Topics
Integers
- Natural Numbers
- Whole Numbers
- Negative and Positive Numbers
- Integers
- Representation of Integers on the Number Line
- Ordering of Integers
- Addition of Integers
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers
Fractions and Decimals
- Concept of Fraction
- Types of Fractions
- Concept of Proper and Improper Fractions
- Concept of Mixed Fractions
- Concept of Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Using Operator 'Of' with Multiplication and Division
- Multiplication of Fraction
- Division of Fractions
- Concept of Reciprocals or Multiplicative Inverses
- Problems Based on Fraction
- The Decimal Number System
- Comparing Decimal Numbers
- Addition of Decimal Fraction
- Subtraction of Decimal Numbers
- Multiplication of Decimal Numbers
- Division of Decimal Numbers
- Problems Based on Decimal Numbers
Data Handling
Simple Equations
Lines and Angles
The Triangle and Its Properties
- Basic Concepts of Triangles
- Classification of Triangles based on Sides
- Classification of Triangles based on Angles
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Basic Properties of a Triangle
- Right-angled Triangles and Pythagoras Property
Comparing Quantities
- Ratio
- Concept of Equivalent Ratios
- Proportion
- Unitary Method
- Basic Concept of Percentage
- Estimation in Percentages
- Interpreting Percentages
- Conversion between Percentage and Fraction or Decimal
- Ratios to Percents
- Increase Or Decrease as Percent
- Basic Concepts of Profit and Loss
- Profit or Loss as a Percentage
- Calculation of Interest
Congruence of Triangles
- Similarity and Congruency of Figures
- Congruence Among Line Segments
- Congruence of Angles
- Congruence of Triangles
- Criteria for Congruence of Triangles
- Criteria for Similarity of Triangles
- SAS Congruence Criterion
- ASA Congruence Criterion
- RHS Congruence Criterion
- Exceptional Criteria for Congruence of Triangles
Rational Numbers
- Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers
Perimeter and Area
- Basic Concepts in Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangle
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Parallelogram
- Area of a Triangle
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter
- Problems based on Area
Practical Geometry
- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)
Algebraic Expressions
Exponents and Powers
- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Crores
Symmetry
Visualizing Solid Shapes
Definition
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.
Notes
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.
Which congruence criterion do you use in the following?
Given:
∠MLN = ∠FGH
∠NML = ∠GFH
ML = FG
So, ∆LMN ≅ ∆GFH
Solution: ASA Congruence criterion, as two angles and the side included between these angles of ΔLMN, are equal to two angles and the side included between these angles of ΔGFH.
Example
By applying the ASA congruence rule, it is to be established that ∆ABC ≅ ∆QRP and it is given that BC = RP. What additional information is needed to establish the congruence?
For the ASA congruence rule, we need the two angles between which the two sides BC and RP are included. So, the additional information is as follows:
∠B = ∠R and ∠C = ∠P.
Example
In Fig, can you use the ASA congruence rule and conclude that ∆AOC ≅ ∆BOD?
In the two triangles AOC and BOD, ∠C = ∠D..................(each 70° )
Also, ∠AOC = ∠BOD = 30°........................(vertically opposite angles)
So, ∠A of ∆AOC =180° – (70° + 30°) = 80°..................(using angle sum property of a triangle)
Similarly, ∠B of ∆BOD =180° – (70° + 30°) = 80°
Thus, we have ∠A =∠B, AC = BD, and ∠C = ∠D
Now, side AC is between ∠A and ∠C and side BD is between ∠B and ∠D.
So, by ASA congruence rule, ∆AOC ≅ ∆BOD.
Example
Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by the ASA congruence rule. In the case of congruence, write it in symbolic form.
∆DEF, ∠D = 60º, ∠F = 80º, DF = 6 cm.
∆PQR, ∠Q = 60º, ∠R = 80º, QP = 6 cm.
∠D = ∠Q
∠F = ∠R
Side DF = Side PQ
So, ∆DEF ≅ ∆PQR.......(ASA congruence criterion)
Theorem
Theorem (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.
Proof : We are given two triangles ABC and DEF in which:
∠ B = ∠ E, ∠ C = ∠ F
and BC = EF
We need to prove that ∆ ABC ≅ ∆ DEF
For proving the congruence of the two triangles see that three cases arise.
Case (i) : Let AB = DE in following fig.
You may observe that
AB = DE (Assumed)
∠ B = ∠ E (Given)
BC = EF (Given)
So, ∆ ABC ≅ ∆ DEF (By SAS rule)
Case (ii) :
Let if possible AB > DE. So, we can take a point P on AB such that PB = DE. Now consider ∆ PBC and ∆ DEF in following fig.
Observe that in ∆ PBC and ∆ DEF,
PB = DE (By construction)
∠ B = ∠ E (Given)
BC = EF (Given)
So, we can conclude that:
∆ PBC ≅ ∆ DEF, by the SAS axiom for congruence.
Since the triangles are congruent, their corresponding parts will be equal.
So, ∠ PCB = ∠ DFE
But, we are given that
∠ ACB = ∠ DFE So, ∠ ACB = ∠ PCB
This is possible only if P coincides with A.
or, BA = ED
So, ∆ ABC ≅ ∆ DEF (by SAS axiom)
Case (iii) : If AB < DE, we can choose a point M on DE such that ME = AB and repeating the arguments as given in Case (ii), we can conclude that AB = DE and so, ∆ ABC ≅ ∆ DEF.
You know that the sum of the three angles of a triangle is 180°. So if two pairs of angles are equal, the third pair is also equal (180° – sum of equal angles).
So, two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. We may call it as the AAS Congruence Rule.
