 0^{\circ}$ . In other words, a circle is made up of $36 0^{\circ}$ .

Let us take the example of the face of a clock. It is divided into 12 equal parts. The angle that the arms include between each other, say, at 10 'o clock is exactly $\frac{2}{12}$ of the circle, that is $\frac{2}{12}$ of $36 0^{\circ}$ $= \frac{2}{12} \times 36 0^{\circ} = 6 0^{\circ}$ . At 1.00 a.m. or 1.00 p.m. this angle is $3 0^{\circ}$ and at 3.00 a.m. or $3.00 p . m$ . it is $9 0^{\circ}$ .

The turn (or full circle, revolution, rotation, or cycle) is one full circle. $r$ in rpm (revolutions per minute). 1 turn $= 36 0^{\circ}$ $= 2 prad = 400 grad =$ 4 right angles. SPOT LIGHT

The turn from north to east is by a right angle. The turn from north to south is by two right angles. It is called a straight angle.

Rotation Round The Clock

When the minute hand of a clock starts at 12 and reaches at 3 , it has reached quarter past and has made a quarter of a rotation and has turned through an angle of magnitude $9 0^{\circ}$ . At 6 (half past) the minute hand has made $\frac{1}{2}$ of a rotation and turned through an angle of measure $18 0^{\circ}$ . At 9 (quarter to), it has made $(\frac{3}{4})$ three quarter of a rotation and turned through an angle of measure $27 0^{\circ}$ . When the minute hand reaches 12 , it has moved exactly once round the clock, i.e., it has made one rotation and through an angle of measure $36 0^{\circ}$ .

Directions

You are familiar with the concept of direction. There are four main directions North (N), South (S), East (E), and West (W). Jammu is to the North of Delhi, Kolkata is to its East, Rajkot to its West and Cochin to its South. Midway between there are the four sub-directions, namely North-East (N.E.), South-East (S.E.), North-West (N.W.) and South-West (S.W.).

Degree, Minutes and Seconds

A degree is further subdivided into minutes and seconds. We have $1^{\circ} = 60$ minutes and 1 minute $= 60$ seconds. The minutes are denoted by a dash (') and second by double dash (").

$1^{\circ} = 6 0^{'}$ $1^{'} = 6 0^{''}$

Thus $1^{\circ} = 6 0^{'}$ and $1^{'} = 6 0^{''}$ . Note: The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit $= 6 0^{\circ} = \frac{π}{3}$ rad.

4.0Types of Angles

Right Angle

When the clock shows $3 0^{'}$ clock, the angle between its two hands is equal to $9 0^{\circ}$ . This is called a right angle. An angle of magnitude exactly $9 0^{\circ}$ is called a right angle.

A right angle is an angle which is exactly in the shape 'L'.

Straight Angle

When the arms of an angle are opposite rays forming a straight line, the angle thus formed is called a straight angle. $∠ POQ$ is a straight angle and its measure is equal to two right angles, that is $18 0^{\circ}$ . Thus the measure of $∠ POQ = 2 \times 9 0^{\circ} = 18 0^{\circ}$ .

Acute Angle

An angle of magnitude less than a right angle is called an acute angle. $∠ AOB$ is an acute angle. $0^{\circ} < x < 9 0^{\circ}$

Obtuse Angle

An angle of magnitude more than $9 0^{\circ}$ and less than $18 0^{\circ}$ is called an obtuse angle. $∠ AOB$ is an obtuse angle. $9 0^{\circ} < x < 18 0^{\circ}$

Zero Angle

As $OP$ takes positions $OQ_{1}, OQ_{2}, OQ_{3}$ , etc., the angle becomes bigger and bigger. However, when $OP$ has not yet moved, the angle formed between $OP$ and $OQ$ is zero. This angle is called a zero angle.

Complete Angle (360 $^{\circ}$ )

When $OQ$ makes a complete revolution, it covers $36 0^{\circ}$ and again coincides with $OP$ . The angle formed by $OQ$ with is one complete circle, that is $36 0^{\circ}$ . Such an angle is called a complete angle.

Reflex Angle

An angle of magnitude more than $18 0^{\circ}$ and less than $36 0^{\circ}$ is called a reflex angle. Therefore, $∠ A OB$ (in a and b) is a reflex angle. It is more than $18 0^{\circ}$ . Now, $∠ AOP$ is $18 0^{\circ}$ and $∠ AOB$ is greater than that. (Note that angles are usually measured in the anticlockwise direction.) But in (c), $∠ AOB$ is not a reflex angle as the measure of the angle is less than $18 0^{\circ}$ . $18 0^{\circ} < x < 36 0^{\circ}$

Types of Angle	Measure	Figure
Zero angle	$0^{\circ}$
Acute angle	Between $0^{\circ}$ and $9 0^{\circ}$
Right angle	$0^{\circ}$
Obtuse angle	Between $9 0^{\circ}$ and $18 0^{\circ}$
Straight angle	$18 0^{\circ}$
Reflex angle	More than $18 0^{\circ}$

5.0Lines

Perpendicular Lines

When two lines intersect so that four right angles are formed, we say that the lines are perpendicular to each other. The symbol ' 7 ' (a square corner) is used in a diagram to show that $A B$ is perpendicular to CD. The symbol ' $⊥$ ' stands for 'is perpendicular to' and to express the fact that $A B$ is perpendicular to CD . We write $AB ⊥ CD$ .

Parallel Lines

Lines that never meet and are always at equal distance from each other are called parallel lines. Line $AB, CD$ are parallel. We use the symbol '||' for 'parallel to'.

So here we can write AB is parallel to CD or AB || CD.

A triangle is a polygon with the least number of sides.

6.0Polygons

Polygons are simple closed figures that consist of line segments joining in turn, so that each line segments intersect exactly two other line segments at their end points. Types of polygons Polygons are classified according to the number of sides they have :

Name	Number of Sides
Triangle	3
Quadrilateral	4
Pentagon	5
Hexagon	6
Heptagon	7
Octagon	8
Nonagon	9
Decagon	10

Regular Polygon

A regular polygon is a polygon with all its sides and all its angles equal. Note: The sum of interior angles of a $n$ sided polygon is equal to $(2 n - 4) \times 9 0^{\circ}$ .

A line joining the non adjacent vertices of a polygon is called a diagonal of polygon.

The triangle 6 shown in figure and another is ADB, ACB, $BDC, ADC$ , so that total number of triangles is 10 .

7.0Triangle

A 3-sided polygon, is called triangle.

Classification of Triangles

Triangles are classified either with reference to their sides or to their angles. On the basis of side

Scalene Triangle

A scalene triangle is one that has all sides unequal.

Isosceles Triangle

An isosceles triangle is one that has two sides equal.

Equilateral Triangle

An equilateral triangle is one that has all sides equal.

On the Basis of Angle

Acute Angled Triangle

An acute angled triangle is one that has all its angles acute.

Obtuse Angled Triangle

An obtuse angled triangle is one that has one of its angles obtuse.

Right Angled Triangle

A right angled triangle is one that has one of its angles right angle.

The hypotenuse in a right angled triangle is the side opposite the right angle and this hypotenuse is the longest side of a right angled triangle.

