ALGEBRAIC EXPRESSIONS - … VII – Std. Algebraic expressions – Class notes I ©Byjus Instructions: This booklet given to you is your Class Notes. Keep filling this sheet as the

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ALGEBRAIC EXPRESSIONS

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VII – Std. Algebraic expressions – Class notes I

©Byjus

Instructions: This booklet given to you is your Class Notes. Keep filling this sheet as the class

proceeds. At the end of this session, you will have your notes ready.

1. Introduction

Algebra has its own language of symbols. Like any other language, you first must learn

the basic vocabulary and the correct ways to structure sentences.

(Refer to the subtopic Introduction _ About algebra)

Question1. What is the use of algebraic expressions in mathematics?

Question2. Write 3 examples for simple algebraic expressions.

(Refer to the subtopic Introduction _ Algebra and Arithmetic)

Question3. Write an expression to find the perimeter of the following figure.

Figure 1

2. Definition of a Variable

Question4. Write down the difference between a constant and a variable.

Classify the following as constants and variables.

a. 5

b. 1

2

c. Temperature at different times of a day

d. Number of months in an year

e. Height of students in a class

f. 2x

g. 4a

(Refer to the subtopic Definition of variables _ Variables Constants and Terms)

3. Formation of Algebraic Expressions

Question5. How is the expression 5𝑥3 − 6𝑦2 formed?

Hint: 𝑥2 is obtained by multiplying 𝑥 by itself.

Question6. Write algebraic expressions for the following conditions.

a) Twice a number, decreased by forty-one.

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b) Number 5 added to three times the product of numbers m and n.

(Refer to the subtopic Formation of algebraic expressions _ Formation of algebraic expressions)

a. Terms of an Expression

Question7. Define terms of an expression.

(Refer to the subtopic Like and Unlike terms _ Terms)

Question8. Write down the terms of the expression 4𝑥2 − 5𝑥𝑦 + 2𝑥.

Question9. “Terms are added to get an expression”. Why not “Terms are added or subtracted to get an

expression” ?

(i) Factors of a term

Question10. Factors of a term are separated by ______________ (Addition/Multiplication).

Question11. Complete the flow chart.

FactorsTermsExpression

5x2 -4xy

_____

_____

_____

_____

_____

_____

_____

_____

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(ii) Coefficients

Question12. Write down the terms of the following expressions and their coefficients.

Expression Terms Coefficients

−4𝑥3 + 2𝑦2 − 2𝑥

128𝑚4 − 56𝑛7 + 5𝑥

6𝑥3

5𝑦

−4𝑧3

4. Like and Unlike Terms

Question13. Write down the algebraic factors of the following terms.

Terms Algebraic factors

−104𝑥𝑦

6𝑥22𝑦2

5𝑥𝑦

−10𝑥2𝑦

−2𝑥2𝑦2

12𝑥𝑦2

Helping hand: Algebraic factors contain only variable.

FactorsTermsExpression

_______

-4m2

_____

_____

_____

8

m

n

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Question14. Classify the terms in Question13 as follows.

Question15. Write the difference between like terms and unlike terms. Give examples.


Following steps help to decide if the given terms are like terms or unlike terms.

Consider the algebraic factors. Ignore the numerical coefficients.

Check the variables in the terms. They must be same.

Check the powers of each variable in the terms. Powers of each variable should be

same.


5. Monomial, Binomial, Trinomial and Polynomial

Question16. Write down the number of unlike terms in the following expressions.

Expression Number of unlike terms

𝑎2 + 𝑏2 − 8𝑎2

𝑥𝑦 + 𝑥 − 𝑦 + 5𝑥

𝑥10

𝑥2 + 𝑦3 − 𝑧2 + 𝑥𝑦 + 2𝑦3

Question17. Using the table in Question16, define monomial, binomial, trinomial and polynomial.

Give examples.

(Refer to the subtopic Monomial, Binomial, Trinomial and Polynomial _ What is a Monomial,

Binomial, Trinomial and Polynomial)

6. Addition and Subtraction of Algebraic Expressions

Note: Sum of two or more like terms is a like term with a numerical coefficient equal to

the sum of the numerical coefficients of all the like terms. Similarly, the difference

between two like terms is a like term with a numerical coefficient equal to the difference

between the numerical coefficients of the two like terms.

(Refer to the subtopic Addition and subtraction of algebraic expressions _ Addition and

subtraction)

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Question18. Complete the following table.

Algebraic Expressions Result

2𝑥 + 5𝑥 7𝑥

(𝑦 + 3𝑦) + (2𝑧 + 𝑧) _____ + ______

3𝑥 + 2𝑦 + 4𝑥 + 5𝑥 ____ + _____

𝑎 + 2𝑏 + 3𝑐 + 6𝑎 + 5𝑏 + 4𝑐 _____ + _____ + _____

4𝑦2 − 3𝑦2

5𝑥 − 10 − (−𝑥2 + 10𝑥 − 9)

Question19. What should be added to 𝑎2 + 𝑎𝑏 + 𝑏2 to obtain 2𝑎2 + 3𝑎𝑏.

Hint: 2 is added to 3 to get 5.

(Refer to the subtopic Addition and subtraction of algebraic expressions _ Addition and

subtraction)

7. Expressions with Variables

Question20. Find the value of the following expressions.

Expression When Value of the expression

25 − 𝑥2

𝑥 = 5

3𝑎2 − 3𝑎 + 2

𝑎 = 2

𝑥3 + 2𝑥2 − 𝑥

𝑥 = 2

𝑎2 + 6𝑎𝑏 + 𝑏2

𝑎 = 2, 𝑏 = 1

𝑥2 − 𝑦2

𝑥 = 3, 𝑦 = 2

Question21. Value of the expression 2𝑦2 + 𝑦 + 𝑎 is 5 when 𝑦 = 1. Find the value of 𝑎.

