Introduction
Previously we have studied about perimeter and area of different plane geometrical figures. We know that the magnitude of region bounded by plane figures is area. In this chapter we will learn more about area of plane and solid figures (like triangle quadrilateral, cube, cuboid, etc.) 
Parallel Lines
Two lines are said to be parallel if the distance among them always remains same at each and every point. The parallel lines never intersect each other.
In other words we can say that if two lines do not have any common point than they are said to be parallel. In the figure I and m are parallel lines.
Concept of Transversal
Transversal is a line which intersects two or more parallel lines. In the figure, n is a transversal line.
Alternate Interior Angles
In the above figure
\[\angle 3\]and \[\angle 5,\text{ }\angle 4\]and \[\angle 6\]are alternate interior angles.
Alternate Exterior Angles
In the above figure \[\angle 7\]and \[\angle 2\]are alternate exterior angles.
Corresponding Angles
\[\angle 2\] and \[\angle 3\] in the above figure are corresponding angles.
They are also the angles on the same side of transversal.
Properties of Angles
When the parallel lines are intersected by a transversal:
Using the figure below which one of the following statements is true?
(a) Z ABD and \[\angle ABC\]are adjacent angles
(b) \[\angle ABD\]and \[\angle DBC\]are complementary
(c) \[\angle DBC\]is half of the measure of \[\angle ABC\]
(d) \[\angle ABC\]and Z DBC are congruent
(e) None of these
Answer: (b)
Explanation
From the figure only option (b) is correct because\[\angle ABD+\angle DBC={{90}^{O}}\]
Find the value of x in the figure given below.
(a)\[15{}^\circ \]
(b) \[20{}^\circ \]
(c)\[9{}^\circ \]
(d) \[15{}^\circ \]
(e) None of these
Answer: (d)
Explanation
From the figure \[5x{}^\circ +5x{}^\circ +2x{}^\circ =180{}^\circ \]
\[\Rightarrow 12x{}^\circ =180{}^\circ \Rightarrow X{}^\circ =15{}^\circ \]
Find the difference between two angles in the figure given below.
(a)\[15{}^\circ \]
(b) \[50{}^\circ \]
(c)\[70{}^\circ \]
(d) \[20{}^\circ \]
(e) None of these
Answer: (b)
Read the following statements.
(i) \[\angle PQT\] and \[\angle TQS\] are adjacent angles
(ii) more... 
Angle
If two rays have common end point then the inclination between two rays is called angle. In the figure O is the vertex, \[\overline{OP}\] and \[\overline{OQ}\] are called arm of the angle. It is represented by notation\[\angle \].
Types of angle
Acute Angle
Tangle whose measure is more than \[0{}^\circ \] and less than \[90{}^\circ .\]
Right Angle
The angle of measure \[90{}^\circ \]
Obtuse Angle
The angle whose measure is more than 90° and less than \[180{}^\circ .\]
Straight Angle
The angle whose measure is \[180{}^\circ \]
Reflex Angle
The angle whose measure is more than \[180{}^\circ \]and less than \[360{}^\circ .\]
Complete Angle
The angle whose measure is 360°.
Equal Angles
Two angles are said to be equal if they are of same measure.
Complementary Angles
If the sum of measure of two angles is \[{{90}^{o}}\] then they are said to be complementary angles .e.g \[75{}^\circ \] and \[15{}^\circ \] are complementary angles and they are said to be complement of each other.
Supplementary Angles
If the sum of measure of two angles is \[180{}^\circ \]then they are said to be supplementary angles, e.g \[107{}^\circ \] and \[73{}^\circ \] are said to be supplement of each other.
Which one of the following statements is not true?
(i) A line segment has finite length
(ii) A line has only one dimension
(iii) A line \[\overleftrightarrow{AB}\] and \[\overleftrightarrow{BA}\]represents the same
(iv) A ray \[\overleftrightarrow{AB}\]and \[\overleftrightarrow{BA}\]represents the same
(a) i, ii
(b) ii and iii
(c) Only iv
(d) iii and iv
(e) None of these
Answer: (c)
Explanation
\[\]and \[\]are different rays. They are started from different end points A and B respectively.
Therefore, option (c) is correct and rest of the options is incorrect.
In the following AD is the bisector of \[\angle EAF\]
Which one of the following statements is incorrect?