In a triangle largest angle is opposite largest side and smallest angle is opposite to smallest side.
In an isosceles triangle the angles opposite to equal sides are equal.
In an equilateral triangle all the angles are of $6 0^{\circ}$ .
In a right angled triangle the two smallest angles add to $9 0^{\circ}$ .

Angle Sum Property of Triangle

The sum of the interior angles of a triangle is $18 0^{\circ}$ .

8.0Quadrilaterals

Definition of Quadrilaterals A 4-sided polygon is called quadrilateral. Convex quadrilateral A quadrilateral in which the measure of each angle is less than $18 0^{\circ}$ , is called a convex quadrilateral.

Concave Quadrilateral

A quadrilateral in which the measure of one of the angles is more than $18 0^{\circ}$ is called a concave quadrilateral.

Angle Sum Property of Quadrilateral

The sum of the interior angles of quadrilateral is $36 0^{\circ}$ .

9.0Properties of Special Quadrilaterals

Trapezium

A quadrilateral having exactly one and only one pair of parallel sides is called a trapezium. ABCD is a trapezium in which $AB ∥ DC$ .

(a)

(b)

A trapezium is said to be an isosceles trapezium if its non-parallel sides are equal. PQRS is an isosceles trapezium in which $PQ ∥ SR$ and $PS = QR$ . The diagonal of an isosceles trapezium are always equal.

Parallelogram

A quadrilateral in which both pairs of opposite sides are parallel, is called a parallelogram.

Properties of a parallelogram (i) The opposite sides of a ||gm are equal and parallel. (ii) The opposite angles of a |lgm are equal. (iii) The diagonals of a |gm bisect each other.

Thus, in a parallelogram $A BC D$ , we have

(i) $AB = DC, AD = BC$ and $AB ∣∣ DC, AD ∣∣ BC$ (ii) $∠ BAD = ∠ BCD$ and $∠ ABC = ∠ ADC$ (iii) If the diagonals AC and BD intersect at 0 , then $OA = OC$ and $OB = 0 D$ .

Rhombus

A parallelogram in which all the sides are equal is called a rhombus.

Properties of a rhombus (i) The opposite sides of a rhombus are parallel. (ii) All the sides of a rhombus are equal. (iii) The opposite angles of a rhombus are equal. (iv) The diagonals of a rhombus bisect each other at right angles.

Thus, in a rhombus ABCD, we have

(i) $AB ∣ ∣ DC$ and AD $∣ ∣ BC$ . (ii) $AB = BC = CD = DA$ . (iii) $∠ DAB = ∠ BCD$ and $∠ ABC = ∠ CDA$ . (iv) Let the diagonals AC and BD intersect at 0 . Then, $OA = 0 OC, OB = OD$ and $∠ AOB = ∠ COD$ $= ∠ BOC = ∠ AOD = 1$ right angle .

Rectangle

A parallelogram in which each angle is a right angle is called a rectangle.

Properties of a rectangle (i) Opposite sides of a rectangle are equal and parallel. (ii) Each angle of a rectangle is $9 0^{\circ}$ . (iii) Diagonals of a rectangle are equal.

Thus, in a rectangle ABCD, we have

(i) $AB = DC, AD = BC$ and $AB ∥ DC, AD ∣∣ BC$ . (ii) $∠ A = ∠ B = ∠ C = ∠ D =$ right angle. (iii) Diagonal $AC =$ diagonal BD .

Square

A parallelogram in which all the sides are equal and each angle is a right angle is called a square.

Properties of a square (i) The sides of a square are all equal. (ii) Each angle of a square is $9 0^{\circ}$ . (iii) The diagonals of a square are equal and bisect each other at right angles.

Thus, in a square ABCD, we have

(i) $AB = BC = CD = DA$ (ii) $∠ A = ∠ B = ∠ C = ∠ D = 9 0^{\circ}$ . (iii) Diagonal $AC =$ diagonal BD . (iv) Let the diagonals AC and BD intersect at 0 .

Then, $OA = OC, OB = OD$ and $∠ AOB = ∠ COD = ∠ BOC = ∠ AOD = 1$ right angle .

Kite

A quadrilateral which has two pairs of equal adjacent sides but unequal opposite sides is called a kite. ABCD is a kite in which

$CB = C D$ and $A B = A D$ but $A D \neq = BC$ and $A B \neq = C D$ .

10.0Solid Figures

Three Dimensional Shapes

The figures such as triangles, squares, rectangles, quadrilaterals, polygons etc., have only the length and the breadth; they do not have the height or depth, and hence they are called as two dimensional figures. We can only see these shapes, but can not handle them. But the solids can be handled and the properties of these can be experienced. These solid shapes have length, breadth, and height and hence they are called as three-dimensional or 3D shapes. Solid figure A closed figure which lies in more than one plane is called a space figure or solid figure.

Face

The surface of a solid is called its face.

Edge

An edge is a line segment that is the intersection of two faces.

Vertex

A vertex in a solid shape is the point where the edges meet.

Euler's formula

If a polyhedron has F number of faces, V number of vertices and E number of edges then $F + V - E = 2$

Cuboid

Solids such as a wooden box, a match box, a brick, a book, an almirah, etc. are all in the shape of a cuboid. Some of these shape are given below in the diagram.

Cube

A cuboid whose length, breadth and height are equal is called a cube.

Cylinder

Objects such as a circular pillar, a circular pipe, a test tube, a circular storage tank, a measuring jar etc. are in the shape of cylinder.

A cylinder has a curved lateral surface and two circular faces at its ends. It has no corner or vertex. It has two plane faces, namely, the top and the base. The distance between its end faces is called its length.

Sphere

An object which is in the shape of a ball is said to have the shape of a sphere. A sphere has curved surface, it has no vertex and no edge.

Cone

Objects such as an ice-cream cone, a conical tent, a conical vessel etc. are in the shape of a cone.

A cone has plane circular end as the base and a curved surface tapering into a point, called its vertex. It has no circular edge and one vertex.

11.0Pyramid

A pyramid is a solid where base is a plane rectilinear figure and whose side faces are triangles having a common vertex, called the vertex of the pyramid. The length of perpendicular drawn from the vertex of a pyramid to its base is called the height of the pyramid. The side faces of a pyramid are called its lateral faces.

Square Pyramid

A solid whose base is a square and whose side faces are triangles having a common vertex is called a square pyramid. A square pyramid $O A BC D$ with 0 as vertex, the square $A BC D$ as its base and $OP$ as its height. A square pyramid has 4 lateral triangular faces and 8 edges.

Triangular Pyramid

A solid whose base is a triangle and whose side faces are triangles having a common vertex is called a triangular pyramid. A triangular pyramid $O A BC$ with $O$ as vertex and $△ A BC$ as its base.

A triangular pyramid has 3 triangular lateral faces, one triangular base and 6 edges.

12.0Prism

Prisms are polyhedra whose top and base are congruent polygons and the other faces are parallelograms.

