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  • Diversity in the Living World | Study

    Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Content … Key Terms Topic Terminology Term Important Tables Table: . Assessments Test Your Learning readmore

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  • Life Processes | Study

    Mind Map Overal Idea Content Speed Notes Quick Coverage Biology is the study of living things. All living things are called organisms, both plants and animals are living organisms. But how we decide whether something is living or non-living depends on 7 lifeprocesses. If something is living it will carryout the 7 life processes below. readmore

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    Biology is the study of living things. All living things are called organisms, both plants and animals are living organisms. But how we decide whether something is living or non-living depends on 7 lifeprocesses. If something is living it will carryout the 7 life processes below. (Scroll down till end ..)

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  • Mindful Eating: A Path to a Healthy Bodys | Study

    Mind Map Overal Idea Content Speed Notes Quick Coverage Content : (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content Content … Key Terms Topic Terminology Term Important Tables Table: . Assessments Test Your Learning readmore

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  • Number Play | Assess

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  • Number Play | Study

    Mind Map Overal Idea Content Speed Notes Quick Coverage We have discussed multiples, divisors, factors and have seenhow to identify factors and multiples. We have discussed and discovered thefollowing: (a) A factor of a number is an exactdivisor of thatnumber. (Scroll down till end of the page) Study Tools Audio, Visual & Digital Content (b) readmore

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    We have discussed multiples, divisors, factors and have seenhow to identify factors and multiples. We have discussed and discovered thefollowing: (a) A factor of a number is an exactdivisor of thatnumber. (Scroll down till end of the page)

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    (b) Every number is a factor of itself. 1 is a factor ofevery number.

    (c) Every factor ofa number isless than or equal tothe given number.

    (d) Every number isa multiple ofeach of itsfactors.

    (e) Every multiple ofa given number is greater thanor equal tothat number.

    (f) Every number is a multiple of itself.

    We have learnt that – (a) The number otherthan 1, withonly factors namely 1 and thenumber itself, isa prime number. Numbers that have more than two factors are called composite numbers. Number 1is neither prime nor composite.

    (b) The number 2is the smallest prime number andis even. Every prime number other than 2 isodd.

    (c) Two numbers withonly 1 asa common factor are called co-prime numbers.

    (d) If a number is divisible byanother number thenit is divisible by each of the factors of that number.

    (e) A number divisible by two co-prime numbers is divisible by their product also.

    We have discussed how we can find just by looking at a number, whether it is divisible by small numbers 2,3,4,5,8,9 and 11.

    We have explored the relationship between digits of thenumbers and theirdivisibility by different numbers.

    (a) Divisibility by 2,5and 10 canbe seen byjust the lastdigit.

    (b) Divisibility by 3and 9 ischecked by finding the sum ofall digits.

    (c) Divisibility by 4 and 8is checked bythe last 2and 3 digits respectively.

    (d) Divisibility of11 is checked by comparing thesum of digits at odd andeven places.

    We have discovered that if twonumbers are divisible by a number then their sum and difference are also divisible by that number.

    We have learnt that – (a) The Highest Common Factor (HCF) of two ormore given numbers is the highest of their common factors.

    (b) The Lowest Common Multiple (LCM) of two ormore given numbers is the lowest of their common multiples.

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  • QUADRILATERALS | Study

    Mind Map Overal Idea Content Speed Notes Quick Coverage Quadrilateral Any closed polygon with four sides, four angles and four vertices are called Quadrilateral. It could be regular or irregular. (Sroll down to continute till the end …) Study Tools Audio, Visual & Digital Content Quadrilateral Quadrilateral is a closed figure with four sides. Characteristics readmore

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    Quadrilateral

    Any closed polygon with four sides, four angles and four vertices are called Quadrilateral. It could be regular or irregular. (Sroll down to continute till the end …)

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    Quadrilateral

    Quadrilateral

    Quadrilateral is a closed figure with four sides.