8. What is an Equation?

Question22. Complete the following table with formulas for perimeter and area.

Perimeter

Equilateral triangle (Side length is 𝑙)

Square (Side length is 𝑙)

Regular pentagon (Side length is 𝑙)

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Area

Square (Side length is 𝑙)

Rectangle (length is 𝑙, breadth is 𝑏)

Triangle (Base is 𝑏 and height is ℎ)

Question23. Pratik went for jogging in a square shaped park .He covered 212 meters after completing

one round of running. Find the value of 𝑎 if length of one side of the park is (2𝑎 + 3) meters.

(Refer to the subtopic What is an equation_About equation)

9. Solving an Equation

Question24. Find the value of x in the equation 10 + 2𝑥 = 16

Helping hand: While solving an equation,

Same number can be added to (or subtracted from) both sides of the equation.

Both sides of the equation can be multiplied or divided by same non- zero number.

(Refer to the subtopic Solving an equation _ Solving equation)

10. Algebra as Patterns

a. Number patterns

Question25. If a natural number is denoted as 𝑛, then

(a) Its successor is ______.

(b) Even numbers are denoted as ________.

(c) Odd numbers are denoted as _______.

Question26. Following table shows different patterns and the number of matchsticks used to make it.

Match them accordingly.

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(Refer to the subtopic Algebra as patters_Algebra as patterns)

b. Pattern in geometry

Number of diagonals which can be drawn from one vertex of a triangle is ______

Number of diagonals which can be drawn from one vertex of a rectangle are ______

Number of diagonals which can be drawn from one vertex of a pentagon is ______

In general, number of diagonals which can be drawn from one vertex of a polygon

having 𝑛 sides is ______.

11. Word Problems

Question27. “A number multiplied by 2, then decreased by 41 is 3”. Write an equation for the

statement and find the number.

(Refer to the subtopic Word problems _ Word problems)

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VII – Std. Algebraic expressions – Homework

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Instructions: This booklet given to you is your Homework Sheet. Solve these problems at home. You shall be exam-ready if you can finish all the problems.

Questions from 1 to 5 are MCQ’s.

1. If 6 is added to three times the product of two numbers m and n, then expression will be

A) 𝑚𝑛 + 6

B) 3𝑚𝑛 + 6

C) 3 + 6𝑚𝑛

D) 𝑚 + 6 + 3𝑛

2. Coefficient of term 𝑥𝑦2 in expression 2𝑥𝑦2 + 𝑥3– 𝑥5 + 𝑥𝑦2is?

A) 1

B) 2

C) 3

D) -3

3. Value of expression 2𝑎 − 2𝑏 − 4 + 𝑎 + 𝑏 if 𝑎 = 2 and 𝑏 = 3

A) 1

B) 3

C) 2

D) -1

4. Observe the pattern given below

1,4,9,16,25......

Then term at 𝑛𝑡ℎposition will be given by expression

A) 𝑛 + 3

B) 𝑛 − 3

C) 2𝑛

D) 𝑛2

5. Value of the expression (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏) when 𝑎 = 3 and 𝑏 = 2 is

A) 13 C) 17

B) 15 D) 16

Questions from 6 to 20 are subjective questions.

6. Find the sum of the following expressions

𝑡 − 𝑡2 − 𝑡3 − 14; 15𝑡3 + 13 + 9𝑡 − 8𝑡2; 12𝑡2 − 19 − 24𝑡𝑎𝑛𝑑 4𝑡 − 9𝑡2 + 19𝑡3.

7. Will the value of 12𝑥 + 3 for 𝑥 = −4 be greater than or less than 15? Explain.

8. Each symbol given below represents an algebraic expression written in it:

Find the value of

9. Sonu and Raju have to collect different kinds of leaves for science project. They went to park where

Sonu collected 12 leaves and Raju collected 𝑥 leaves. After sometime Sonu lost 3 leaves and Raju

collects

2𝑥 more leaves. Write an algebraic expression to find the total number of leaves collected by both

of them.

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VII – Std. Algebraic expressions – Homework

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10. Rohit’s mother gave him Rs. 3𝑥𝑦2 and his father gave him Rs. 5(𝑥𝑦2 + 2).Out of this total money,

he spent Rs. (10 − 3𝑥𝑦2) on his birthday party. How much money is left with him?

11. Two adjacent sides of a rectangle are 5𝑥2 − 3𝑦2 and 𝑥2 − 2𝑥𝑦.Find its perimeter.

12. Simplify (a + b) (2a – 3b + c) – (2a – 3b) c.

13. The perimeter of a triangle is 6p2 - 4p + 9 and two of its adjacent side are p2 - 2p + 1 and 3p2 - 5p + 3.

Find third side of triangle.

14. Diya is 3 times as old as her cousin. The total of their ages is 36 years. How old is Diya’s cousin?

15. The number of children at the library was 3 times the number of adults. The total number of people

at the library was 48. How many children were at the library?

16. Numerical coefficients of terms of expression 4𝑥2 + 3𝑥𝑦 − 4𝑥2𝑦 are __,__,__ respectively.

17. When a monomial is multiplied with a binomial, resultant is a binomial. Explain with an example.

18. Sum of coefficients of the expression 4𝑥3 − 𝑎𝑥2 + 2𝑦2 + 𝑧 is 5. Find the value of the constant 𝑎.

19. What is the number of diagonals that can be drawn from one vertices of a polygon having ‘n’ sides?

What if the polygon is an octagon?

20. Simplify the expression 2𝑥𝑦 + 9𝑦𝑥 − 4𝑦𝑥. Is it a monomial? Why?

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LINEAR EQUATIONS IN

ONE VARIABLE

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VII – Std. Linear equations in one variable – Class notes

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1. Introduction

Pratik and Rashid were playing a game. Rashid asked Pratik to think about a number and

multiply that number with 2 and add 5 to the product. Then he asked Rashid to tell the result.