Pentagonal prism

13.0Numerical Ability

Convert into mm: (i) 3.9 cm (ii) 176.5 cm (iii) 3.8 dm

Solution (i) $1 cm = 10 mm$ $3.9 cm = 3.9 \times 10 mm = 39 mm$ (ii) $1 cm = 10 mm$ $176.5 cm = 176.5 \times 10 mm = 1765 mm$ (iii) $1 dm = 10 cm$ $3.8 cm = 3.8 \times 10 cm = 38 cm$ $= 38 \times 10 mm = 380 mm$

Convert: (i) 5.03 m into m and cm (ii) 1.24 km in km and m .

Explanation (i) $5.03 m = 5 m + 0.03 m$

= 5 m + 0.03 \times 100 cm = 5 m + 3 cm = 5 m 3cm

(ii) $1.24 km = 1 km + 0.24 km$ $= 1 km + 0.24 \times 1000 m$ $= 1 km + 240 m = 1 km 240 m$

If $B$ is the midpoint of $A C$ and $C$ is mid point of $B D$ . where $A, B, C, D$ lie on a straight line, say why $A B = C D$ ?

Solution Since $B$ is the mid-point of $A C$ . $AB = BC$ Since $C$ is the mid-point of $B D$ . $BC = CD$ From equation (1) and (2), we may find that $A B = C D$

If $A B = 5 cm, BC = 3 cm$ then show that $A C = BC + A B$ and point $B$ is lying between A & C.

Explanation Given that, $AB = 5 cm$ $BC = 3 cm$ $AC = 8 cm$ It can be observed that $A C = A B + BC$ Clearly, point $B$ is lying between $A$ and $C$ .

Find the measure of the angle shown in each figure. (First estimate with your eyes and then find the actual measure with a protractor).

Solution The measures of the angles shown in the above figure are $4 0^{\circ}, 13 0^{\circ}, 6 5^{\circ}, 13 5^{\circ}$ respectively.

Which angle has a large measure? First estimate and then measure. Measure of angle $A =$ Measure of angle $B =$

Explanation Measure of angle $A = 4 0^{\circ}$ Measure of angle $B = 6 8^{\circ}$ $∠ B$ has the greater measure than $∠ A$ .

(i) Through what angle does the minute hand of clock turn in 45 minutes, and the hour hand in 30 minutes? (ii) What rotation is needed to turn (a) From North to South-West in a clockwise direction? (b) From South-West to South-East in a counter clockwise direction?

Explanation (i) In one hour the minute hand completes a full circle of $36 0^{\circ}$ . Therefore in 45 minutes it goes through an angle equal to $\frac{45}{60} \times 36 0^{\circ}$ ar $27 0^{\circ}$ . In one hour, the hour hand (ii) turns through an angle of $\frac{1}{12}$ of $36 0^{\circ}$ or $3 0^{\circ}$ . Therefore in 30 minutes it turns through $1 5^{\circ}$ . (a) Adding a turn of $18 0^{\circ}$ from North to South and of $4 5^{\circ}$ from South to South-West, we get $18 0^{\circ} +$ $4 5^{\circ}$ , or $22 5^{\circ}$ . (b) The turn from South-West to South-East is equal to $4 5^{\circ} + 4 5^{\circ}$ , or $9 0^{\circ}$ .

Find the angles between the hands of a clock at (i) 7 O'clock, (ii) 3: $300$ 'clock.

Solution On the clock dial the angle between the hands pointing to any two adjacent numericals is equal to $\frac{1}{12} \times 36 0^{\circ}$ , or $3 0^{\circ}$ .
(i) At $7 0^{'}$ clock, $∠ a = 5 \times 3 0^{\circ} = 15 0^{\circ}$ . (ii) At 3:30 $0^{'}$ clock, $∠ b = 2 \times 3 0^{\circ} + 1 5^{\circ} = 7 5^{\circ}$

Classify the angles whose magnitude are given below : (i) $12 2^{\circ}$ (ii) $1 7^{\circ}$ (iii) $18 2^{\circ}$ (iv) $0^{\circ}$ (v) $179.9 9^{\circ}$ (vi) $90. 5^{\circ}$ (vii) $36 0^{\circ}$

Explanation (i) Angle is greater than $9 0^{\circ}$ so $12 2^{\circ}$ is an obtuse angle. (ii) Angle $1 7^{\circ}$ is less than $9 0^{\circ}$ so it is an acute angle. (iii) Angle $18 2^{\circ}$ is greater than $18 0^{\circ}$ so it is a reflex angle. (iv) Zero angle (v) Obtuse angle (vi) Obtuse angle (vii) Complete angle

Which of the following are models for perpendicular lines: (i) The adjacent edges of a table top. (ii) The lines of a railway track. (iii) The line segments forming the letter 'L' (iv) The letter V.

Solution (i) The adjacent edges of a tabletop are perpendicular to each other. (ii) The lines of a railway track are parallel to each other. (iii) The line segments forming the letter L are perpendicular to each other. (iv) The sides of letter $V$ are inclined at some acute angle on each other. hence, (a) and (c) are the models for perpendicular lines and (b) is the model for parallel lines.

Let $\overline{PQ}$ be the perpendicular to the line segment $\overline{XY}$ . Let $\overline{PQ}$ and $\overline{XY}$ intersect in the point $A$ . What is the measure of $∠ P A Y$ ?

Explanation
From the figure, it can be easily observed that the measure of $∠ PAY$ is $9 0^{\circ}$ .

Study the figure and answer the following questions (i) Name the equilateral triangles. (ii) Name the isosceles triangles. (iii) Name the scalene triangles.
(iv) Name the acute triangles. (v) Name the obtuse triangles. (vi) Name the right triangles.

Explanation (i) $△ ABC$ (ii) $△ ADE, △ ADC, △ AEB$ (iii) $△ ABD, △ ABM, △ AEC, △ AMC, △ AME, △ ADM$ (iv) $△ ABC, △ ADC, △ AEB, △ ADE$ (v) $△ ADB, △ AEC$ (vi) $△ AMB, △ AMD, △ AME, △ AMC$

Find the angles of a triangle which are in the ratio $2 : 3 : 4$ .

Solution Let the measures of the given angles be (2x), (3x), (4x). Then as we know that the sum of the angles of the triangle is $18 0^{\circ}$ . $∴ 2 x + 3 x + 4 x = 18 0^{\circ}$ $\Rightarrow 9 x = 18 0^{\circ} \Rightarrow x = 2 0^{\circ}$ Hence the measures of the given angles are $2 x = 2 \times 2 0^{\circ} = 4 0^{\circ}$ and $3 x = 3 \times 2 0^{\circ} = 6 0^{\circ}$ and $4 x = 4 \times 2 0^{\circ} = 8 0^{\circ}$

10. The angles of a quadrilateral are in the ratio 1:2:3:4. Find the measure of each of the four angles.

Solution Let the measure of the angles of the given quadrilateral be $x, (2 x), (3 x)$ and (4x). Then, $x + 2 x + 3 x + 4 x = 36 0^{\circ}$ [ $∴$ The sum of the angles of a quadrilateral is $36 0^{\circ}$ ] $\Rightarrow 10 x = 36 0^{\circ}$ $\Rightarrow x = 3 6^{\circ}$ Hence the required angles are $3 6^{\circ}, 7 2^{\circ}, 10 8^{\circ}$ and $14 4^{\circ}$ .

