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Tag: Class 7
Simple Equations | Study
Lines and Angles | Study
CBSE 7 | Mathematics – Study – Free
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Class 7 | Mathematics | All In One
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English Version Simple Equations | Speed Notes
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An equation isa condition ona variable suchthat two expressions in the variable should have equalvalue.
Thevalue of thevariable for whichthe equation issatisfied is called the solution ofthe equation.
An equation remains the same if the LHSand the RHSare interchanged. (Scroll down to continue …)
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In case ofthe balanced equation, if we add the same number to both thesides, or subtract the same number from both the sides,
or
multiply both sidesby the same number, or divide both sidesby the samenumber, the balance remains un disturbed,
i.e.,the value of the LHS remains equal to the value of the RHS The above property gives a systematic method of solving an equation.
We carry out a series of identical mathematical operations on the two sides of the equation in such a waythat on oneof the sides we get justthe variable. Thelast step isthe solution of the equation.
Transposing means moving to the other side.
Transposition of a number has the same effect as adding same number to (or subtracting the same number from) both sides of the equation.
Whenyou transpose a number fromone side ofthe equation tothe other side, you change itssign.
For example, transposing +3 fromthe LHS tothe RHS in equation x + 3 = 8 gives x = 8 – 3 (= 5).
We can carry out the transposition of an expression in thesame way as the transposition of a number.
We havelearnt how to construct simple algebraic expressions corresponding to practical situations.
Wealso learnt how,using the technique of doing thesame mathematical operation (for example adding the samenumber) on bothsides, we could build an equation starting fromits solution.
Further, we also learnt that we could relate a given equation tosome appropriate problem/puzzlefrom the equation. practical situation and build a practical word.
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Lines and Angles | Study
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We recall that
(i) A line-segment has two end points.
(ii) A ray has only one end point (its vertex);and
(III) A line has no end points on either side.
An angle is formed when two lines (or rays or line-segments) meet.
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Important Formulas | Lines and Angles
When two lines l and m meet, we say they intersect; the meeting point is calledthe point of intersection.
When lines drawnon a sheet of paper do not meet, howeverfar produced, we call them to be parallel lines.
Point: A point name a location.
Line: A line is perfectlystraight and extends forever in both direction.
Line segment: A line segmentis the part of a line betweentwo points.
Ray: A ray is part of a line that starts at one point and extendsforever in one direction.
Intersecting lines: Two or more lines that have one and only one point in common.
The common point where all the intersecting lines meet is called the point of
intersection.
Transversal: A line intersects two or more lines that lie in the same plane in distinct points.
Parallel lines: Two lineson a plane that nevermeet. They distance apart.
Complementary Angles: Two angles whose measures add to 90^O Supplementary Angles: Two angles whose measures add to 180 ^o
Adjacent Angles: Two angles have a common vertex and a
common interiorpoints.
Linear pairs: A pair of adjacentangles whose non-common sides are oppositerays. Vertically Opposite Angles: Two angles formed by two intersecting lines have common arm.
Angles made by Transversal: When two lines are intersecting by a transversal, eight anglesare formed.
Transversal of Parallel Lines: If two parallel lines are intersected by a transversal, each pair of:
Corresponding angles are congruent. Alternateinterior angles are congruent. Alternateexterior angles are congruent.
If the transversal is perpendicular to the parallellines, all of the angles formed are congruent to 90 o angles.
1. A linewhich intersects two or more given lines at distinct points is called a transversal to the given lines.
2. Lines in a plane areparallel if theydo not intersect when produced indefinitely in either direction.
3. The distance between two intersecting lines is zero.
4. The distance between two parallel lines is thesame everywhere andis equal tothe perpendicular distance between them.
5. If two parallel lines are intersected by a transversal then (i) pairs ofalternate (interior orexterior) angles are equal. (ii) pairs of corresponding angles are equal. (iii) interior angles onthe same sideof the transversal are supplementary. 6. If twonon-parallel lines areintersected by transversal then none of (i), (ii) and (iii) hold true in 5. 7. If twolines are intersected by a transversal, thenthey are parallel ifany one of thefollowing is true: (i) The angles of a pair of corresponding angles are equal. (ii) The angles of a pairof alternate interior angles are equal. (iii) The angles of a pairof interior angles on the sameside of the transversal are supplementary.
