1
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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VII – Std. Algebraic expressions – Class notes I
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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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VII – Std. Algebraic expressions – Class notes I
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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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VII – Std. Algebraic expressions – Class notes I
©Byjus
Question14. Classify the terms in Question13 as follows.
Question15. Write the difference between like terms and unlike terms. Give examples.
(Refer to the subtopic Like and Unlike terms _ Terms)
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.
(Refer to the subtopic Like and Unlike terms _ Terms)
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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VII – Std. Algebraic expressions – Class notes I
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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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VII – Std. Algebraic expressions – Class notes I
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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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VII – Std. Algebraic expressions – Class notes I
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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)
9
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VII – Std. Algebraic expressions – Homework
©Byjus
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?
12
LINEAR EQUATIONS IN
ONE VARIABLE
13
VII – Std. Linear equations in one variable – Class notes
©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
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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VII – Std. Linear equations in one variable – Class notes
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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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VII – Std. Linear equations in one variable – Class notes
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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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VII – Std. Linear equations in one variable – Class notes
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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
©Byjus
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
19
EXPONENTS AND POWERS
EXPONENTS AND
POWERS
20
VII – Std. Exponents and Powers – Class notes
©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
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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VII – Std. Exponents and Powers – Class notes
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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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VII – Std. Exponents and Powers – Class notes
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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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VII – Std. Exponents and Powers – Class notes
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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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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. 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
Questions from 6 to 20 are subjective questions.
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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(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
28
LINES AND ANGLES
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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 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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VII – Std. Lines and Angles – Class notes
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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
38
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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. 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
Questions from 6 to 20 are subjective questions.
6. Find the value of ∠1, ∠2 and ∠3.
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VII – Std. Lines and Angles – Homework
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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
©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 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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VII – Std. Triangles and its properties – Class notes
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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.
Figure 21:Triangle 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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VII – Std. Triangles and its properties – Class notes
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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.
Figure 24:Triangle ABC
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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VII – Std. Triangles and its properties – Class notes
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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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VII – Std. Triangles and its properties – Class notes
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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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VII – Std. Triangles and its properties – Class notes
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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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VII – Std. Triangles and its properties – Class notes
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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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VII – Std. Triangles and its properties – Class notes
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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
53
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VII – Std. Triangles and its properties – 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. 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?
Figure 40:Triangle ABC
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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VII – Std. Triangles and its properties – Homework
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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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VII – Std. Triangles and its properties – Homework
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Figure 47
15. In the given figure, prove that:
Figure 48
(i) ∆XYZ ≅ ∆XPZ
(ii) YZ = P