11. The three angles of a quadrilateral are $7 0^{\circ}, 6 0^{\circ}$ and $11 0^{\circ}$ . Find the fourth angle.

Explanation Sum of all the angles of a quadrilateral is $36 0^{\circ}$ . Fourth angle $= 36 0^{\circ} - (7 0^{\circ} + 6 0^{\circ} + 11 0^{\circ}) = 12 0^{\circ}$

12. Two sides of a ||gm are in the ratio $4 : 3$ . If its perimeter is 56 cm , find the lengths of its sides.

Solution Suppose the sides of the $∥ g m$ is 4 x and 3 x and as we know that in a |gm, opposite sides are equal. So other two sides will be 4 x and 3 x . Perimeter $=$ Sum of its all sides $\Rightarrow 56 = 4 x + 3 x + 4 x + 3 x$ $\Rightarrow 56 = 14 x$ $\Rightarrow x = \frac{56}{14} = 4$ So, the other two sides are $4 x = 4 \times 4 = 16 cm$ and $3 x = 3 \times 4 = 12 cm$ .

13. Specify the type of quadrilateral $A BC D$ in each case, given the following information. (i) $AB ∣∣ CD, BC ∣∣ AD, ∠ DAB = 6 0^{\circ}$ (ii) $AB ∥ CD, AD ∥ BC, ∠ DAB = ∠ ABC$ (iii) $AB ∣∣ CD, AD ∣∣ CB, AB = CD, BC = AD, DA ⊥ AB$ (iv) $AO = OC, DO = OB, DB ⊥ AC, ∠ DAB = 6 3^{\circ}, 0$ being the point of intersection of diagonals. (v) $AB = CD, CD = AD = BC$

Explanation (i) Parallelogram
(ii) Rectangle
(iii) Rectangle
(iv) Rhombus
(v) Rhombus

14. Two sides of a ||gm are in the ratio $4 : 3$ . If its perimeter is 56 cm , find the lengths of its sides.

Solution Suppose the sides of the $∥ g m$ is 4 x and 3 x and as we know that in a |gm, opposite sides are equal. So other two sides will be 4 x and 3 x . Perimeter $=$ Sum of its all sides $\Rightarrow 56 = 4 x + 3 x + 4 x + 3 x$ $\Rightarrow 56 = 14 x$ $\Rightarrow x = \frac{56}{14} = 4$ So, the other two sides are $4 x = 4 \times 4 = 16 cm$ and $3 x = 3 \times 4 = 12 cm$ .

15. Specify the type of quadrilateral $A BC D$ in each case, given the following information. (i) $AB ∣∣ CD, BC ∣∣ AD, ∠ DAB = 6 0^{\circ}$ (ii) $AB ∥ CD, AD ∥ BC, ∠ DAB = ∠ ABC$ (iii) $AB ∣∣ CD, AD ∣∣ CB, AB = CD, BC = AD, DA ⊥ AB$ (iv) $AO = OC, DO = OB, DB ⊥ AC, ∠ DAB = 6 3^{\circ}, 0$ being the point of intersection of diagonals. (v) $AB = CD, CD = AD = BC$

Explanation (i) Parallelogram
(ii) Rectangle
(iii) Rectangle
(iv) Rhombus
(v) Rhombus

16. Which type of solid shape is a (i) Dice (ii) Gas pipe (iii) Football (iv) Brick (v) Ice-cream cone (vi) Kaleidoscope

Explanation (i) Cube (ii) Cylinder (iii) Sphere (iv) Cuboid (v) Cone (vi) Triangular prism

17. A polyhedron has 4 faces and 6 edges. How many vertices will it have?

Solution As we know that For a polyhedron, $F + V - E = 2$ here $F = 4, E = 6$ $\Rightarrow 4 + V - 6 = 2$ $\Rightarrow V - 2 = 2$ $\Rightarrow V = 2 + 2 = 4$ So, the polyhedron has 4 vertices.

14.0Memory Map

Algebraic Expressions

An expression, its terms and factors of the terms can be represented by a tree diagram to make it easily comprehensible to you......

Data Handling

The word data means information in the form of numerical figures or a set of given facts. E.g. The percentage of marks scored by 10 students.......

The Triangles and its Properties

A closed figure formed by joining three non-collinear points is called a triangle. The three sides and three angles of a triangle are collectively known as elements of the triangle......

Visualising Solid Shapes

A solid is any enclosed three-dimensional shape, i.e., it has 3 dimensions- length, width and height, whereas there are some common (flat) shapes which can be easily drawn on paper. They have only.....

Fractions

Fractions having the same denominator are called like fractions, whereas fractions having different denominator are called unlike fractions......

Perimeter and Area

Mensuration : The process, art or the act of measuring is called mensuration. Anything that can be measured is said to be mensurable.......

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Understanding Elementary Shapes

1.0Measuring Line Segments

When we measure a line segment, we measure its length or distance from one end point to the other. When we measure the length, we must know the units of measurement. Today the metric system is used at most universally. The standard unit in this system is metre.

Conversion of Units of Length

Conversion	Equivalent
10 millimetres (mm)	= 1 centimetre (cm)
10 centimetres (cm)	= 1 decimetre (dm)
10 decimetres (dm)	= 1 metre (m) = 1000 mm
10 metres (m)	= 1 decametre (dam)
10 decametres (dam)	= 1 hectometre (hm)
10 hectometres (hm)	= 1 kilometre (km) = 1000 m

However most often we use the following units. 10 millimetres = 1 centimetres 100 centimetres $= 1$ metres 1000 metres $= 1$ kilometre $1 cm = 0.3937$ inch. By using a ruler: The ruler has centimetre and millimetre marks on one edge and other edge is divided into inches.

One small division $=$ One-tenth of one $cm = 0.1 cm = 1 mm$ Two small division $=$ Two-tenths of one centimetre $= 0.2 cm = 2 mm$

There is a global standard, the international system of units (SI), the modern form of the metric system.

The distance between the endpoints of a line segment is called its length.

2.0Comparison of Line Segments

Comparison by Observation

The simplest way of comparing two line segments is to observe their lengths. Here we can easily observe that line segment 'b' is placed directly below line segment 'a'. Line segment 'a' extends further to the right.

But if the lengths are almost equal then comparison by observation is not easy.

Comparison by Divider

Let us compare the line segments $\overline{AB}$ and $\overline{CD}$ , using a divider.

Place one point of the divider on A and open the other leg of the divider until the other point coincides with B . This measures the length of $\overline{AB}$ . Now take the divider as it is and place one point of the divider at C and the other point along $\overline{CD}$ . We will observe: (i) If the other point touches $\overline{CD}$ exactly at D , then $\overline{AB} = \overline{CD}$ . (ii) If the other point of the divider is beyond the point D on $\overline{CD}$ , then $\overline{AB} > \overline{CD}$ . (iii) If the other point is between $C$ and $D$ on $\overline{CD}$ , then $\overline{AB} < \overline{CD}$ .

