Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Geometry ↓


Congruence of Triangles


In geometry, a triangle is a special shape with three sides and three angles. Understanding whether two triangles are similar in size and shape is an important concept in mathematics. This is where the idea of "congruence" comes in. The simple meaning of congruence is This means that the triangles are similar in size and shape.

Two triangles are said to be congruent if the three corresponding sides and the three corresponding angles are exactly equal. When triangles are congruent, they can be placed on top of each other without any gaps or overlaps. This feature helps us measure the angles in various geometrical problems. and allows properties to be easily predicted.

Criteria for triangle congruence

There are several ways to prove that two triangles are congruent. These are known as triangle congruence criteria. Here, we will explore the main criteria:

1. Side-side-side (SSS) criterion

According to the SSS criterion two triangles are similar if the three sides of one triangle are equal to the three sides of the other triangle. Imagine two triangles whose sides are 5 cm, 7 cm and 9 cm long. If the sides of the two triangles are of these lengths then the sum of the sides of the triangles is equal to the sum of the sides of the other triangle., then they are isomorphic according to the SSS criterion.

Visual example:

A B C side 7 cm side 9 cm side 5 cm

2. Side-angle-side (SAS) criterion

The SAS criterion states that two triangles are similar if two sides and the angle between them in one triangle are equal to two sides and the angle between them in the other triangle. It is important that the angle is between the two sides being compared. It has been happening.

Visual example:

A B C Angle 60° side 7 cm side 5 cm

3. Angle-side-angle (ASA) criterion

According to the ASA criterion, if two angles and the side between them in a triangle are equal to two angles and the side between them in another triangle, then those triangles are congruent.

Visual example:

A B C Angle 45° Angle 70° side 7 cm

4. Angle-angle-side (AAS) criterion

Similar to ASA, the AAS criterion indicates that two triangles are congruent if two angles and a non-included (not between two angles) side in one triangle are equal to two angles and a non-included side in the other triangle.

Visual example:

A B C Angle 45° Angle 70° side 8 cm

5. Right angle-hypotenuse-side (RHS) criterion

This criterion applies only to right-angled triangles. Two right-angled triangles are congruent if the hypotenuse and another side of one triangle are equal to the hypotenuse and another side of the other triangle.

Visual example:

A B C Ear side 6 cm

Note: When two triangles are similar, it is possible to map one onto the other by rotation, reflection, and translation. This means that if you rearrange or flip one triangle, it will perfectly overlap the other triangle.

Examples of triangle congruence

Example 1: SSS congruence

Consider triangles ABC and DEF where:

  • Side AB = Side DE
  • Side BC = Side EF
  • Side CA = Side FD
Triangle ABC
A
,
,
B-------C

Triangle DEF
D
,
,
E----F

Since three sides of ABC are equal to the three sides of DEF, therefore, triangles ABC and DEF are congruent by the SSS criterion.

Example 2: SAS conformance

Triangles XYZ and RST have the following properties:

  • Side XY = Side RS
  • Angle YZ = Angle ST
  • Side YZ = Side ST
Triangle XYZ
X----Y
,
Jade

Triangle RST
R----S
,
Tea

This shows that according to SAS the triangles are congruent because the two sides and the angle between them are equal.

Why congruence matters

Understanding congruence is important because it helps in proving the properties of shapes geometrically, solving problems involving symmetry, and many construction tasks. Congruent triangles are used in real-life applications such as architecture, engineering, computer graphics, and more. are also original.

Properties of congruent triangles

When we talk about congruent triangles, certain properties are always true:

  • Corresponding angles are equal.
  • Corresponding sides are equal in length.
  • Congruent triangles have the same area.
  • The perimeter of congruent triangles is also the same.

These can be used effectively to solve various mathematical problems and real-life questions.

Conclusion

In conclusion, congruence of triangles is an important concept in geometry that helps determine similarity of shapes and sizes. By mastering the criteria for establishing triangle congruence, we can effectively address a wide range of geometric challenges. Remember, congruence is all about comparing corresponding sides and angles, ensuring that one shape can theoretically be placed on another without any changes. Now that you have a basic understanding of triangle congruence, then you are well equipped to delve further into the fascinating world of geometry.


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Congruence of Triangles

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