According to Rashid, the final number is 25.

Question1. Can you help Pratik in finding the number? Can you represent the given condition

mathematically?

Question2. Define Variable.

Question3.What is the difference between a variable and aconstant?

a. What is an Equation?

An equation is a condition on a variable. The condition is that two expressions should

have equal value. Note that atleast one of the two expression must contain the variable.

An equation remains same, when the expressions in LHS and RHS areinterchanged.

(Refer to the subtopic Introduction_Introduction – I)

Note: If there is some sign other than the equality sign between the LHS and the RHS, it is

not an equation. Thus, 2x + 2 > 65 is not an equation.

Question4. What is the importance of an equality sign in an equation?

Question5. What is the difference between an equation and an algebraic expression?

Question6. What are the mathematical operations involved in mathematical expressions?

Mathematical equations are not just useful, many are quite beautiful.

Did you know?

Pythogorean Theorem, which every geometry student learn.

This equation describes how, for any right – angled triangle, the

square of the length of the hypotenuse(c) equals the sum of the

squares of the lengths of the other two sides (a and b).

Thus 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐

Question7. Write the following statements in the form of equations:

(i) The sum of 4 times 𝑥 and 11 is 42

(ii) 2 times of 𝑥 is equal to 4 times of 3

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Question8. Convert the following equations to statement form:

(i) 3x + 4 = -1

(ii) 2x – 8 = 0

Question9. Sum of two number is 56. One number is 8 more than the other number. Set up an

equation for the following condition?

Did you know?

Euler’s equation, This simple equation explains about the

nature of spheres. “It says that if you cut the surface of a sphre up

into faces, edges and vertices, and let F be the number of faces,

E the number of edges and V the number of vertices, you

will always get 𝑽 − 𝑬 + 𝑭 = 𝟐

b. Linear Equation In one variable:

An equation which has highest power of variable as 1.

Note: There is no restriction on the number of variable to classify an equation as linear,

but these variables should appear in power 1 only.

Question10. Find out which of the following are examples of Linear Equation In One variable.

2x + 4, 2xy – 3, x2, (x + 1)(x – 1), (2x + 1), (yzx – 9)

(Refer to the subtopic Introduction_Introduction – I)

2. Solving Linear Equations

In an equation, the expression on LHS and the expression on RHS are equal and this is true

only for one value of the variable which is called the solution of the equation.

How do we solve these equations to get the value of the variable?

a. Solving equations which have linear expressions on one side and numbers on the

other side.

(i) By adding, subtracting, multiplying or dividing by the same number on either

side.

The expressions on either side of the equation are always equal to each other. So,

If some mathematical operations are performed on both sides of an equation, the

equality does not change.

Question11. Multiply the following equations by 2 and check whether they are same before and after.

1. 3x + 4 = 2x – 2

2. 2y + 1 = y – 1

Hint: Check whether the solution remains same or not.

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Note: When both sides of an equation are multiplied, added, subtracted or divided by the same non –

zero quantity, the equality does not change. However, division by zero is not permitted, as it is not

defined.

Question12. Solve the following equations:

(i) 2x – 6 = -8

(ii) 3x + 2 = -7

(ii) Transposition method.

In this method, the number or a variable is transposed to other side. It is same as

adding or subtracting the number from both sides. In doing so, the sign of the

number has to be changed. What applies to the numbers also applies to

expressions.

Question13. Solve by transpose method, 3p -10 = 5

Question14.Solve 2𝑝

3=

5

2

Question15.Sum of digits of two digits number is 8. The digit in tens place is thrice the digit in unit

place. Find the number.

b. Solving equations having variables on both sides.

Question 16. Find the solution of 4x + 12 = 2x – 2

Question 17. Find the solution of 2x + 1 = 3x – 5

Question18.Sum of digits of a two digit number is 9. If 27 is added to the number, the digits of the

original number gets interchanged. Find the original number.

(Refer to the subtopic Solving Linear Equations_Solving Linear Equation)

c. Cross multiplication method:

Question 19.Solve 3𝑥

2=

5𝑥

4

Question20. Solve 𝑥−2

3 =

𝑥−5

2

d. Application of liner equations to practical problems:

Many day-to-day problems can be solved by framing equations. These problems involve

relations among variables and numbers. The process consists of two parts, formulation

and solution.

Question 21.The perimeter of a rectangle is 36 cm. If the length of the rectangle is 4 cm. Find its

length.

3. Equations Reducible to Linear Form

Question22. State whether the following equation is linear. If yes, solve it.

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𝑥+1

𝑥−3 =

1

2

There are some equations which are not linear but we can put them in the form of linear

equation by doing some mathematical operations e.g. by multiplying both the sides of the

equation by a suitable expression.

Question23.The ratio of the present age ofManas and his wife is 4 : 3. After 4 years, the ratio of their

ages will be 9 : 7. What is the present age of Manas?

Question24.The denominator of a rational number is greater than its numerator by 5. If the number is

increased by 10 and the denominator is decreased by 3, the new number becomes 2. Find the original

number.

(Refer to the subtopic Equations Reducible to Linear Form_Question 1)

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VII – Std. Linear equations in one variable – Homework

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Instructions: This booklet given to you is your Homework Sheet. Solve these problems at home. You

shall be exam-ready if you can finish all the problems.

Questions from 1 to 5 are MCQ’s

1. Sum of three consecutive multiple of 6 is 666. One of these multiple is

A) 210

B) 234

C) 222

D) 240

2. 3x + 4 = 0 is

A) A linear polynomial

B) A binomial

C) A linear equation in one variable

D) None of these

3. What should be added to rational number 7

3 to get

17

6 ?