Comparison by Tracing

We can also compare two-line segments, say $\overline{AB}$ and $\overline{CD}$ , by tracing one of them and overlapping the traced line on the other, with one endpoint coinciding. We can easily make out which line is longer, which is shorter, or whether they are both equal. $\overline{AB}$ is placed on $\overline{CD}$ , with the endpoints C and A coinciding. Since the other two endpoints $B$ and $D$ do not coincide, we can say that

(i) $\overline{AB}$ is not equal to $\overline{CD}$ . (ii) $\overline{AB}$ is shorter than $\overline{CD}$ as the endpoint B of $\overline{AB}$ falls short of D , the endpoint of $\overline{CD}$ .

$500 cm < 1 km$

3.0Measure of Angles

The magnitude or measure of the angle is the measure of rotation. Suppose a ray OP starts rotating around 0 , from the fixed position OA to different position $OP_{1}, OP_{2}, OP_{3}$ etc. then measure of the angle will equal the measure of this rotation.

Measuring Angles Using Protractor

The word angle comes from the Latin word angulus, meaning "a corner".

The protractor is an instrument used to measure angles and draw angles of required magnitude. Suppose you have to measure the angle BAC. Place the protractor such that its centre falls on the vertex A of the angle and its horizontal edge (zero line) on the arm AC. Now look at the protractor to find out which line of division on the rim falls on the arm AB .

Read the degree measure from the protractor, use the anticlockwise, i.e. the inner scale. Thus, by measurement $∠ BAC = 6 0^{\circ}$ .

Degree Measure of An Angle

$OQ$ rotates from position $OP$ . When it has made one complete rotation, it reaches $OP$ again. We say that the angle thus formed is 360 degrees. It is written as $36 0^{\circ}$ . In other words, a circle is made up of $36 0^{\circ}$ .

Let us take the example of the face of a clock. It is divided into 12 equal parts. The angle that the arms include between each other, say, at 10 'o clock is exactly $\frac{2}{12}$ of the circle, that is $\frac{2}{12}$ of $36 0^{\circ}$ $= \frac{2}{12} \times 36 0^{\circ} = 6 0^{\circ}$ . At 1.00 a.m. or 1.00 p.m. this angle is $3 0^{\circ}$ and at 3.00 a.m. or $3.00 p . m$ . it is $9 0^{\circ}$ .

The turn (or full circle, revolution, rotation, or cycle) is one full circle. $r$ in rpm (revolutions per minute). 1 turn $= 36 0^{\circ}$ $= 2 prad = 400 grad =$ 4 right angles. SPOT LIGHT

The turn from north to east is by a right angle. The turn from north to south is by two right angles. It is called a straight angle.

Rotation Round The Clock

When the minute hand of a clock starts at 12 and reaches at 3 , it has reached quarter past and has made a quarter of a rotation and has turned through an angle of magnitude $9 0^{\circ}$ . At 6 (half past) the minute hand has made $\frac{1}{2}$ of a rotation and turned through an angle of measure $18 0^{\circ}$ . At 9 (quarter to), it has made $(\frac{3}{4})$ three quarter of a rotation and turned through an angle of measure $27 0^{\circ}$ . When the minute hand reaches 12 , it has moved exactly once round the clock, i.e., it has made one rotation and through an angle of measure $36 0^{\circ}$ .

Directions

You are familiar with the concept of direction. There are four main directions North (N), South (S), East (E), and West (W). Jammu is to the North of Delhi, Kolkata is to its East, Rajkot to its West and Cochin to its South. Midway between there are the four sub-directions, namely North-East (N.E.), South-East (S.E.), North-West (N.W.) and South-West (S.W.).

Degree, Minutes and Seconds

A degree is further subdivided into minutes and seconds. We have $1^{\circ} = 60$ minutes and 1 minute $= 60$ seconds. The minutes are denoted by a dash (') and second by double dash (").

$1^{\circ} = 6 0^{'}$ $1^{'} = 6 0^{''}$

Thus $1^{\circ} = 6 0^{'}$ and $1^{'} = 6 0^{''}$ . Note: The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. 1 Babylonian unit $= 6 0^{\circ} = \frac{π}{3}$ rad.

4.0Types of Angles

Right Angle

When the clock shows $3 0^{'}$ clock, the angle between its two hands is equal to $9 0^{\circ}$ . This is called a right angle. An angle of magnitude exactly $9 0^{\circ}$ is called a right angle.

A right angle is an angle which is exactly in the shape 'L'.

Straight Angle

When the arms of an angle are opposite rays forming a straight line, the angle thus formed is called a straight angle. $∠ POQ$ is a straight angle and its measure is equal to two right angles, that is $18 0^{\circ}$ . Thus the measure of $∠ POQ = 2 \times 9 0^{\circ} = 18 0^{\circ}$ .

Acute Angle

An angle of magnitude less than a right angle is called an acute angle. $∠ AOB$ is an acute angle. $0^{\circ} < x < 9 0^{\circ}$

Obtuse Angle

An angle of magnitude more than $9 0^{\circ}$ and less than $18 0^{\circ}$ is called an obtuse angle. $∠ AOB$ is an obtuse angle. $9 0^{\circ} < x < 18 0^{\circ}$

Zero Angle

As $OP$ takes positions $OQ_{1}, OQ_{2}, OQ_{3}$ , etc., the angle becomes bigger and bigger. However, when $OP$ has not yet moved, the angle formed between $OP$ and $OQ$ is zero. This angle is called a zero angle.

Complete Angle (360 $^{\circ}$ )

When $OQ$ makes a complete revolution, it covers $36 0^{\circ}$ and again coincides with $OP$ . The angle formed by $OQ$ with is one complete circle, that is $36 0^{\circ}$ . Such an angle is called a complete angle.

Reflex Angle

An angle of magnitude more than $18 0^{\circ}$ and less than $36 0^{\circ}$ is called a reflex angle. Therefore, $∠ A OB$ (in a and b) is a reflex angle. It is more than $18 0^{\circ}$ . Now, $∠ AOP$ is $18 0^{\circ}$ and $∠ AOB$ is greater than that. (Note that angles are usually measured in the anticlockwise direction.) But in (c), $∠ AOB$ is not a reflex angle as the measure of the angle is less than $18 0^{\circ}$ . $18 0^{\circ} < x < 36 0^{\circ}$

Types of Angle	Measure	Figure
Zero angle	$0^{\circ}$
Acute angle	Between $0^{\circ}$ and $9 0^{\circ}$
Right angle	$0^{\circ}$
Obtuse angle	Between $9 0^{\circ}$ and $18 0^{\circ}$
Straight angle	$18 0^{\circ}$
Reflex angle	More than $18 0^{\circ}$

5.0Lines

Perpendicular Lines

When two lines intersect so that four right angles are formed, we say that the lines are perpendicular to each other. The symbol ' 7 ' (a square corner) is used in a diagram to show that $A B$ is perpendicular to CD. The symbol ' $⊥$ ' stands for 'is perpendicular to' and to express the fact that $A B$ is perpendicular to CD . We write $AB ⊥ CD$ .

Parallel Lines

Lines that never meet and are always at equal distance from each other are called parallel lines. Line $AB, CD$ are parallel. We use the symbol '||' for 'parallel to'.

So here we can write AB is parallel to CD or AB || CD.

A triangle is a polygon with the least number of sides.

