Mahesh

Students can go through AP Board 7th Class Maths Notes Chapter 5 Triangle and Its Properties to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 5 Triangle and Its Properties

→ Triangles can be classified according to properties of their sides and angles. Based on sides, triangles are of three types.

→ Equilateral triangle: A triangle in which all the three sides are equal is called an equilateral triangle. In △ABC
AB = BC = CA, also ∠A = ∠B = ∠C In an equilateral triangle each angle is equal to 60°.

→ Isosceles triangle: A triangle in which two sides are equal is called an isosceles triangle.
In △PQR
PQ = PR also ∠Q = ∠R

The non-equal side in an isosceles triangle may be taken as base of the triangle.

→ Scalene triangle: A triangle in which no two sides are equal is called a scalene triangle.

In △BAT
BA ≠ AT ≠ BT also ∠B ≠ ∠A ≠ ∠T.

→ Based on angles, triangles can be classified into three types.

→ Acute angled triangle: A triangle in which all the three angles are acute is called an acute-angled triangle.

In △TAP,
∠T, ∠A, ∠P are acute angles.

→ Obtuse angled triangle: A triangle in which one angle is obtuse is called an obtuse angled triangle.

In △FAN,
∠A is obtuse angle.
A triangle cannot have more than one obtuse angle,

→ Right angled triangle: A triangle in which one angle is a right angle is called a right angled triangle.

In △COT
ZO is right angle (i.e) 90°.
A triangle cannot have more than one right angle,

→ Right angled isosceles triangle: A triangle in which one angle is right angle and two sides are equal is called a right angled isosceles triangle.

In △POT,
PO = OT and ∠O = 90° also
∠P = ∠T = 45°

→ Family of triangles – Flow chart

→ Relation between sides of a triangle
In any triangle the sum of the lengths of any two sides is greater than the length of the third side.

In △TIN,
TI + IN > TN; TN + NI > TI; TI + TN > IN
Also the difference between lengths of any two sides of the triangle is less than the length of the third side.
In △TIN, TI > TN – NI; IN > TI – TN; TN > IN – TI

→ Altitutes of a triangle
The length of a line segment drawn from a vertex to its opposite side and is perpendicular to it is called an altitude or height of the triangle. An altitude can be drawn from each vertex.

Altitude of a triangle may be in its interior or exterior.

→ Medians of a triangle
A line segment joining a vertex and the mid-point of its opposite side is called a median.

A triangle has three medians.
The medians of a triangle are concurrent.
The point of concurrence of medians of a triangle is called the centroid of the triangle.
In △ABC, D, E and F are mid-points of the sides AB, BC and AC.
AE, CD and BF are mid-points.
G is the centroid.

→ Angle – sum property of a triangle: The sum of interior angles of a triangle is equal to 180° or two right angles.

In △BET,
∠B + ∠E + ∠T = 180°

→ Exterior angle of a triangle

→ When one side of a triangle is produced, the angle thus formed is called an exterior angle.

In △COL; the side OL is produced to D.
∠CLD is an exterior angle.
The exterior angle of a triangle is equal to the sum of the interior opposite angles.
∠COL + ∠OCL = ∠CLD

Students can go through AP Board 7th Class Maths Notes Chapter 4 Lines and Angles to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 4 Lines and Angles

→ Complementary angles: If the sum of two angles is 90°, the angles are called complementary angles.
Eg: 55°, 35° are complementary angles.

→ Linear pair of angles: When a ray stands on a line, the pair of angles thus formed are called linear pair of angles and their sum is 180°. In the figure ∠x and ∠y are called linear pair of angles.

→ Adjacent angles: Two angles with a common vertex and a common arm are called adjacent angles. Here the non-common arms lie on either sides of the common arm.
Eg: ∠AOB and ∠BOC are adjacent angles.

→ Supplementary angles : Two angles are said to be supplementary if their sum is 180°. Eg : (100°, 80°), (110°, 70°), (60°, 120°)

→ Vertically opposite angles: Two angles are said to be vertically opposite angles if they are formed by two intersecting lines and are not adjacent.
Eg: ∠AOC and ∠BOD are vertically opposite angles.

→ A line which intersects two or more lines at distinct points is called a transversal. In the figures ‘t’ is a transversal.

In the figure (i) t is not a transversal as it doesn’t intersect other two lines at two distinct points.

→ When a transversal intersects a pair of lines, 8 angles are formed.
Here l, m are two lines and ‘t’ is a transversal.

The angles are ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8.
∠3, ∠4, ∠5, ∠6 are called interior angles and ∠1, ∠2, ∠7 and ∠8 are called exterior angles.