A) 3

2

B) 1

2

C) −1

2

D) None of these

4. The perimeter of a rectangle is 24 cm. If the length of the rectangle is 3cm, then its breadth is

A) 9 cm

B) 10 cm

C) 11 cm

D) 12 cm

5. The ratio of Vimal’s age and Arun’s age is 3: 5 and sum of their ages is 80 years. The ratio of

their ages after 10 years will be

A) 3 : 2

B) 1 : 2

C) 2 : 1

D) 2 : 3

Questions from 6 to 20 are subjective questions

6. Solve : 2𝑥+1

3𝑥−2 =

9

10

7. Two numbers are in ratio of 5 : 3. If they differ by 18, what are the numbers?

8. Sum of four consecutive odd numbers is 40. Find the numbers.

9. A number is twice another number. If their sum is 96, what are the numbers?

10. The difference between two numbers is 18. If their sum is 86, what are the numbers?

11. When a number is multiplied by 4 and then diminished by 7, the result is 65. What is the

number?

12. The sum of two numbers is 45 and their ratio is 7: 8. Find the numbers.

13. Sheela is now 15 years older than her younger brother Sanjay. Ten years from now Sheela

will be twice as old as Sanjay. Find the present age of each.

14. The denominator of a rational number is greater than its numerator by 3. Find the fraction.

15. Solve : 4 + 3x = 2

5 ( 6x – 2)

16. Find three consecutive even numbers whose sum is 96.

17. A box of sweets is divided among 24 children. They will get 5 sweets each. How many would

each get, if the number of children is reduced by 4.

18. Solve the equation: 𝑥

2 =

4

5 ( x + 10 )

19. One number is three times another. If the larger number is subtracted from 60, the result is 5

less than the smaller number subtracted from 55. Find the numbers.

20. Solve the equation: 5 ( x + 43)

2 =

2 (3x + 4)

3

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EXPONENTS AND POWERS

EXPONENTS AND

POWERS

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VII – Std. Exponents and Powers – Class notes

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1. Introduction

Question1. Can you explain how addition of the same number many times and multiplication of two

numbers are connected?

Hint: Think about adding the number 5, six times and multiplying 5 with 6.

Question2. Is there any difference between the two terms: 2 × 2 × 2 × 2 and 24? Explain.

(Refer to the videoclip of subtopic Introduction_Exponents and Powers – I)

Question3. Evaluate:

a) 24

b) 42

c) 102

(Refer to the videoclip of subtopic Introduction_Exponents and Powers – II)

2. Visualisation of Powers and Exponents

A power is the product of multiplying a number by itself. The number which is being

multiplied is called the base and how many times the number is multiplied is called the

exponent. The exponent is written on the right top of the base.

Did you know?

The term power was first used by Greek

mathematician Euclid for the square

of a line.

Misconception: ‘Power’ and ‘Exponent’ are same.

Clarification:

Figure 2:Exponential form of a number

In the above figure, ‘𝑥’ is base.

‘𝑎’ is exponent.

𝑥𝑎 as a whole is power.

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Question4. A small cubical box is considered as 2 units. Can you find the number of cubical box

present in a large one if it is considered as 24?

(Refer to the videoclip of subtopic Visualisation of Powers and Exponents_Visualisation- II )

a. Value of 𝒂𝟏

By the definition, the exponent shows how many times the base is multiplied by itself.

Here it's multiplied only one time, that is why it equals itself.

∴ 𝑎1 = 𝑎

b. Meaning of negative exponents.

In a number 𝑎𝑛, it is not always necessary that 𝑛 should be a natural number, 𝑛 can be

even negative number also. The reciprocal of 𝑎𝑛 is written as 𝑎−𝑛.

Note: 𝑎−𝑚 is the multiplicative inverse of 𝑎𝑚 because if you multiply 𝑎−𝑚and 𝑎𝑚, you

end upgetting 1 as product.

3. Laws of Exponents

Did you know?

The word exponent was introduced in 1544 by Michael Stifel

a. Law of Multiplication

If m and n are natural numbers, then

𝑎𝑚 = 𝑎 × 𝑎 × 𝑎 × … … … 𝑚 times

𝑎𝑛 = 𝑎 × 𝑎 × 𝑎 × … … … . 𝑛 times

Now, 𝑎𝑚 × 𝑎𝑛 = (𝑎 × 𝑎 × 𝑎 × … … 𝑚 times) × (𝑎 × 𝑎 × 𝑎 × … … 𝑛 times)

= 𝑎 × 𝑎 × 𝑎 × 𝑎 × … . ( _______)times

= ________

Thus, 𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛

This is the law of multiplication.

Note: The law of multiplication can be extended to more powers. For example, 𝑎𝑚 × 𝑎𝑛 × 𝑎𝑝 =

𝑎𝑚+𝑛+𝑝

Question5. Simplify (64)2

3 × (64)1

3

(Refer to the videoclip of subtopic Laws of Exponents_Laws of Exponents – I)

Note: (−𝑎)𝑚= 𝑎𝑚 ; if 𝑚 is an even number.

(−𝑎)𝑚= −𝑎𝑚 ; if 𝑚 is an odd number.

b. Law of division

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If 𝑚 and 𝑛 are natural number , where 𝑚 ˃ 𝑛, then

𝑎𝑚

𝑎𝑛=

𝑎 × 𝑎 × 𝑎 × … … . 𝑚 times

𝑎 × 𝑎 × 𝑎 × … … . . 𝑛 times

On cancelling n factors in the numerator with those in the denominator, we will be left

with

𝑚 – 𝑛 factors in the numerator , since 𝑚 ˃ 𝑛.

Therefore, the quotient becomes 𝑎 × 𝑎 × 𝑎 × … . . (𝑚– 𝑛) times.

𝑎𝑚

𝑎𝑛= 𝑎𝑚−𝑛

When 𝑛 ˃ 𝑚, there will be more factors in the denominator than in the numerator.

𝑎𝑚

𝑎𝑛 =

1

𝑎𝑛−𝑚

When 𝑚 = 𝑛, Number of factors of numerator and denominators are same.