6.0Polygons

Polygons are simple closed figures that consist of line segments joining in turn, so that each line segments intersect exactly two other line segments at their end points. Types of polygons Polygons are classified according to the number of sides they have :

Name	Number of Sides
Triangle	3
Quadrilateral	4
Pentagon	5
Hexagon	6
Heptagon	7
Octagon	8
Nonagon	9
Decagon	10

Regular Polygon

A regular polygon is a polygon with all its sides and all its angles equal. Note: The sum of interior angles of a $n$ sided polygon is equal to $(2 n - 4) \times 9 0^{\circ}$ .

A line joining the non adjacent vertices of a polygon is called a diagonal of polygon.

The triangle 6 shown in figure and another is ADB, ACB, $BDC, ADC$ , so that total number of triangles is 10 .

7.0Triangle

A 3-sided polygon, is called triangle.

Classification of Triangles

Triangles are classified either with reference to their sides or to their angles. On the basis of side

Scalene Triangle

A scalene triangle is one that has all sides unequal.

Isosceles Triangle

An isosceles triangle is one that has two sides equal.

Equilateral Triangle

An equilateral triangle is one that has all sides equal.

On the Basis of Angle

Acute Angled Triangle

An acute angled triangle is one that has all its angles acute.

Obtuse Angled Triangle

An obtuse angled triangle is one that has one of its angles obtuse.

Right Angled Triangle

A right angled triangle is one that has one of its angles right angle.

The hypotenuse in a right angled triangle is the side opposite the right angle and this hypotenuse is the longest side of a right angled triangle.

In a triangle largest angle is opposite largest side and smallest angle is opposite to smallest side.
In an isosceles triangle the angles opposite to equal sides are equal.
In an equilateral triangle all the angles are of $6 0^{\circ}$ .
In a right angled triangle the two smallest angles add to $9 0^{\circ}$ .

Angle Sum Property of Triangle

The sum of the interior angles of a triangle is $18 0^{\circ}$ .

8.0Quadrilaterals

Definition of Quadrilaterals A 4-sided polygon is called quadrilateral. Convex quadrilateral A quadrilateral in which the measure of each angle is less than $18 0^{\circ}$ , is called a convex quadrilateral.

Concave Quadrilateral

A quadrilateral in which the measure of one of the angles is more than $18 0^{\circ}$ is called a concave quadrilateral.

Angle Sum Property of Quadrilateral

The sum of the interior angles of quadrilateral is $36 0^{\circ}$ .

9.0Properties of Special Quadrilaterals

Trapezium

A quadrilateral having exactly one and only one pair of parallel sides is called a trapezium. ABCD is a trapezium in which $AB ∥ DC$ .

(a)

(b)

A trapezium is said to be an isosceles trapezium if its non-parallel sides are equal. PQRS is an isosceles trapezium in which $PQ ∥ SR$ and $PS = QR$ . The diagonal of an isosceles trapezium are always equal.

Parallelogram

A quadrilateral in which both pairs of opposite sides are parallel, is called a parallelogram.

Properties of a parallelogram (i) The opposite sides of a ||gm are equal and parallel. (ii) The opposite angles of a |lgm are equal. (iii) The diagonals of a |gm bisect each other.

Thus, in a parallelogram $A BC D$ , we have

(i) $AB = DC, AD = BC$ and $AB ∣∣ DC, AD ∣∣ BC$ (ii) $∠ BAD = ∠ BCD$ and $∠ ABC = ∠ ADC$ (iii) If the diagonals AC and BD intersect at 0 , then $OA = OC$ and $OB = 0 D$ .

Rhombus

A parallelogram in which all the sides are equal is called a rhombus.

Properties of a rhombus (i) The opposite sides of a rhombus are parallel. (ii) All the sides of a rhombus are equal. (iii) The opposite angles of a rhombus are equal. (iv) The diagonals of a rhombus bisect each other at right angles.

Thus, in a rhombus ABCD, we have

(i) $AB ∣ ∣ DC$ and AD $∣ ∣ BC$ . (ii) $AB = BC = CD = DA$ . (iii) $∠ DAB = ∠ BCD$ and $∠ ABC = ∠ CDA$ . (iv) Let the diagonals AC and BD intersect at 0 . Then, $OA = 0 OC, OB = OD$ and $∠ AOB = ∠ COD$ $= ∠ BOC = ∠ AOD = 1$ right angle .

Rectangle

A parallelogram in which each angle is a right angle is called a rectangle.

Properties of a rectangle (i) Opposite sides of a rectangle are equal and parallel. (ii) Each angle of a rectangle is $9 0^{\circ}$ . (iii) Diagonals of a rectangle are equal.

Thus, in a rectangle ABCD, we have

(i) $AB = DC, AD = BC$ and $AB ∥ DC, AD ∣∣ BC$ . (ii) $∠ A = ∠ B = ∠ C = ∠ D =$ right angle. (iii) Diagonal $AC =$ diagonal BD .

Square

A parallelogram in which all the sides are equal and each angle is a right angle is called a square.

Properties of a square (i) The sides of a square are all equal. (ii) Each angle of a square is $9 0^{\circ}$ . (iii) The diagonals of a square are equal and bisect each other at right angles.

Thus, in a square ABCD, we have

(i) $AB = BC = CD = DA$ (ii) $∠ A = ∠ B = ∠ C = ∠ D = 9 0^{\circ}$ . (iii) Diagonal $AC =$ diagonal BD . (iv) Let the diagonals AC and BD intersect at 0 .

Then, $OA = OC, OB = OD$ and $∠ AOB = ∠ COD = ∠ BOC = ∠ AOD = 1$ right angle .

Kite

A quadrilateral which has two pairs of equal adjacent sides but unequal opposite sides is called a kite. ABCD is a kite in which

$CB = C D$ and $A B = A D$ but $A D \neq = BC$ and $A B \neq = C D$ .

10.0Solid Figures

Three Dimensional Shapes

The figures such as triangles, squares, rectangles, quadrilaterals, polygons etc., have only the length and the breadth; they do not have the height or depth, and hence they are called as two dimensional figures. We can only see these shapes, but can not handle them. But the solids can be handled and the properties of these can be experienced. These solid shapes have length, breadth, and height and hence they are called as three-dimensional or 3D shapes. Solid figure A closed figure which lies in more than one plane is called a space figure or solid figure.

Face

The surface of a solid is called its face.

Edge

An edge is a line segment that is the intersection of two faces.

Vertex

A vertex in a solid shape is the point where the edges meet.

Euler's formula

If a polyhedron has F number of faces, V number of vertices and E number of edges then $F + V - E = 2$

Cuboid

Solids such as a wooden box, a match box, a brick, a book, an almirah, etc. are all in the shape of a cuboid. Some of these shape are given below in the diagram.

Cube

A cuboid whose length, breadth and height are equal is called a cube.

Cylinder

Objects such as a circular pillar, a circular pipe, a test tube, a circular storage tank, a measuring jar etc. are in the shape of cylinder.

A cylinder has a curved lateral surface and two circular faces at its ends. It has no corner or vertex. It has two plane faces, namely, the top and the base. The distance between its end faces is called its length.

Sphere

An object which is in the shape of a ball is said to have the shape of a sphere. A sphere has curved surface, it has no vertex and no edge.