→ When a transversal intersects a pair of lines the following pairs of angles are called corresponding angles.

i) One angle is interior and the other is exterior.
ii) Not adjacent angles.
iii) Two angles are on the same side of the transversal.

→ The following pairs are called alternate interior angles
i) both are interior
ii) not-adjacent
iii) either sides of the transversal

→ The following pairs are called alternate exterior angles
i) both are exterior
ii) not-adjacent
iii) on the either sides of the transversal

→ The following pairs are called interior angles on the same side of the transversal.

→ When a pair of parallel lines intersected by a transversal the pairs of corresponding angles are equal.
∠1 = ∠5
∠2 = ∠6
∠3 = ∠7
∠4 = ∠8

The pairs of alternate interior angles are equal.
∠3 = ∠5
∠4 = ∠6
The pairs of alternate exterior angles are equal.
∠1 = ∠7
∠2 = ∠8
The interior angles on the same side of the transversal are supplementary.
∠3 + ∠6 = 180°
∠4 + ∠5 = 180°
The exterior angles on the same side of the transversal are supplementary.
∠1 + ∠8 = ∠2 + ∠7 = 180°

→ Conversely when a transversal intersects a pair of lines in such way
i) making pairs of corresponding angles equal
(or)
ii) making alternate interior angles equal.
(or)
iii) making alternate exterior angles equal.
(or)
iv) making angles on the same side of the transversal interior/exterior supplementary then the lines are parallel.

Students can go through AP Board 7th Class Maths Notes Chapter 3 Simple Equations to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 3 Simple Equations

→ Simple equations help in solving various problems in daily life.
Eg: After 5 years Ramesh’s age is 15 years. What is his present age?
Solution. Let Ramesh’s present age be x years
After 5 years Ramesh’s age = x + 5
By problem, x + 5 = 15
x = 15 – 5 = 10 years
∴ Ramesh’s present age = 10 years

→ To balance an equation
a) Same number can be added on both sides.
b) Same number can be subtracted from both sides.
c) Multiply both sides with same number.
d) Divide both sides with same number.
So that the equality remains unaltered.

→ An equation remains same if the L.H.S and R.H.S are interchanged.

→ To solve a simple equation we transform term from one side to another.
While transforming term from one side to another
‘+’ quantity becomes ‘-‘ quantity
‘-‘ quantity becomes ‘+’ quantity
‘×’ quantity becomes ‘÷’ quantity
‘÷’ quantity becomes ‘×’ quantity
(i.e.) when the terms are transposed they get opposite signs and the term which multiplies one side, divides the other side.

Students can go through AP Board 7th Class Maths Notes Chapter 2 Fractions, Decimals and Rational numbers to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 2 Fractions, Decimals and Rational numbers

→ A proper fraction is a fraction that represents a part of a whole i.e., a fraction in which numerator is less than the denominator is called a proper fraction.
Eg: \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{2}{3}\), \(\frac{5}{6}\), \(\frac{8}{13}\),….. etc

→ An improper fraction is a fraction that represents a whole or more than a whole i.e., a fraction in which the numerator is more than or equal to the denominator is called an improper fraction.
Eg: \(\frac{5}{3}\), \(\frac{4}{3}\), \(\frac{8}{7}\), \(\frac{11}{5}\), ….. etc

→ Fractions can be represented pictorially.
Eg:

→ Like fractions can be compared by their numerators.

→ Unlike fractions can be compared by converting them into like fractions.

→ An equivalent fraction of a given fraction can be obtained by multiplying its numerator and denominator by same number.
Eg: Equivalent fraction for \(\frac{3}{5}\) is

→ To multiply a fraction with a whole number; we take the product of the numerator and the whole number as the new numerator, keeping the denominator the same.
Eg:

→ Product of the fractions = \(\frac{\text { Product of Numerators }}{\text { Product of Denominators }}\)
Eg: \(\frac{5}{3}\) × \(\frac{4}{7}\) = \(\frac{20}{21}\)

→ In mathematical computation ‘of’ means multiplication.
Eg: \(\frac{1}{3}\) of 24 = \(\frac{1}{3}\) × 24 = 8

→ The product of two proper fractions is less than each of the fraction in multiplication.
Eg:

→ The product of a proper and improper fraction is less than the improper fraction and greater than the proper fraction.
Eg: \(\frac{3}{4}\) × \(\frac{7}{5}\) = \(\frac{21}{20}\)
Here \(\frac{3}{4}\) < \(\frac{21}{20}\) and \(\frac{7}{5}\) > \(\frac{21}{20}\)