𝑎𝑚

𝑎𝑚 =

𝑎 ×𝑎 ×𝑎 ×…….𝑚 𝑡𝑖𝑚𝑒𝑠

𝑎 ×𝑎 ×𝑎 ×……..𝑚 𝑡𝑖𝑚𝑒𝑠 = 1

(Refer to the videoclip of subtopic Laws of Exponents_Laws of Exponenets – I)

(i) The vaueof 𝑎0

Using the above case 3 we can prove 𝑎0 = 1

Proof : 𝑎0 = 𝑎1−1 =𝑎1

𝑎1 = 𝑎

𝑎= 1

Question6. Simplify (3−7 ÷ 3−10) × 3−5

Hint: For any non zerointeger 𝑎, 𝑎−𝑚 = 1

𝑎𝑚 where a is a positive integer.

c. Law of powers

If 𝑚 and 𝑛 are natural number, then

(𝑎𝑚)𝑛 = 𝑎𝑚 × 𝑎𝑚 × … . . 𝑛 times

= 𝑎𝑚 + 𝑚 + 𝑚 + …….𝑛 times

= 𝑎𝑚𝑛

The law of powers is,(𝑎𝑚)𝑛 = 𝑎𝑚𝑛

Question7. Simplify [{(3)−2}2]−1

d. Powers of a product

If 𝑛 is a natural number, then

(𝑎𝑏)𝑛 = (𝑎𝑏) × (𝑎𝑏) × … . 𝑛 times

∴ 𝑎0 = 1

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= (𝑎 × 𝑎 × 𝑎 × … . 𝑛 times) × (𝑏 × 𝑏 × 𝑏 × … . 𝑛 times)

= 𝑎𝑛 × 𝑏𝑛

Therefore,

(𝑎𝑏)𝑛 = 𝑎𝑛 × 𝑏𝑛

Question8. Simplify 3−4 × 2−4 × 5−4

e. Power of the quotient

(𝑎

𝑏)

𝑛

=𝑎

𝑏×

𝑎

𝑏× … … . 𝑛 𝑓𝑎𝑐𝑡𝑜𝑟𝑠

= 𝑎𝑛

𝑏𝑛

Note:All the laws of exponents are applicable for negative and fractional indices.

Question9. Simplify (5

8)

−7× (

8

5)

−5

4. Questions on Law of Exponents

Question10. Find the value of (15559)0.

(Refer to the videoclip of subtopic Laws of Exponents_Laws of Exponents – II)

Question11. What is the value of (256

81)

−1

4?

Question12. If 𝑥 = 2 and 𝑦 = 4, then what is the value of (𝑥)2𝑦 –𝑥

2?

Question13. Solve 2𝑥 + 23 = 24

Question14. Evaluate [{(5-1)-1}-1]-1

Question15. Find the value of (16

81)−

1

4 × (27

64)−

2

3 × 20.

(Refer to the videoclip of subtopic Questions Laws of Exponents_Question-I)

5. Use of Exponents

Exponents are used to express very large numbers or very small numbers effectively. The

number in the new form is called standard form.

(Refer to the subtopic Uses of Exponenets _ Questions on Standard Form- I)

Question16. Express the following numbers in their standard form:

a) 0.000000132

b) 0.00000000000542

c) 0.00000000000089

(Refer to the videoclip of subtopic Uses of Exponenets_Questions on Standard Form- I)

Question17. Express the following numbers in their normal form:

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a) 3.02 × 10-9

b) 11.26 × 10-4

(Refer to the videoclip of subtopic Uses of Exponenets_Questions on Normal Form)

Note: Exponents are used to compare very large numbers and very small numbers.

For example: To compare the diameter of the sun and diameter of the moon

Diameter of the sun = 1.4 × 109km

Diameter of the moon = 3474 km

Hence, we can conclude that sun is larger than moon but when seen from earth, they appeared to be

of same size.

(Refer to the videoclip of subtopic Uses of Exponenets_Application -I)

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VII – Std. Powers and Exponents – Homework

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Questions from 1 to 5 are MCQ’s.

1. Which one is greater 72 or 27?

A) 72

B) 27

C) Both are equal

D) None of these

2. Find the value of (√3)0.

A) 1

B) 0

C) 2

D) 1.732

3. Find the value of ((22) × (2-2)) × 22.

A) 2

B) 1

C) 0

D) 4

4. Find the value of 32 × 42

24

A) 16

B) 9

C) 3

D) 4

5. Find the value of 16 × 2-4.

A) 1

B) 32

C) 64

D) None of these


6. Can you tell which one is greater: (62), (2-4), (493

2)?

7. Write exponential form of 64 × 64 × 64 × 64 taking base 2.

8. Simplify and write the answer in exponential form: (7-2 × 74) ÷ (3432

3)

9. Simplify and write the answer in exponential form:[(22)3 × 36] × 56

10. Simplify : 24 × 33 × 16

32

16 × 2−1

11. Find the value of (64 × 1

36 ) ÷ 62

12. Simplify: 𝑎𝑏2 ×𝑎2𝑏 × 𝑎3𝑏3

𝑎𝑏

13. Find the value of (20 × 30) ÷ 1

2−2

14. Say true or false and justify your answer:

10 × 1011 = 10011

100 = 3330

15. Find the value of (100 × 2-1) ÷ (24 ÷ 2-2)

16. Express the following in standard form:

(i) 8976.32

(ii) 3,456,000

(iii) 90876.01

17. Write the number from each of the following expanded form:

(i) 5 × 103 + 2 ×102 + 3 × 10 + 2

(ii) 7 × 105 + 2 × 102 + 9

18. Express the number appearing in the following statement in standard form:

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VII – Std. Powers and Exponents – Homework

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(i) The mass of earth is 5467000000000 kg

(ii) The diameter of the sun is 1392000 km

19. Simplify the following: (42)2 × 16

12

× 64

−13

64

20. Simplify the following : 25−1

2 × 52 × 1

(52)−1

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LINES AND ANGLES

29

VII – Std. Lines and Angles – Class notes

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1. Introduction to Geometry

Geometry is a subject in mathematics that focuses on the study of shapes, sizes, relative

configurations, and spatial properties. Derived from the Greek word meaning "earth

measurement," geometry is one of the oldest sciences. It was first formally organized by the

Greek mathematician Euclid around 300 BC. Geometry has been the subject of countless

developments.