    QUADRILATERAL

    Characteristics of a quadrilateral

    Angle Sum Property of a Quadrilateral:  

    Qudrilateral is a four sided closed figure.

    Sum of all angles of a quadrilateral is 360°.

    Types Of Quadrilaterals

    Classification of quadrilaterals
    Classification of quadrilaterals

    Quadrilaterals are broadly classified into three categories as:

    (i) Kite

    (ii) Trapezium

    (ii) Parallelogram

    Kite:

    Kite

    (i) Kite has no parallel sides

    (ii) Kite has a pair of equal adjacent sides.

    (ii) It is not a parallelogram

    Characteristics Of Kite:

    Perimeter Of Square

    Area Of Kite

    Trapezium:

    quadrilateral ABCD

    Trapezium is a quadrilateral with the following characteristics:

    (i) One pair of opposite sides is parallel to each other.

    (ii) The other pair of opposite sides may not be parallel to each other.

    Characteristics Of Trapezium

    (i) Sum of all angles of a quadrilateral is 360°.

    (ii) One pair of opposite sides is parallel to each other.

    (iii) The other pair of opposite sides need not be parallel to each other.

    Types Of Trapezium:

    Quadrilaterals are broadly classified into two categories as:

    (i) Isosceles Trapezium.

    (ii) Scalene Trapezium.

    (i) Right Trapezium.

    Isosceles Trapezium:

    Isosceles Trapezium is a quadrilateral with the following characteristics:

    (i) One pair of opposite sides is parallel to each other.

    (ii) The other pair of opposite sides are equal.

    (iii) The other pair of opposite sides need not be parallel to each other.

    Isosceles Trapezium is a trapezium with the following characteristics:

    (i) One pair of opposite sides is parallel to each other.

    (ii) The other pair of opposite sides are equal.

    (iii) The other pair of opposite sides need not be parallel to each other.

    Characteristics Of Isosceles Trapezium

    (i) Sum of all angles of a quadrilateral is 360°.

    (ii) One pair of opposite sides is parallel to each other.

    (iii) The other pair of opposite sides are equal.

    (iv) The other pair of opposite sides need not be parallel to each other.

    Scalene Trapezium:

    • Scalene trapezium: Classified by the length of the legs or the measurement of their angles.

    Characteristics Of Scalene Trapezium

    Right Trapezium:

    • Right trapezium: Has one pair of parallel sides and one pair of right angles.

    Characteristics Of Right Trapezium

    Perimeter Of Trapezium

    Area Of Trapezium

    Parallelogram:

    Parallelogram is a quadrilateral with the following characteristics:

    (i)  Two pairs of opposite sides are parallel to each other.

    (ii) Two pairs of opposite sides are equal in length.

    Characteristics of a parallelogram

    (i) Sum of all angles of a Parallelogram is 360°.

    (ii)  Two pairs of opposite sides are parallel to each other.

    (ii) Two pairs of opposite sides are equal in length.

    (ii) Two pairs of opposite angles are equal.

    (iii) Diagonals bisect each other.

    (iv) Diagonals need not be equal to each other.

    (v) Diagonals divide it into two congruent triangles.

    Types Of Parallelogram

    Parallelograms are broadly classified into three categories as:

    (i) Rectangle

    (ii) Rhombus

    (iii) Square

    Perimeter Of Parallelogram

    Area Of Parallelogram

    Rectangle:

    Rectangle is a quadrilateral with the following characteristics:

    (i) Two pairs of opposite sides are parallel to each other.

    (ii) Two pairs of opposite sides are equal in length.

    (iii) All four angles are right angles. (each angle is 90 o).

    Characteristics Of Rectangle 

    (i) Sum of all angles of a quadrilateral is 360°.

    (ii)  Two pairs of opposite sides are parallel to each other.

    (ii) Two pairs of opposite sides are equal in length.

    (iii) All four angles are right angles. (each angle is 90 o).

    (iii) Diagonals bisect each other.

    (iv) Diagonals are equal to each other.