→ The product of two improper fractions is greater than each of the fractions.
Eg: \(\frac{7}{5}\) × \(\frac{3}{2}\) = \(\frac{21}{20}\)
Here \(\frac{7}{5}\) < \(\frac{21}{10}\) and \(\frac{3}{2}\) < \(\frac{21}{20}\)

→ To divide a whole number with a fraction; multiply the whole number by the reciprocal of the given fraction and vice versa.
Eg:

→ To divide one fraction by another, multiply the first fraction with the reciprocal of 2nd fraction.
Eg: \(\frac{3}{5}\) ÷ \(\frac{5}{8}\) = \(\frac{3}{5}\) × \(\frac{8}{5}\)

→ To multiply a decimal by 10,100,1000,, we move the decimal point in the number to the right side as many places as there are zeros in the numbers 10, 100, 1000 ……
Eg:
1.125 × 10 = 11.25
1.125 × 100 = 112.5
1.125 × 1000 = 1125
1.255 × 10,000 = 12,550

→ To multiply two decimal numbers.
i) multiply them as whole numbers.
ii) count the total number of digits in decimal places and add them.
iii) place the decimal point in the product by counting the sum of digits from its right most place.
Eg: 6.25 × 3.14
i) 625 × 314 = 196250
ii) sum of the number of digits in decimal places = 2 + 2 = 4
iii) 19.6250

→ To divide a decimal number by numbers like 10,100,1000, …… etc. we shift the decimal point in the decimal number to the left by as many places as there are zeros in 10, 100,1000 etc.
Eg: 435.873 ÷ 10 = 43.5873
4551.3 ÷ 100 = 45.513
8374.2 ÷ 1000 = 8.3742
24.82 ÷ 1000 = 0.02482

→ To divide a decimal number by a whole number
i) divide them as whole numbers
ii) place the decimal point in the quotient as in the decimal number.
Eg: 86.5 ÷ 5
i) 865 ÷ 5 = 173
ii) 17.3
To divide a decimal number by another,
i) shift the decimal to the right by equal number of places in both to convert the denominator to a whole number.
ii) divide them as in above
Eg: 6.25 ÷ 2.5

→ The numbers in the form \(\frac{p}{q}\) where p, q are integers and q ≠ 0 are called rational numbers.

→ The set of rational numbers is represented by Q.

→ Q includes all integers, positive fractional numbers and negative fractional numbers.

→ All rational numbers can be represented on a number line.

Students can go through AP Board 7th Class Maths Notes Chapter 1 Integers to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 1 Integers

→ Number System:
Natural Numbers:
a) Counting numbers 1, 2, 3, 4, 5, 6, …… are called natural numbers.
b) The set of all natural numbers can be represented by N = {1, 2, 3, 4, 5, ……}

→ Whole Numbers:
a) If we include ‘O’ among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5, …… are called whole numbers.
b) The set of whole numbers can be represented by W = {0, 1, 2, 3, ……}
c) Clearly, every natural number is a whole number but ‘O’ is a whole number which is not a natural number.

→ Integers:
a) All counting numbers and their negatives including zero are known as integers.
b) The set of integers can be represented by Z or I = {……, -4, -3, -2,-1, 0, 1, 2, 3, 4, ……}

• Positive Integers:
  The set I⁺ = {1, 2, 3, 4, ……} is the set of all positive integers. Clearly positive integers and natural numbers are same.
• Negative Integers:
  The set I^– = {-1, -2, -3, ……} is the set of all negative integers. ‘0’ is neither positive nor negative.
• Non-Negative Integers:
  The set {0, 1, 2, 3, ……} is the set of all non-negative integers.

→ Properties of integers:
For any three integers a, b, c
i) a + b is also an integer – closure property w.r.t addition.
ii) a – b is also an integer – closure property w.r.t subtraction.
iii) a . b is also an integer – closure property w.r.t multiplication.
iv) a + b = b + a – commutative law w.r.t addition. ‘
v) a . b = b . a – commutative law w.r.t multiplication.
vi) a + (b + c) = (a + b) + c – associative law w.r.t addition.
a . (b . c) = (a . b). c – associative law w.r.t multiplication.
vii) a + 0 = 0 + a = a – identity w.r.t addition.
viii) a . 1 = 1 . a = a – identity w.r.t multiplication.
ix) a.(b + c) = a.b + a.c – distributive property.
x) a ÷ 0 is not defined
a ÷ 1 = a
0 ÷ a = 0 (a ≠ 0)

→ On a number line when you add a positive integer you move right side on the number line; and if a negative integer is added you move to the left side on the number line.