(a) \[\angle EAD\]is an acute angle
(b) \[\angle BAE\]is an obtuse angle
(c)\[\angle FAD\] and \[\angle DAE\]are complement to each other
(d) \[\angle CAD\]and \[\angle DAE\]are not complement to each other
(e) None of these
Answer: (d)
Explanation
Since \[\angle EAD=\angle CAD=45\]degree hence, option (a) is correct.
\[\angle BAE\]is more than 90° therefore, it is obtuse hence, option (b) is also correct. more... 
Some Terms Related to Lines
Point
It is a dimensionless figure which represents exact position. It is represented by a fine dot. We denote point by capital letters like A, B, C..........
Line Segment
It is the straight path between two points. In other words we can say that it has two end points and is of finite length.
Ray
When line segment extends infinitely in one direction is a ray. Simply we can say that a ray has one end point and definite length.
Line
When both end of line segment extended infinitely is known as a line. Simply we can say that line has no end point and definite length
Concurrent Lines
If three or more than three lines intersect at a point then the lines are known a concurrent lines,
Intersecting Lines
When two lines having one common point is called intersecting lines.
Introduction
In our daily life we observe different geometrical shapes. These geometrical shapes are not only the matter of study of mathematics but are directly related with our daily, life basic geometrical figures which make these geometrical shapes are lines and angles. 
Introduction
Percentage is a fraction whose denominator is 100. The numerator of the such fraction is called the rate percent. For example 15 percent means \[\frac{15}{100}\] and denoted by 15 %.
Percentage
a % means \[\frac{a}{100}\] and simplify it. e.g. \[45%=\frac{45}{100}=\frac{9}{20}\]
For conversion of fraction \[\frac{p}{q}\] as percentage, we simply multiply it by 100 and put the sign of % or mathematically we can write \[\frac{p}{q}=\left( \frac{p}{q}\times 100 \right)%\]
The population of a village is \[4500.{{\left( \frac{11}{18} \right)}^{th}}\] of them are males and the rest are females. If 40 % of the females are married, then the number of married females is:
(a) 1750
(b) 700
(c) 750
(d) 900
(e) None of these
Answer: (b)
Explanation
No. of males\[=\frac{11}{18}\times 4500=2750\]
No. of females \[=\text{ }4500-2750=1750\]
Thus no. of married females
\[=\text{ }40\text{ }%\text{ }\times 1750=700\]
In a class of 38 girls, 3 are absent, 20 % of the remainder have failed to do the home work. Find the number of girls who did their homework.
(a) 7
(b) 25
(c) 28
(d) 30
(e) None of these
Solution: (c)
No. of girls present \[=38-3=35\]
No. of girls who did not do home work \[=20%\times 35=7\]
No. of girls who did their home work \[=35-7=28.\] 
Application Based Problem on Percentage
The following are the points to remember to solve the problem related to variation in the price of an article.
In the new budget, the price of petrol increased by 25 %. By how much % person should reduce his consumption so that his expenditure is not affected
(a) 10%
(b) 20%
(c) 25%
(d) 30%
(e) None of these
Answer: (b)
Explanation
Reduction in consumption \[\left( \frac{25}{100+25}\times 100 \right)%=20%\]
Due to reduction of \[6\frac{1}{4}%\] in the price of sugar, a man is able to buy 1 kg more sugar for Rs 120. The reduced rate of sugar is:
(a) Rs 8 per kg
(b) Rs 6.5 per kg
(c) Rs 7.5 per kg
(d) Rs 9 per kg
(e) None of these
Answer: (c)
Explanation
Suppose original rate of sugar \[Rs\,x\] per kg.
Reduced rate \[=\left[ \left( 100-\frac{25}{4} \right)\times \frac{1}{100}\times x \right]=\frac{15x}{16}\]
According to question, \[\frac{120}{\frac{15x}{16}}-\frac{120}{x}=1\]
\[\Rightarrow \frac{128}{x}-\frac{120}{x}=1\Rightarrow x=8\]
Reduced rate \[=\frac{15}{16}\times 8=Rs\,7.5\] per kg
Problem Based on the Population of a Locality
Suppose the present population of a locality be 'A' and let it increases by \[x\text{ }%\]per annum then
The population of a town is 176400. If it increases at the rate of 5 % per annum then what was its population 2 years ago?
(a) 166400
(b) 154600
(c) 160000
(d) 166000
(e) None of these
Answer: (c)
Explanation
According to question
Population of city before y years, if it increases at the rate of \[x%=\frac{A}{{{\left( 1+\frac{x}{100} \right)}^{y}}}\]
\[\therefore \]The population if city \[=\frac{176400}{{{\left( 1+\frac{5}{100} \right)}^{2}}}=176400\left( \frac{20}{21}\times \frac{20}{21} \right)=160000\]
In a certain year the population of London is 200000. If it increases at the rate of 6.5 % per annum then what will be its population after 2 years?