    (v) Diagonals of a rectangle divide it into two congruent triangles.

    Conclusions:

    1. Every Rectangle is a Parallelogram. But Every Parallelogram need not to be a Rectangle.

    Condition for a rhombus to be a square:

    If all four angles of a parallelogram are right angles. (each angle is 90 o), the parallelogram becomes a Rectangle.

    Perimeter Of Rectangle

    Area Of Recatangle 

    Rhombus:

    Rhombus is a quadrilateral with the following characteristics:

    (i)  Two pairs of opposite sides are parallel to each other.

    (ii) All four sides are equal in length.

    Characteristics Of Rhombus

    (i) Sum of all angles of a quadrilateral is 360°.

    (ii)  Two pairs of opposite sides are parallel to each other.

    (ii) All four sides are equal in length.

    (ii) Two pairs of opposite angles are equal.

    (iii) Diagonals bisect each other.

    (iv) Diagonals need not be equal to each other.

    (v) Diagonals divide a Rhombus into two congruent triangles.

    Conclusions:

    1. Every Rhombus is a Parallelogram. But Every Parallelogram need not to be a Rhombus.

    Condition for a rhombus to be a square:

    If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.

    Perimeter Of Rhombus

    Area Of Rhombus 

    Square:

    Square is a quadrilateral with the following characteristics:

    (i)  Two pairs of opposite sides are parallel to each other.

    (ii) All four sides are equal in length.

    (iii) All four angles are right angles. (each angle is 90 o).

    Characteristics Of Square

    (i) Sum of all angles of a quadrilateral is 360°.

    (ii)  Two pairs of opposite sides are parallel to each other.

    (iii) All four sides are equal in length.

    (iv) All four angles are right angles. (each angle is 90 o).

    (v) Diagonals bisect each other.

    (vi) Diagonals need not be equal to each other.

    (vii) Diagonals divide a Rhombus into two congruent triangles.

    Conclusions:

    1. Every square is a Rhombus. But Every Rhombus need not to be a square.

    Condition for a rhombus to be a square:

    If all the angles of a rhombus are right angles (euqal to 90o), the rhombus becomes a square.

    2. Every Square is a prallelogram. But Every prallelogram need not to be a square.

    Condition for a prallelogram to be a square:

    (i) If all the angles of a parallelogram are right angles (euqal to 90o), and all the sides of a parallelogram are equal in length, the parallelogram becomes a square.

    3. Every Square is a rectangle. But Every Rectangle need not to be a square.

    Condition for a Rectangle to be a square:

    If all the sides of a Rectangle are equal in length, the Rectangle becomes a square.

    If all the sides of a parallelogram are equal, the parallelogram becomes a Rhombus.

    Perimeter Of Square

    Area Of Square

    Important Points To Remember

    IMPORTANT POINTS TO REMEMBER
    • The diagonals of a parallelogram are equal if and only if it is a rectangle.
    • If a diagonal of a parallelogram bisects one of the angles of the parallelogram then it also bisects the opposite angle.
    • In a parallelogram, the bisectors of any two consecutive angles intersect at a right angle.
    • The angle bisectors of a parallelogram form a rectangle.

    Mid Point Theorem

    A line segment joining the mid points of any two sides of a triangle is parallel to the third side and length of the line segment is half of the parallel side.

    Converse Of Mid Point Theorem

    A line through the midpoint of a side of a triangle parallel to another side bisects the third side.

    Intercept Theorem

    If there are three parallel lines and the intercepts made by them on one transversal are equal then the intercepts on any other transversal are also equal.

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    Angle Sum Property of a Quadrilateral

    The sum of the four angles of a quadrilateral is 360°

    Angle Sum Property of a Quadrilateral

    If we draw a diagonal in the quadrilateral, it divides it into two triangles.  

    And we know the angle sum property of a triangle i.e. the sum of all the three angles of a triangle is 180°.

    The sum of angles of ∆ADC = 180°.

    The sum of angles of ∆ABC = 180°.