Cone

Objects such as an ice-cream cone, a conical tent, a conical vessel etc. are in the shape of a cone.

A cone has plane circular end as the base and a curved surface tapering into a point, called its vertex. It has no circular edge and one vertex.

11.0Pyramid

A pyramid is a solid where base is a plane rectilinear figure and whose side faces are triangles having a common vertex, called the vertex of the pyramid. The length of perpendicular drawn from the vertex of a pyramid to its base is called the height of the pyramid. The side faces of a pyramid are called its lateral faces.

Square Pyramid

A solid whose base is a square and whose side faces are triangles having a common vertex is called a square pyramid. A square pyramid $O A BC D$ with 0 as vertex, the square $A BC D$ as its base and $OP$ as its height. A square pyramid has 4 lateral triangular faces and 8 edges.

Triangular Pyramid

A solid whose base is a triangle and whose side faces are triangles having a common vertex is called a triangular pyramid. A triangular pyramid $O A BC$ with $O$ as vertex and $△ A BC$ as its base.

A triangular pyramid has 3 triangular lateral faces, one triangular base and 6 edges.

12.0Prism

Prisms are polyhedra whose top and base are congruent polygons and the other faces are parallelograms.

Pentagonal prism

13.0Numerical Ability

Convert into mm: (i) 3.9 cm (ii) 176.5 cm (iii) 3.8 dm

Solution (i) $1 cm = 10 mm$ $3.9 cm = 3.9 \times 10 mm = 39 mm$ (ii) $1 cm = 10 mm$ $176.5 cm = 176.5 \times 10 mm = 1765 mm$ (iii) $1 dm = 10 cm$ $3.8 cm = 3.8 \times 10 cm = 38 cm$ $= 38 \times 10 mm = 380 mm$

Convert: (i) 5.03 m into m and cm (ii) 1.24 km in km and m .

Explanation (i) $5.03 m = 5 m + 0.03 m$

= 5 m + 0.03 \times 100 cm = 5 m + 3 cm = 5 m 3cm

(ii) $1.24 km = 1 km + 0.24 km$ $= 1 km + 0.24 \times 1000 m$ $= 1 km + 240 m = 1 km 240 m$

If $B$ is the midpoint of $A C$ and $C$ is mid point of $B D$ . where $A, B, C, D$ lie on a straight line, say why $A B = C D$ ?

Solution Since $B$ is the mid-point of $A C$ . $AB = BC$ Since $C$ is the mid-point of $B D$ . $BC = CD$ From equation (1) and (2), we may find that $A B = C D$

If $A B = 5 cm, BC = 3 cm$ then show that $A C = BC + A B$ and point $B$ is lying between A & C.

Explanation Given that, $AB = 5 cm$ $BC = 3 cm$ $AC = 8 cm$ It can be observed that $A C = A B + BC$ Clearly, point $B$ is lying between $A$ and $C$ .

Find the measure of the angle shown in each figure. (First estimate with your eyes and then find the actual measure with a protractor).

Solution The measures of the angles shown in the above figure are $4 0^{\circ}, 13 0^{\circ}, 6 5^{\circ}, 13 5^{\circ}$ respectively.

Which angle has a large measure? First estimate and then measure. Measure of angle $A =$ Measure of angle $B =$

Explanation Measure of angle $A = 4 0^{\circ}$ Measure of angle $B = 6 8^{\circ}$ $∠ B$ has the greater measure than $∠ A$ .

(i) Through what angle does the minute hand of clock turn in 45 minutes, and the hour hand in 30 minutes? (ii) What rotation is needed to turn (a) From North to South-West in a clockwise direction? (b) From South-West to South-East in a counter clockwise direction?

Explanation (i) In one hour the minute hand completes a full circle of $36 0^{\circ}$ . Therefore in 45 minutes it goes through an angle equal to $\frac{45}{60} \times 36 0^{\circ}$ ar $27 0^{\circ}$ . In one hour, the hour hand (ii) turns through an angle of $\frac{1}{12}$ of $36 0^{\circ}$ or $3 0^{\circ}$ . Therefore in 30 minutes it turns through $1 5^{\circ}$ . (a) Adding a turn of $18 0^{\circ}$ from North to South and of $4 5^{\circ}$ from South to South-West, we get $18 0^{\circ} +$ $4 5^{\circ}$ , or $22 5^{\circ}$ . (b) The turn from South-West to South-East is equal to $4 5^{\circ} + 4 5^{\circ}$ , or $9 0^{\circ}$ .

Find the angles between the hands of a clock at (i) 7 O'clock, (ii) 3: $300$ 'clock.

Solution On the clock dial the angle between the hands pointing to any two adjacent numericals is equal to $\frac{1}{12} \times 36 0^{\circ}$ , or $3 0^{\circ}$ .
(i) At $7 0^{'}$ clock, $∠ a = 5 \times 3 0^{\circ} = 15 0^{\circ}$ . (ii) At 3:30 $0^{'}$ clock, $∠ b = 2 \times 3 0^{\circ} + 1 5^{\circ} = 7 5^{\circ}$

Classify the angles whose magnitude are given below : (i) $12 2^{\circ}$ (ii) $1 7^{\circ}$ (iii) $18 2^{\circ}$ (iv) $0^{\circ}$ (v) $179.9 9^{\circ}$ (vi) $90. 5^{\circ}$ (vii) $36 0^{\circ}$

Explanation (i) Angle is greater than $9 0^{\circ}$ so $12 2^{\circ}$ is an obtuse angle. (ii) Angle $1 7^{\circ}$ is less than $9 0^{\circ}$ so it is an acute angle. (iii) Angle $18 2^{\circ}$ is greater than $18 0^{\circ}$ so it is a reflex angle. (iv) Zero angle (v) Obtuse angle (vi) Obtuse angle (vii) Complete angle

Which of the following are models for perpendicular lines: (i) The adjacent edges of a table top. (ii) The lines of a railway track. (iii) The line segments forming the letter 'L' (iv) The letter V.

Solution (i) The adjacent edges of a tabletop are perpendicular to each other. (ii) The lines of a railway track are parallel to each other. (iii) The line segments forming the letter L are perpendicular to each other. (iv) The sides of letter $V$ are inclined at some acute angle on each other. hence, (a) and (c) are the models for perpendicular lines and (b) is the model for parallel lines.

Let $\overline{PQ}$ be the perpendicular to the line segment $\overline{XY}$ . Let $\overline{PQ}$ and $\overline{XY}$ intersect in the point $A$ . What is the measure of $∠ P A Y$ ?

Explanation
From the figure, it can be easily observed that the measure of $∠ PAY$ is $9 0^{\circ}$ .

Study the figure and answer the following questions (i) Name the equilateral triangles. (ii) Name the isosceles triangles. (iii) Name the scalene triangles.
(iv) Name the acute triangles. (v) Name the obtuse triangles. (vi) Name the right triangles.

Explanation (i) $△ ABC$ (ii) $△ ADE, △ ADC, △ AEB$ (iii) $△ ABD, △ ABM, △ AEC, △ AMC, △ AME, △ ADM$ (iv) $△ ABC, △ ADC, △ AEB, △ ADE$ (v) $△ ADB, △ AEC$ (vi) $△ AMB, △ AMD, △ AME, △ AMC$

Find the angles of a triangle which are in the ratio $2 : 3 : 4$ .