→ On the number line if you subtract a positive integer you move to the left side and if you subtract a negative integer you move to the right side.

→ Product of any two positive integers or any two negative integers is always a positive integer.

→ Product of a positive integer and a negative integer is always a negative integer (i.e.,) two integers with opposite signs always give a negative product.

→ Product of even number of negative integers is always a positive integer.

→ Product of odd number of negative integers is always a negative integer.

Students can go through Andhra Pradesh SCERT AP State Board Syllabus 7th Class Maths Notes Pdf in English Medium and Telugu Medium to understand and remember the concepts easily. Besides, with our AP State 7th Class Maths Notes students can have a complete revision of the subject effectively while focusing on the important chapters and topics. Students can also read AP Board 7th Class Maths Solutions for exam preparation.

AP State Board Syllabus 7th Class Maths Notes

• Chapter 1 Integers Notes
• Chapter 2 Fractions, Decimals and Rational numbers Notes
• Chapter 3 Simple Equations Notes
• Chapter 4 Lines and Angles Notes
• Chapter 5 Triangle and Its Properties Notes
• Chapter 6 Ratio – Applications Notes
• Chapter 7 Data Handling Notes
• Chapter 8 Congruency of Triangles Notes
• Chapter 9 Construction of Triangles Notes
• Chapter 10 Algebraic Expressions Notes
• Chapter 11 Exponents Notes
• Chapter 12 Quadrilaterals Notes
• Chapter 13 Area and Perimeter Notes
• Chapter 14 Understanding 3D and 2D Shapes Notes
• Chapter 15 Symmetry Notes

These AP State Board Syllabus 7th Class Maths Notes provide an extra edge and help students to boost their self-confidence before appearing for their final examinations.

Students can go through AP Board 9th Class Maths Notes Chapter 15 Proofs in Mathematics to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 15 Proofs in Mathematics

→ The sentences that can be judged on some criteria, no matter by what process for their being true or false are statements.

→ Mathematical statements are of a distinct nature from general statements. They cannot be proved or justified by getting evidence while they can be disproved by finding a counter example.

→ Making mathematical statements through observing patterns and thinking of the rules that may define such patterns.
A hypothesis is a statement of idea which gives an explanation to a sense of observation.

→ A process which can establish the truth of a mathematical statement based purely on logical arguments is called a mathematical proof.

→ Axioms are statements which are assumed to be true without proof.

→ A conjecture is a statement we believe is true based on our mathematical intuition, but which we are yet to prove.

→ A mathematical statement whose truth has been established or proved is called a theorem.

→ The prime logical method in proving a mathematical statement is deductive reasoning.

→ A proof is made up of a successive sequence of mathematical statements.

→ Beginning with given (Hypothesis) of the theorem and arrive at the conclusion by means of a chain of logical steps is mostly followed to prove theorems.

→ The proof in which, we start with the assumption contrary to the conclusion and arriving at a contradiction to the hypothesis is another way that we establish the original conclusion is true is another type of deductive reasoning.

→ The logical tool used in the establishment of the truth of an un-ambiguous statement is called deductive reasoning.

→ The reasoning which is based on examining of variety of cases or sets of data discovering pattern and forming conclusion is called Inductive reasoning.

Students can go through AP Board 9th Class Maths Notes Chapter 14 Probability to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 14 Probability

→ If in an experiment all the possible outcomes are known in advance and none of the outcomes can be predicted with certainty, then such an experiment is called a random experiment.
Eg: Tossing a coin; throwing a die, drawing a card from deck of cards.

→ The possible outcomes of a trial are called events.

→ Events are said to be equally likely if there is no reason to expect any one in preference to other. Thus equally likely events mean outcome is as likely to occur as any other outcome.

→ To measure the chance of its happening numerically we classify them as follows.

→ Certain: Something that must happen

→ More likely: Something that would occur with great chance

→ Equally likely: Something that have the same chance of occurring

→ Less likely: Something that would occur with less chance

→ Impossible: Something that cannot happen

→ Probability of an event = \(\frac{\text { Number of favourable outcomes for the event }}{\text { Number of total possible outcomes }}\)

Students can go through AP Board 9th Class Maths Notes Chapter 12 Circles to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 12 Circles

→ A collection of all points in a plane which are at a fixed distance from a fixed point in the sapie plane is called a circle. The fixed point is called the centre and the fixed distance is called the radius of the circle.

→ A line segment joining any two points on the circle is called a chord.