This discussion primarily focuses on the properties of lines, points, and angles.

Question1. Differentiate between a line and a line segment.

Line Line Segment

Question2. Can you identify the number of angles present in the given figure?

Figure 3

Question3. What is the minimum number of lines or line segments required to form an angle?

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Question4. Can you name all the angles present in the given figure?

Figure 4

(Refer to the subtopic Introduction to Geometry_Lines and Angles)

Question5. What are Triangular Numbers ? Explain briefly.

(Refer to the subtopic Introduction to Geometry_Geometry as Number Patterns)

2. Intersecting Lines and Pairs of Angles

Misconception: Size of an angle depends on the length of its arms or its orientation.

Clarification:

Figure 5: Angle

In the above figure, both the angles are equal to 45° irrespective of length of their arms or

orientation.

a. Types of Angles

(i) Acute angle :

An acute angle is an angle which is smaller than 90°.

Question6. Find the total number of acute angles present in the given figure and name them.

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Figure 6

(ii) Obtuse Angle:

Obtuse angle is an angle which is _______________ than 90°.

Question7. What is a right angle? Explain with figure.

Question8. What do you mean by a straight angle? Explain.

Question9. What is the measure of a whole(complete) angle?

b. Complementary Angles :

When the sum of the measure of two angle is 90°, then the angles are called

complementary angles.

Question10. Can you pair the following angles as complementary angles? How many pairs are

possible?

Figure 7

Note: If two angles are complementary, each angle is said to be the complement of other angle and

vice versa.

Did you know?

The word angle comes from the latin word

angulus, meaning “corner”

Question 11. Can two obtuse angles be complement of each other? Explain why?

Question12. The measure of two complementary angles are given as 2x + 10° and 3x - 20.Find the

measure of each angle.

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c. Supplementary Angles:

Pair of angles whose sum is equal to 180° are known as supplementary angles.

Question13. What will be the measure of the supplement of each of the following:

(i) 58°

(ii) 112°

(iii) 145°

Question14. Pair of supplementary angles are given as 4x + 45° and 6x + 35°. Find the measure of

each of the angles.

Note: If the two angles are supplementary, the their sum should be equal to 180°.

d. Adjacent Angles

Two angles are said to be adjacent if:

(i) They have a common vertex.

(ii) They have a common arm.

(iii) The non- common arm are on either side of the common arm.

Question15. State whether angles marked in the figure are adjacent or not. Why?

Figure 8

Question16. Can two right angles be adjacent angles?

Did you know?

GreatMathematician Eudemus first used the

concept of an angle, who regarded an angle as a

deviation from a straight line.

e. Linear Pair

Linear pair of angles is a pair of adjacent angles which are supplementary.

Question17. Find which of the following is a linear pair and explain why?

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Figure 9

f. Vertically Opposite Angles

Question18. Take two pencils and tie them with the help of a rubber band in the middle. Can you

name vertically opposite angles if AB and CD are written on the end of the pencil?

Note: When two lines intersect, the vertically opposite angles are equal.

Question19. In the given figure, find the value of x.

Figure 10

g. Intersecting Lines

Question20. Can you list out what do all have common in the following:

(i) Y

(ii) X

(iii) H

Question21. Name all the intersecting lines present in the given figure.

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Figure 11

Question22. Find the measure of the angles made by intersecting lines(sides) at the vertices of an

equilateral triangle.

(Refer to the subtopic Intersecting Lines and Pair of Angles_Types and Theorem on Angles)

3. Parallel Lines and a Transversal

a. Transversal Lines

You might have seen railway tracks crossing two or more railway tracks. These giveyou

an idea of tranversal.

A line that intersects two or more lines at distinct points is called a ‘transversal’.

Question23. How many transversals can you draw for the given two lines p and q?

Figure 12

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b. Angles Made by a Transversal

Question24. Name the pair of angles in the given figure:

Figure 13

Note: Corresponding angles include ,(i) different vertices (ii) are on the same side of transversal

Question25. Explain the condition required for two angles to be alternate interior angles.

Question26. Name the pair of angles in each figure.

Figure 14

c. Transversal of Parallel Lines

Question27. Define parallel lines and explain with an example.

If two parallel lines are cut by a transversal, each pair of corresponding angles are equal

in

measure.

If two parallel lines are cut by a transversal, each pair of alternate angles are equal.

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If two parallel lines are cut by a transversal, then each pair of interior angles on the same

side of the transversal are supplementary.

Question28. In the following figure, is 𝑙 parallel to 𝑚?

Figure 15

(Refer to the subtopic Parallel lines and a Transversal_The Basics)

4. Basic Properties of Triangle

Theorem 1: Sum of the angles of a triangle is 180°

Consider a triangle PQR, draw a line XY through P such that XY∥QR

Figure 16: Triangle PQR

Proof: XPY is a straight line.

So, ∠4+∠1+∠5 = __________. ------- (1)

But XY∥QR; PQ, PR are transversals.

∠4 = ∠2. Similarly ∠5 = ________.(Why?) (By ___________________)

Substituting the values in (1)

We get, ___+∠1+ _____ = 180°

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(Refer to the subtopic Basic properties of a triangle_Angle sum property)

Question29.If the angles of a triangle are in the ratio 2 : 3 : 4, determine all the angles of

triangle. Question30.What is an exterior angle of a triangle?