    By adding both we get ∠A + ∠B + ∠C + ∠D = 360°

    Hence, the sum of the four angles of a quadrilateral is 360°.

    Example

    Find ∠A and ∠D, if BC∥ AD and ∠B = 52° and ∠C = 60° in the quadrilateral ABCD.

    quadrilateral ABCD

    Solution:

    Given BC ∥ AD, so ∠A and ∠B are consecutive interior angles.

    So ∠A + ∠B = 180° (Sum of consecutive interior angles is 180°).

    ∠B = 52°

    ∠A = 180°- 52° = 128°

    ∠A + ∠B + ∠C + ∠D = 360° (Sum of the four angles of a quadrilateral is 360°).

    ∠C = 60°

    128° + 52° + 60° + ∠D = 360°

    ∠D = 120°

    ∴ ∠A = 128° and ∠D = 120 °.

    Types of Quadrilaterals

    S No. QuadrilateralPropertyImage
    1.
    Kitea. No Parallel Sides
    b. Two pairs of adjacent sides are equal.
    Kite
    2.TrapeziumOne pair of opposite sides is parallel.Trapezium
    3.ParallelogramBoth pairs of opposite sides are parallel.Parallelogram
    3.Rectanglea. Both the pair of opposite sides are parallel.
    b. Opposite sides are equal.c.
    All the four angles are 90°.
    Rectangle
    4.Squarea. All four sides are equal.
    b. Opposite sides are parallel.
    c. All the four angles are 90°.
    Square
    5.Rhombusa. All four sides are equal.
    b. Opposite sides are parallel.
    c. Opposite angles are equal.d.
    Diagonals intersect each other at the centre and at 90°.
    Rhombus

    Remark: A square, Rectangle and Rhombus are also a parallelogram.

    Properties of a Parallelogram

    Parallelogram

    Theorem 1: When we divide a parallelogram into two parts diagonally then it divides it into two congruent triangles.

    ∆ABD ≅ ∆CDB

     In a parallelogram, opposite sides will always be equal

    Theorem 2: In a parallelogram, opposite sides will always be equal.

    Theorem 3: A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.

    A quadrilateral will be a parallelogram if each pair of its opposite sides will be equal.

    Here, AD = BC and AB = DC

    Then ABCD is a parallelogram.

    Theorem 4: In a parallelogram, opposite angles are equal.

     In a parallelogram, opposite angles are equal.

    In ABCD, ∠A = ∠C and ∠B = ∠D

    Theorem 5: In a quadrilateral, if each pair of opposite angles is equal, then it is said to be a parallelogram. This is the reverse of Theorem 4.

    Theorem 6: The diagonals of a parallelogram bisect each other.

    The diagonals of a parallelogram bisect each other.

    Here, AC and BD are the diagonals of the parallelogram ABCD.

    So the bisect each other at the centre.

    DE = EB and AE = EC

    Theorem 7: When the diagonals of the given quadrilateral bisect each other, then it is a parallelogram.

    This is the reverse of the theorem 6.

    The Mid-point Theorem

    1. If a line segment joins the midpoints of the two sides of the triangle then it will be parallel to the third side of the triangle.

    Triangle

    If AB = BC and CD = DE then BD ∥ AE.

    2. If a line starts from the midpoint of one line and that line is parallel to the third line then it will intersect the midpoint of the third line. 

    Triangle

    If D is the midpoint of AB and DE∥ BC then E is the midpoint of AC.

    Example

    Prove that C is the midpoint of BF if ABFE is a trapezium and AB ∥ EF.D is the midpoint of AE and EF∥ DC.

    Trapezium

    Solution:

    Let BE cut DC at a point G.

    Now in ∆AEB, D is the midpoint of AE and DG ∥ AB.

    By midpoint theorem, G is the midpoint of EB.

    Again in ∆BEF, G is the midpoint of BE and GC∥ EF.

    So, by midpoint theorem C is the midpoint of BF.

    Hence proved.

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