Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Geometry → Triangles ↓


Congruence of Triangles


In geometry, the concept of congruence is important when two figures are exactly the same in size and shape. Congruence applies to all geometric shapes, not just triangles, but here we are focusing on triangles. In simple words, when two triangles are congruent, they have the same size and shape, although their orientation may be different.

Understanding congruent triangles

Two triangles are said to be congruent when all their corresponding sides and angles are equal. Think of it like two identical pieces cut from the same sheet of paper; no matter how you rotate or flip one of them, they will always be congruent. They will fit perfectly on top of each other.

Basic properties

  • If two triangles are congruent, then all their corresponding sides are of equal length.
  • The corresponding angles of two congruent triangles are equal.

Symbols and notation

In geometry, we use the ≅ sign to indicate congruence. For example, if triangle ABC is equilateral to triangle DEF, we write:

 △ABC ≅ △DEF

Illustrating similar triangles

Let's look at a simple visual example of similar triangles. Consider the following triangles:

In this example, both triangles are the same in shape and size. They're just positioned differently, but if you move one over the other, they'll overlap perfectly.

Conditions for triangle congruence

There are several specific rules or conditions under which we can determine whether two triangles are similar. These rules work because instead of checking all sides and all angles, we rely on a specific set of sides and angles. You can. These conditions are known as congruence criteria:

1. Side-side-side (SSS) criterion

If three sides of a triangle are equal to three sides of another triangle, then those triangles are congruent.

In this case, if AB = DE, BC = EF, and CA = FD, then triangles ABC and DEF are congruent.

2. Side-angle-side (SAS) criterion

If two sides and the angle between them of a triangle are equal to two sides and the angle between them of another triangle, then the two triangles are congruent.

In this case, if AB = DE, BC = EF, and ∠ABC = ∠DEF, then triangles ABC and DEF are congruent.

3. Angle-side-angle (ASA) criterion

If two angles and their included side of a triangle are equal to two angles and their included side of another triangle, then those triangles are congruent.

In this case, if ∠ABC = ∠DEF, BC = EF, and ∠BCA = ∠EFD, then triangles ABC and DEF are congruent.

4. Angle-angle-side (AAS) criterion

If two angles and a nonconnected side of a triangle are equal to two angles and the corresponding nonconnected side of another triangle, then the triangles are congruent.

In this case, if ∠ABC = ∠DEF, ∠BCA = ∠EFD, and CA = FD, then triangles ABC and DEF are congruent.

5. Right angle hypotenuse side (RHS) criterion

This criterion applies specifically to right triangles. If the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, then the triangles are congruent.

In this case, if AC = DF and AB = DE, then triangles ABC and DEF are congruent.

Examples of congruent triangles

Let us use some simple examples to understand these principles in a deeper way:

Example 1

Suppose we have two triangles ABC and DEF where:

  • AB = 5 cm, BC = 7 cm, CA = 4 cm
  • DE = 5 cm, EF = 7 cm, FD = 4 cm

Since all the three corresponding sides are equal by SSS criterion:

 △ABC ≅ △DEF

Example 2

In triangles PQR and JHK, we have:

  • PQ = 6 cm and JH = 6 cm
  • ∠PQR = 90° and ∠JHK = 90°
  • QR = 8 cm and HK = 8 cm

This satisfies RHS criterion for congruent triangles:

 △PQR ≅ △JHK

Practical importance of congruence

Similar triangles are widely used in fields requiring precise measurements, such as various fields of engineering, architecture, and design. Their properties allow us to replicate shapes accurately and predictably, thereby ensuring stability and symmetry in structures it is ensured.

Conclusion

The study of similar triangles forms a fundamental basis in geometry, providing a manual for understanding and predicting the structure of all polygonal shapes. Whether creating building models or solving complex geometric puzzles, congruence criteria are important in classrooms and professional classrooms. The applications serve as essential tools for both.


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Congruence of Triangles

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