Theorem 2:An exterior angle of a triangle is equal to the sum of the corresponding two

interior opposite angles.Consider a triangle PQR, extend QR to S. ∠PRS is an exterior angle.

Figure 17: Exterior angle

Proof:∠3 +∠4 = 180°--------(1)(Since they are __________________)

Also ∠1+∠2+∠3 = ________ -----------(2)(By __________________________)

From (1) and (2),

∠4= ∠1+∠2

(Refer to the subtopic Basic properties of a triangle_Exterior angle property)

Question31.In the given figure, sides BA and CB of ΔABC are produced to point D and E

respectively. If ∠DAC = 135° and ∠ABE = 110°. Find ∠ACB.

Figure 18: Triangle ABC

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VII – Std. Lines and Angles – Homework

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1. Which of the following is a pair of complementary angles?

A) 45°, 65°

B) 125°, 55°

C) 55°, 35°

D) 90°, 90°

2. Which of the following is a pair of supplementary angles?

A) 45°, 65°

B) 125, 55°

C) 45°, 45°

D) None of these

3. Two angles can be supplementary if both of them are :

A) Acute angles

B) Obtuse angles

C) Right angles

D) None of these

4. Which of following is a pair of vertically opposite angles?

A) 1 and 2

B) 2 and 3

C) 1 and 4

D) 1 and 5

5. Which of the following is a pair of linear angles?

A) 1 and 3

B) 1 and 5

C) 3 and 4

D) 2 and 4


6. Find the value of ∠1, ∠2 and ∠3.

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7. Find the value of x.

8. Find the value of x.

9. Find the value of x°.

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10. Find the complement of 2

5 of

1

3 of a right angle.

11. In the given figure, ∠1 and ∠2 are supplementary angles. If ∠1 is decreased, what changes

should take place in ∠2 so that both the angles still remain supplementary.

12. In the adjoining figure, name the following pairs of angles.

(i) Obtuse vertically opposite angles

(ii) Adjacent complementary angles

(iii) Adjacent angles that do not form a linear pair

13. In the adjoining figure, p is parallel to q. Find the unknown angles.

14. Find the value of x in the figure if 𝑙 is parallel to 𝑚.

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15. Find the value of x and y in the given figure.

16. Find the value of x if 𝑙 and 𝑚 are parallel.

17. Find the value of x if 𝑙 and 𝑚 are parallel.

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18. Find the measure of all angles of ∆ABC.

19. Find the value of all angles in the given triangle.

20. Find the value of x in the given figure.

44

TRIANGLES AND ITS

PROPERTIES

45

VII – Std. Triangles and its properties – Class notes

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1. Introduction to Triangles

Question1. Find the total number of triangles in the given figure.

Figure 19: Triangle

Question2. How many angles are there in a triangle?

Question3. Can you classify triangles based on sides ?

Question4. Classify triangles based on angles?

Question5. Given below is ∆ABC, answer the following questions regarding ∆ABC.

Figure 20:Triangle ABC

(i) Side opposite to ∠ABC = ____________________

(ii) Angle opposite to side BC = ___________________

a. Medians of a Triangle

Median of a triangle is a line segment joining a vertex to the mid point of its opposite side.

Question6. How many medians can a triangle have ?

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Note: In case of isosceles triangle and equilateral triangle, a median bisects any angle at a vertex

whose adjacent sides are equal in length.

b. Altitute of a Triangle

Question7. Draw some lines which connect the vertex A and side BC of ∆ABC.


An altitude of a triangle is a line through a vertex and perpendicular to a line cointaining the

base.

Figure 22:Trinagle ABC

Hence, AD is the altitude of ∆ABC

Question8. How many altitudes a triangle can have ?

Question9. Can altitude and median be same for a triangle?

c. Angle sum property

Have you ever thought, what would be the sum of all angles of a triangle?

The sum of measure of the interior angles of a triangle, is _______ .

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Figure 23:Triangle

∠A + ∠B + ∠C = _______ .

Question10. Find the measure of the third angle ∠C, if ∠A = 75° and ∠B = 45°.

(Refer to the subtopic Basic properties of a triangle_Angle Sum Property-ASP)

Question11. Prove that the measure of each angle of an equilateral triangle is 60°.

Helping Hand : Apply angle sum property of the triangle

d. Exterior Angle property- EAP

Question12. What is an exterior angle of a triangle.

EAP- Measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-

adjacent interior angle.

Question13. Find the value of ∠X in the given figure.


2. Congruence of Triangles

Let’s do one activity in order to understand the concept of congruency:

Take two bangles of same set and try to place one on another. Write down your observation in

the given box.

Question14. Define congruent figures.

Two triangles are congruent, if they have exactly the same shape and same size. In other words,

if on placing one over the other, they coincide. In case of triangles, we have six parameters which

are to be compared. i.e. we compare three sides and three angles of one triangle with the other.

Two triangles are said to be congruent only when all these parameters are equal.

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Figure 25:Triangle PQR and Triangle ABC

When ∆PQR is congruent to ∆ABC, Symbolicallywewrite ∆PQR ≅ ∆ABC.

Notice that when ∆PQR ≅ ∆ABC, then sides of ∆PQR fall on corresponding equal sides of

∆ABC and so is the case for the angles.

i.e. PQ coincide with AB, QR coincide with BC and RP coincide with CA; ∠P coincide

with∠A, ∠Q coincide with∠B and ∠Rcoincide with∠C. From the definition it is clear that in

congruent triangles, all sides and angels of one triangle are equal to all sides and angels of

another triangle. But it is not necessary to check all the sides and angles to decide whether

triangles are congruent or not. It is sufficient to check any 3 components of the triangles to

decide. Thus we have conditions for the congruency such as SAS, ASA, SSS, RHS and AAS.

a. Criteria for Congruence of triangles:

(i) SAS congruence rule:

In the triangles ABC and PQR given below,

sideAB = side _____ ,side_____ = side PR and ∠BAC = ∠_______ .