Solution Let the measures of the given angles be (2x), (3x), (4x). Then as we know that the sum of the angles of the triangle is $18 0^{\circ}$ . $∴ 2 x + 3 x + 4 x = 18 0^{\circ}$ $\Rightarrow 9 x = 18 0^{\circ} \Rightarrow x = 2 0^{\circ}$ Hence the measures of the given angles are $2 x = 2 \times 2 0^{\circ} = 4 0^{\circ}$ and $3 x = 3 \times 2 0^{\circ} = 6 0^{\circ}$ and $4 x = 4 \times 2 0^{\circ} = 8 0^{\circ}$

10. The angles of a quadrilateral are in the ratio 1:2:3:4. Find the measure of each of the four angles.

Solution Let the measure of the angles of the given quadrilateral be $x, (2 x), (3 x)$ and (4x). Then, $x + 2 x + 3 x + 4 x = 36 0^{\circ}$ [ $∴$ The sum of the angles of a quadrilateral is $36 0^{\circ}$ ] $\Rightarrow 10 x = 36 0^{\circ}$ $\Rightarrow x = 3 6^{\circ}$ Hence the required angles are $3 6^{\circ}, 7 2^{\circ}, 10 8^{\circ}$ and $14 4^{\circ}$ .

11. The three angles of a quadrilateral are $7 0^{\circ}, 6 0^{\circ}$ and $11 0^{\circ}$ . Find the fourth angle.

Explanation Sum of all the angles of a quadrilateral is $36 0^{\circ}$ . Fourth angle $= 36 0^{\circ} - (7 0^{\circ} + 6 0^{\circ} + 11 0^{\circ}) = 12 0^{\circ}$

12. Two sides of a ||gm are in the ratio $4 : 3$ . If its perimeter is 56 cm , find the lengths of its sides.

Solution Suppose the sides of the $∥ g m$ is 4 x and 3 x and as we know that in a |gm, opposite sides are equal. So other two sides will be 4 x and 3 x . Perimeter $=$ Sum of its all sides $\Rightarrow 56 = 4 x + 3 x + 4 x + 3 x$ $\Rightarrow 56 = 14 x$ $\Rightarrow x = \frac{56}{14} = 4$ So, the other two sides are $4 x = 4 \times 4 = 16 cm$ and $3 x = 3 \times 4 = 12 cm$ .

13. Specify the type of quadrilateral $A BC D$ in each case, given the following information. (i) $AB ∣∣ CD, BC ∣∣ AD, ∠ DAB = 6 0^{\circ}$ (ii) $AB ∥ CD, AD ∥ BC, ∠ DAB = ∠ ABC$ (iii) $AB ∣∣ CD, AD ∣∣ CB, AB = CD, BC = AD, DA ⊥ AB$ (iv) $AO = OC, DO = OB, DB ⊥ AC, ∠ DAB = 6 3^{\circ}, 0$ being the point of intersection of diagonals. (v) $AB = CD, CD = AD = BC$

Explanation (i) Parallelogram
(ii) Rectangle
(iii) Rectangle
(iv) Rhombus
(v) Rhombus

14. Two sides of a ||gm are in the ratio $4 : 3$ . If its perimeter is 56 cm , find the lengths of its sides.

Solution Suppose the sides of the $∥ g m$ is 4 x and 3 x and as we know that in a |gm, opposite sides are equal. So other two sides will be 4 x and 3 x . Perimeter $=$ Sum of its all sides $\Rightarrow 56 = 4 x + 3 x + 4 x + 3 x$ $\Rightarrow 56 = 14 x$ $\Rightarrow x = \frac{56}{14} = 4$ So, the other two sides are $4 x = 4 \times 4 = 16 cm$ and $3 x = 3 \times 4 = 12 cm$ .

15. Specify the type of quadrilateral $A BC D$ in each case, given the following information. (i) $AB ∣∣ CD, BC ∣∣ AD, ∠ DAB = 6 0^{\circ}$ (ii) $AB ∥ CD, AD ∥ BC, ∠ DAB = ∠ ABC$ (iii) $AB ∣∣ CD, AD ∣∣ CB, AB = CD, BC = AD, DA ⊥ AB$ (iv) $AO = OC, DO = OB, DB ⊥ AC, ∠ DAB = 6 3^{\circ}, 0$ being the point of intersection of diagonals. (v) $AB = CD, CD = AD = BC$

Explanation (i) Parallelogram
(ii) Rectangle
(iii) Rectangle
(iv) Rhombus
(v) Rhombus

16. Which type of solid shape is a (i) Dice (ii) Gas pipe (iii) Football (iv) Brick (v) Ice-cream cone (vi) Kaleidoscope

Explanation (i) Cube (ii) Cylinder (iii) Sphere (iv) Cuboid (v) Cone (vi) Triangular prism

17. A polyhedron has 4 faces and 6 edges. How many vertices will it have?

Solution As we know that For a polyhedron, $F + V - E = 2$ here $F = 4, E = 6$ $\Rightarrow 4 + V - 6 = 2$ $\Rightarrow V - 2 = 2$ $\Rightarrow V = 2 + 2 = 4$ So, the polyhedron has 4 vertices.

Related Article:-

Algebraic Expressions

Data Handling

The Triangles and its Properties

Visualising Solid Shapes

Fractions

Perimeter and Area

Join ALLEN!

Understanding Elementary Shapes

1.0Measuring Line Segments

Conversion of Units of Length

2.0Comparison of Line Segments

Comparison by Observation

Comparison by Divider

Comparison by Tracing

3.0Measure of Angles

Measuring Angles Using Protractor

Degree Measure of An Angle

Rotation Round The Clock

Directions

Degree, Minutes and Seconds

4.0Types of Angles

Right Angle

Straight Angle

Acute Angle

Obtuse Angle

Zero Angle

Complete Angle (360∘)

Reflex Angle

5.0Lines

Perpendicular Lines

Parallel Lines

6.0Polygons

Regular Polygon

7.0Triangle

Classification of Triangles

Scalene Triangle

Isosceles Triangle

Equilateral Triangle

On the Basis of Angle

Acute Angled Triangle

Obtuse Angled Triangle

Angle Sum Property of Triangle

8.0Quadrilaterals

Concave Quadrilateral

Angle Sum Property of Quadrilateral

9.0Properties of Special Quadrilaterals

Trapezium

Parallelogram

Rhombus

Rectangle

Square

Kite

10.0Solid Figures

Three Dimensional Shapes

Face

Edge

Vertex

Euler's formula

Cuboid

Cube

Sphere

Cone

11.0Pyramid

Square Pyramid

Triangular Pyramid

12.0Prism

13.0Numerical Ability

14.0Memory Map

Table of Contents

Related Article:-

Algebraic Expressions

Data Handling

The Triangles and its Properties

Visualising Solid Shapes

Fractions

Perimeter and Area

Join ALLEN!

Understanding Elementary Shapes

1.0Measuring Line Segments

Complete Angle (360 $^{\circ}$ )

Complete Angle (360 $^{\circ}$ )