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 → Triangles ↓


Types of Triangles


Triangles are one of the simplest and most fundamental shapes in geometry. They consist of three sides and three corners. The language and logic of triangles have been an essential foundation in the study of nature, engineering, art, and science. In this detailed lesson, we will explore the different types of triangles based on their sides and angles. By understanding these types, students can gain insight into how this simple shape forms the cornerstone of geometry and other applications.

1. Classification based on arms

Triangles can be classified based on the length of their sides. There are three main types of triangles:

1.1 Equilateral triangle

An equilateral triangle has all three sides of equal length. Because of this, the three angles are also equal, each measuring 60 degrees.

ABC

If AB = BC = CA, then triangle ABC is equilateral.

1.2 Isosceles triangle

An isosceles triangle has at least two sides of equal length. This also means that the angles opposite to those sides are equal.

ABC

If AB = AC, then ∠B = ∠C. Triangle ABC is isosceles.

1.3 Scalene triangle

All the sides of a scalene triangle are of different lengths. As a result, the three angles are also different.

ABC

If AB ≠ BC ≠ CA, then triangle ABC is scalene.

2. Classification based on angles

Triangles can also be classified according to their interior angles. The main types are:

2.1 Acute triangle

An acute-angled triangle has all three interior angles less than 90 degrees.

ABC

In this type of triangle, each angle is less than 90°.

2.2 Right-angled triangle

A right triangle has one interior angle equal to exactly 90 degrees. The side opposite this angle is called the hypotenuse, and it is the longest side of the triangle.

ABC

The Pythagorean theorem applies:

AB 2 + BC 2 = AC 2

where AC is the hypotenuse.

2.3 Obtuse-angled triangle

An obtuse triangle has one interior angle greater than 90 degrees.

ABC

In an obtuse-angled triangle one angle is greater than 90°.

3. Some interesting facts about triangles

Triangles have some fascinating properties and facts that are important to understand in geometry.

  • The sum of the interior angles of a triangle is always 180 degrees.
  • It is impossible to construct a triangle whose side length is less than the sum of the lengths of the other two sides. This is known as the Triangle Inequality Theorem.
  • The exterior angle of a triangle is equal to the sum of its two opposite interior angles.
  • When two triangles have the same shape but different sizes, they are called similar triangles.
  • When two triangles not only have the same shape but also the same size, they are called congruent triangles.

4. Using knowledge of triangles

Understanding triangles helps solve problems related to design, engineering, construction, and various scientific fields. For example, engineers use triangular structures in building bridges and towers because triangles are naturally stable shapes. Artists use the dynamic sense of triangles to create visual interest in their creations.

Example problems:

Example 1

Given a triangle ABC with side lengths AB = 5, BC = 7, and AC = 10, determine whether the triangle is scalene, isosceles, or equilateral.

Since all sides have different lengths (5 ≠ 7 ≠ 10), triangle ABC is a scalene triangle.

Example 2

Determine the type of triangle whose angle measures are 45 degrees, 45 degrees, and 90 degrees.

Since one angle is 90 degrees, this triangle is a right-angled triangle. Additionally, since the other two angles are equal, this triangle is also isosceles.

Example 3

If a triangle has sides x, x, and y, and it is known that x = 8, y = 6, then identify what type of triangle it is.

Since it has two equal sides (x = x = 8), it is an isosceles triangle.

Example 4

Explain why there cannot be a triangle with sides 3, 4, and 8.

According to the triangle inequality theorem, the sum of any two sides must be greater than the third side. Here 3 + 4 is not greater than 8, so such a triangle cannot exist.

Conclusion

Triangles are a powerful concept in mathematics, and understanding their types is fundamental. Whether you're measuring angles, comparing side lengths, or analyzing real-world problems, recognizing the properties of different triangles enhances understanding and application skills. As you advance in geometry, a basic knowledge of the types of triangles will continue to be an important tool in your mathematical toolkit.


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Types of Triangles

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