(a) 226845
(b) 228645
(c) 224685
(d) 228465
(e) None of these
Answer: (a)
Explanation
Population after 2 more... 
Introduction
In our day to day life we exchange the things with the money with others. During such transaction either we get profit or loss. In this chapter we will frequently use the term profit, loss, profit percent and loss percent.
Terms Related to Profit and Loss
Cost Price
It is the price of an article at which the shopkeeper purchases the goods from manufacturer or wholesaler. In short it can be written as C.P.
Selling Price
It is price of the article at which it is sold by the shopkeeper to the customer. In short it can be written as S.P.
Profit and Profit Percent
If the S.P. of an article is greater than the C.P. then profit will occur and it is the difference between S.P. and C.P.
i.e. Profit = S.P. \[-\]C.P. and profit percent is written as: 
Loss and Loss Percent
If the selling price of an article is less than the cost price then the difference between the cost price and the selling price is called loss.
i.e. Loss = C.P. - S.P. The loss percent is the loss that would be made on a cost price of Rs. 100.
i.e. 
Relation between Profit and Loss
A mobile phone is sold for Rs 3,120 at the loss of 4 %. What will be the gain or loss percent, if it is sold for Rs 3,640?
(a) 10%
(b) 11%
(c) 12%
(d) 13%
(e) None of these
Answer: (c)
Explanation
Here, S.P = Rs 3120, loss % = 4 %
\[\therefore C.P.=\frac{S.P\times 100}{100-Loss%}=Rs.\frac{100\times 3120}{100-4}\] \[=Rs\frac{100\times 3120}{96}=Rs.3250\]
Now, new S.P = Rs 3,640
\[\therefore \] Gain = S.P - C.P = Rs 3,640 - Rs 3,250 = Rs 390
Hence, gain \[%=\frac{Gain}{C.P.}\times 100=\frac{390}{3250}\times 100=12%\]
By selling 42 oranges, a person losses a sum equal to the selling price of 8 oranges. Find the loss percent. more... 
Discount
In our daily life whenever we go to the market, we see banners & big hoardings indicating discount or sale up to 50 % off or buy one get one free. These are different tacts to attract the customers to the market. Shopkeepers want to get maximum rice for their goods and the customers are willing to pay as less as possible. Shopkeepers also offer different types of rebates in order to increase the sales or to finish old or damaged stock. This types of rebate on the price of article is called discount.
Marked Price
The price of an article which is indicated on it is called marked price or list price. In short form, it can be written as M.P. The amount which is to be reduced from marked price is called discount.
Concept Related To Discount
The marked price of a shirt is Rs 940 & the shopkeeper allows a discount of 15 % on it. Find the selling price of the shirt.
(a) Rs 895
(b) Rs 656
(c) Rs 785
(d) Rs 799
(e) None of these
Answer: (d)
Explanation
M.P. of a shirt = Rs 940, Rate of discount = 15%
Discount =15% of \[Rs940=\frac{15}{100}\times 940=Rs141\]
S.P. of the Shirt = M.P.-Discount = RS 940 - Rs 141=Rs 799
The price of a refrigerator is reduced from Rs 9,600 to Rs 8,448 in the winter season. Find the percentage of discount.
(a) 9%
(b) 1%
(c) 11%
(d) 12%
(e) None of these
Answer: (d)
Explanation
According to question M.P = RS 9,600 and S.P = Rs 8,448
Since, discount = M.P. - S.P = Rs 9,600 - Rs. 8,448 = Rs 1,152
Rate of discount
The shopkeeper marked his good at 20 % above the cost price. He sold half of the stock at the marked price, one quarter at a discount of 20 % and rest at the discount of 40 %. Find the total gain or loss percent.
(a) 2% gain
(b) 2 % loss
(c) 15 % gain
(d) 16 % loss
(e) None of these
Answer: (a)
A shopkeeper offers 5 % discount and more... 
Introduction
In our daily life the transaction of money is a common phenomenon in business where transaction involves large amount of money. Money is borrowed from bank Sum from individuals for certain duration of time and at certain rate of interest. The sum further returned to the specified person or bank including the interest on the original sum. Interest is that excess money paid on borrowed amount.
The Interests are of two types:
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