⇒ ∆ ABC ≅ ∆ _________ .

Figure 26:Triangle ABC Figure 27: Triangle PQR

Two triangles are congruent if two sides and the included angle of one triangle are

equal to the two sides and the included angle of the other triangle.

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Question15. In the given figure, AB = CF, EF = BD and ∠AFE = ∠DBC. Prove that ∆AFE ≅ ∆CBD.

Figure 28

(ii) ASA congruence rule:

In ∆ABC and ∆PQR given below,

∠B = ∠____ ,∠______ = ∠R and side ________ = sideQR.

⇒∆_______≅ ∆ PQR.

Figure 29:Triangle ABCFigure Figure 30:Triangle PQR

‘Two triangles are congruent if two angles and the included side of one triangle are

equal to the corresponding two angles and the included side of the other triangle.’

(Refer to the subtopic Congruence of Triangles_Congruency Rule-SAS and ASA)

Question16. Given ∠P = ∠R and PQ = RQ.Prove that ∆PQT ≅ ∆RQS

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Figure 31

(iii) SSS congruence rule:

In ∆ABC and ∆PQR given below,

Side AB = side ____ ,side _____ = side QR and side _______ = side PR.

⇒∆ _______ ≅ ∆ PQR.

Figure 32: Triangle ABC Figure 33:Triangle PQR

‘Two triangles are congruent if three sides of one triangle are equal to the

corresponding three sides of the other triangle’.

(Refer to the subtopic Congruence of Triangles_Congruency Rule-AAS and SSS)

Question17.Given: AB = 4 cm, DC = 4 cm, and AD = BC.Prove that ∆ABC ≅ ∆ADC.

Figure 34:Rectangle ABCD

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(iv) RHS congruence rule:

In right angled triangles ABC and PQR,

Hypotenuse AC = ____________ , side AB = side _______ .

⇒∆ ABC ≅ ∆ ________ .

Figure 35: Triangle ABC Figure 36: Triangle PQR

‘Two right angled triangles are congruent if the hypotenuse and a side of one triangle

are equal to the hypotenuse and the corresponding side of the other triangle.’

(Refer to the subtopic Congruence of Triangles_RHS Rule of Congruency)

Question18.P is any point in interior of the angle ABC such that the perpendiculars drawn from P on

AB and BC are equal. Prove that BP bisects ∠ABC.

Figure 37

3. Inequalities in a Triangle

Question19.Can you construct a triangle taking side length as 3 cm, 3cm and 7 cm?

Were you able to construct one ?

That means there are some conditions to form a triangle.

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Question20. Try constructing a triangle with sides 3 cm, 5 cm and 7 cm.

You know that, if two sides of a triangle are equal then the angles opposite to them are also equal and

conversely. What if the two sides of a triangle are unequal? Which side is larger?

Theorem 1. The sum of any two sides of a triangle is greater than the third side.

(Refer to the subtopic Inequalities in a triangle_Triangle Inequality Theorem)

Theorem 2. In any triangle, the side opposite to the larger angle is longer.

Question21. In a triangle PQR, if ∠P =55°and ∠Q = 65°, find the shortest and largest sides of the

triangle.

(Refer to the subtopic Inequalities in a triangle_Longer Side Theorem)

Theorem 3. If two sides of a triangle are unequal, the angle opposite to the longer side is larger.

(Refer to the subtopic Inequalities in a triangle_Longer Side Theorem)

Question22. In the figure, D is the point on side BC of ∆ABC such that AD = AC. Show that AB ˃

AD.

Figure 38

Question23.AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD.

Show that ∠A ˃ ∠C and ∠B ˃ ∠D.

Figure 39

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VII – Std. Triangles and its properties – Homework

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1. An exterior angle of a triangle is equal to 155° and two interior opposite angles are equal.

Each of these angles is equal to:

A) 75.2°

B) 67.5°

C) 77.5°

D) 90°

2. Two sides of a triangle measure 3 and 7.Which of the following could be the measure of third

side?

A) 11

B) 16

C) 8

D) 19

3. The figure given below shows a right triangle with representation for two angles. What is the

value of x?


A) 35°

B) 20°

C) 25°

D) None of these

4. In triangle ABC, ∠A is obtuse. Which statement is true about the sum of the measures of ∠B

and ∠C?

A) ∠B + ∠C = 90

B) ∠B + ∠C > 90

C) ∠B + ∠C < 90

D) ∠B + ∠C = 180

5. What is y in terms of x?

Figure 41:Triangle

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A) 3

2x

B) 4

3x

C) x

D) 3

4x

Questions from 6 to 15 are subjective questions

6. One angle of a triangle is 60°.The other two angles are in the ratio of 5: 7. Find the two

angles.

7. Calculate the angles of a triangle, if they are in the ratio 4 : 5: 6.

8. If one base angle of an isosceles triangle is double of the vertical angle, find all its angles.

9. In the given figure, prove that :

Figure 42

(i) ∆AOD is congruent to ∆BOC

(ii) AD = BC

10. ABC is an equilateral triangle, AD and BE are perpendicular to BC and AC respectively.

Prove that:

Figure 43

(i) AD = BE

(ii) BD = CE

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11. In the given figure, prove that:

Figure 44

(i) PQ = RS

(ii) PS = QR

12. In the given figure, prove that :

Figure 45

(i) ∆ACB≅ ∆ECD

(ii) AB = ED

13. Find the value of x in the given figure:

Figure 46

14. Find the value of x in the given figure:

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Figure 47

15. In the given figure, prove that:

Figure 48

(i) ∆XYZ ≅ ∆XPZ

(ii) YZ = P

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ALGEBRAIC EXPRESSIONS - … VII – Std. Algebraic expressions – Class notes I ©Byjus Instructions: This booklet given to you is your Class Notes. Keep filling this sheet as the