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Geometry Axioms & Theorems
Class 9
LINES AND ANGLES
Axiom 1: If a rays stands on a line , then the sum of two adjacent angles so formed is 180 0
Axiom 6.2 : If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
Theorem 6.1 : If two lines intersect each other, then the vertically opposite angles are equal.
Axiom 6.3 : If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
Axiom 6.4 : If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.
Theorem 6.2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal
Theorem 6.3 : If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
Theorem 6.4 : If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
Theorem 6.5 : If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
Theorem 6.6 : Lines which are parallel to the same line are parallel to each other.
Theorem 6.7 : The sum of the angles of a triangle is 180º
Theorem 6.8 : If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
TRIANGLES:
Axiom 7.1 (SAS congruence rule) : 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
Theorem 7.1 (ASA congruence rule) : Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle
Theorem 7.2 : Angles opposite to equal sides of an isosceles triangle are equal.
Theorem 7.3 : The sides opposite to equal angles of a triangle are equal
Theorem 7.4 (SSS congruence rule) : If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
Theorem 7.5 (RHS congruence rule) : If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.
Theorem 7.6 : If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater).
Theorem 7.7 : In any triangle, the side opposite to the larger (greater) angle is longer.
Theorem 7.8 : The sum of any two sides of a triangle is greater than the third side
QUADRILATERALS
Theorem 8.1 : A diagonal of a parallelogram divides it into two congruent triangles.
Theorem 8.2 : In a parallelogram, opposite sides are equal.
Theorem 8.3 : If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.
Theorem 8.4 : In a parallelogram, opposite angles are equal.
Theorem 8.5 : If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.
Theorem 8.6 : The diagonals of a parallelogram bisect each other
Theorem 8.7 : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram
Theorem 8.8 : A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.
Theorem 8.9 : The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
Theorem 8.10 : The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.
AREAS OF PARALLELOGRAMS AND TRIANGLES
Theorem 9.1 : Parallelograms on the same base and between the same parallels are equal in area.
Theorem 9.2 : Two triangles on the same base (or equal bases) and between the same parallels are
equal in area
Theorem 9.3 : Two triangles having the same base (or equal bases) and equal areas lie between the same parallels
CIRCLES
Theorem 10.1 : Equal chords of a circle subtend equal angles at the centre.
Theorem 10.2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.
Theorem 10.3 : The perpendicular from the centre of a circle to a chord bisects the chord.
Theorem 10.4 : The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Theorem 10.5 : There is one and only one circle passing through three given non-collinear points.
Theorem 10.6 : Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
Theorem 10.7 : Chords equidistant from the centre of a circle are equal in length.
Theorem 10.8 : The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Theorem 10.9 : Angles in the same segment of a circle are equal
Theorem 10.10 : If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).
Theorem 10.11 : The sum of either pair of opposite angles of a cyclic quadrilateral is 180º.
Theorem 10.12 : If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.
SURFACE AREAS AND VOLUMES
Surface Area of a Cuboid = 2(lb + bh + hl) where l, b and h are respectively the three edges of the cuboid
Surface Area of a Cube = 6a2
Curved Surface Area of a Cylinder = 2πrh
Total Surface Area of a Cylinder = 2πr(r + h)
Curved Surface Area of a Cone
= 1/2 × l × 2πr = πrl
L2= r2 + h2
Total Surface Area of a Cone
= πrl + πr22 = πr(l + r)
Surface Area of a Sphere = 4 π r2
Curved Surface Area of a Hemisphere = 2πr2
Total Surface Area of a Hemisphere = 3πr2
Volume of a Cuboid = base area × height = length × breadth × height
Volume of a Cube = edge × edge × edge = a3
Volume of a Cylinder = πr22
Volume of a Cone = 1/3 πr2h
Volume of a Sphere = 4/3 3 πr3
Volume of a Hemisphere = 2/3πr3
ALGEBRA :
am × an = am+n
(am)×(an) = am+n (am)/(an) = am-n (am)n = amn (am)×(bm) = (ab)m (a0)= am-m = am/am = 1 (am)×(bn) = (ab)m+n am/bm = (a/b)m
Class 8
UNDERSTANDING QUADRILATERALS
Sum of the measures of the external angles of any polygon is 360°.