→ The longest of all chords which passes through the centre is called a diameter.

→ Circles with same radii are called congruent circles.

→ Circles with same centre and different radii are called concentric circles.

→ Diameter of a circle divides it into two semi-circles.

→ The part between any two points on the circle is called an arc.

→ The area enclosed by a chord and an arc is called a segment. If the arc is a minor arc then it is called the minor segment and if the arc is major arc then it is called the major segment.

→ The area enclosed by an arc and the two radii joining the end points of the arc with centre is called a sector.

→ Equal chords of a circle subtend equal angles at the centre.

→ Angles in the same segment are equal.

→ An angle in a semi circle is a right angle.

→ If the angles subtended by two chords at the centre are equal, then the chords are congruent.

→ The perpendicular from the centre of a circle to a chord bisects the chords. The converse is also true.

→ There is exactly one circle that passes through three non-collinear points.

→ The circle passing through the three vertices of a triangle is called a circumcircle.

→ Equal chords are at equal distance from the centre of the circle, conversely chords at equidistant from the centre of the circle are equal in length.

→ Angle subtended by an arc at the centre of the circle is twice the angle subtended by it at any other point on the circle.

→ If the angle subtended by an arc at a point on the remaining part of the circle is 90°, then the arc is a semi circle.

→ If a line segment joining two points subtends same angles at two other points lying on the same side of the line segment, the four points lie on the circle.

→ The sum of pairs of opposite angles of a cyclic quadrilateral are supplementary.

Students can go through AP Board 9th Class Maths Notes Chapter 11 Areas to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 11 Areas

→ The part of a plane enclosed by a simple closed figure is called a planar region corresponding to that figure.

→ The magnitude of a planar region is its area.

→ The unit area is the area enclosed by a unit square i.e. a square of side 1 unit.

→ Area is always expressed in square units.

→ The areas of two congruent figures are equal.

→ Converse of the above is not true, i.e., if two figures have same area, they need not be congruent.

→ The area of a whole figure is equal to sum of the areas of finite parts of it.

→ Area of a rectangle is equal to the product of its length and breadth.

→ Area of a parallelogram is the product of a side and its corresponding altitude.

→ Parallelograms on the same base and between same parallels are equal in area.

→ If a parallelogram and a triangle lie on a same base and between same parallels, then the area of the triangle is half the area of parallelogram.

→ Triangles between same base and between same parallels are equal in area.

Students can go through AP Board 9th Class Maths Notes Chapter 10 Surface Areas and Volumes to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 10 Surface Areas and Volumes

→ Cuboid and cube may be treated as regular prisms having six faces.

→ Solids whose lateral surfaces are parallelograms are called prisms,

→ Solids whole lateral surfaces are triangles are called pyramids.

→ Cube and cuboid are also prisms.

→ Volume of a pyramid = \(\frac{1}{3}\) Area of the base × height
= \(\frac{1}{3}\) of the volume of right prism.

Students can go through AP Board 9th Class Maths Notes Chapter 9 Statistics to understand and remember the concepts easily.

AP State Board Syllabus 9th Class Maths Notes Chapter 9 Statistics

→ The facts or figures which are numerical or otherwise collected with a definite purpose are called data.

→ Statistics is the branch of mathematics which studies about the data and its meaning.

→ The information collected by the investigator with a definite objective is called primary data.

→ The information collected from a source, which had already been recorded lay from registers is called secondary data.

→ Data exists in raw form.

→ Data is classified into groups to make the study easy.

→ The difference between the minimum and maximum values of a data is called the range of the given data.

→ The table showing the actual observations with their frequencies is called ungrouped frequency distribution table or table of weighted observations.

→ Presenting the data in groups with their frequencies is called a grouped frequency distribution table.

→ Mean, Median and Mode are called the measures of central tendency.

(Deviation method: ∑f_id_i – sum of the deviations, A – assumed mean and ∑f_i – sum of the frequencies)
→ Median is the middle observation of a data, when arranged in either ascending/ descending order.

→ When number of observations ‘n’ is odd, the median is \(\left(\frac{n+1}{2}\right)^{th}\) observation.

→ When number of observations ‘n’ is even, the median is \(\left(\frac{n}{2}\right)^{th}\) the average of \(\left(\frac{n}{2}+1\right)^{th}\) observations.

→ Median divides the data into two groups of equal number, one part comprising all values greater and the other comprising all values less than median.

→ Mode is the value of the observation which occurs most frequently, i.e., an observation with the maximum frequency is called mode.

→ If each of the observation is added or multiplied by same quantity, the measure of central tendency also changes accordingly.