The sum of the measures of the three angles of a triangle is 180°.
A parallelogram is a quadrilateral whose opposite sides are parallel
Property: The opposite sides of a parallelogram are of equal length
Property: The opposite angles of a parallelogram are of equal measure.
Property: The adjacent angles in a parallelogram are supplementary
Property: The diagonals of a parallelogram bisect each other (at the point of their intersection, of course!)
Property: The diagonals of a rhombus are perpendicular bisectors of one another
Property: The diagonals of a rectangle are of equal length.
Property: The diagonals of a square are perpendicular bisectors of each other
MENSURATION
1. Area of (i) a trapezium = half of the sum of
the lengths of parallel sides × perpendicular distance between them.
(ii) a rhombus = half the product of its diagonals.
2. Surface area of a solid is the sum of the areas of its faces.
3. Surface area of a cuboid = 2(lb + bh + hl) a cube = 6l 2 a cylinder = 2πr(r + h)
4. Amount of region occupied by a solid is called its volume.
5. Volume of a cuboid = l × b × h a cube = l3 a cylinder = πr 2h 6.
(i) 1 cm3 = 1 mL
(ii) 1L = 1000 cm3
(iii) 1 m3 = 1000000 cm3 = 1000L
EXPONENTS AND POWERS
am × an = am+n
am / an = am-n (am)n = amn (am)×(bn) = (ab)m+n (a0)= am / am = 1 am/am = (a/b)m
Class 7
LINES AND ANGLES
sum of the measures of two angles is 90°, the angles are called complementary angles.
the sum of the measures of two angles is 180°, the angles are called supplementary angles.
These angles are such that:
(i) they have a common vertex;
(ii) they have a common arm;
(iii) the non-common arms are on either side of the common arm.
Such pairs of angles are called adjacent angles. Adjacent angles have a common vertex and a common arm but no common interior points.
A linear pair is a pair of adjacent angles whose non-common sides are opposite rays.
TRIANGLES
An exterior angle of a triangle is equal to the sum of its interior opposite angles.
Statement The total measure of the three angles of a triangle is 1800
A triangle in which all the three sides are of equal lengths is called an equilateral triangle.
A triangle in which two sides are of equal lengths is called an isosceles triangle.
1.The six elements of a triangle are its three angles and the three sides.
2.The line segment joining a vertex of a triangle to the mid point of its opposite side is called a median of the triangle. A triangle has 3 medians.
3.The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes.
4.An exterior angle of a triangle is formed, when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle.
5.A property of exterior angles: The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.
6.The angle sum property of a triangle: The total measure of the three angles of a triangle is 180°.
7. A triangle is said to be equilateral, if each one of its sides has the same length. In an equilateral triangle, each angle has measure 60°
8. A triangle is said to be isosceles, if atleast any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure.
9. Property of the lengths of sides of a triangle: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference between the lengths of any two sides is smaller than the length of the third side.
CONGRUENCE OF TRIANGLES
If two line segments have the same (i.e., equal) length, they are congruent. Also, if two line segments are congruent, they have the same length.
If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are same.
SSS Congruence Criterion:
If under a given correspondence, the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.
SAS Congruence Criterion:
If under a correspondence, two sides and the angle included between them of a triangle are equal to two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.
ASA Congruence Criterion:
If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.
RHS Congruence Criterion:
If under a correspondence, the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.
1 cm2 = 100 mm2
1 m2 = 10000 cm2
1 hectare = 10000 m2
