**Classification of Triangles**

A triangle is a polygon with three sides. It has three sides and three vertices. The sum of the angles of a triangle is 180°.

Triangles can be classified in two ways:

(i) By their sides

(ii) By their angles

**Classification of Triangles by Sides**

**Scalene Triangle:**

A triangle which has no side of equal length, is called a scalene triangle.

In figure, PQ ≠ QR ≠ PR, so ΔPQR is a scalene triangle.

**Isosceles Triangle:**

A triangle which has two sides of equal length, is called an isosceles triangle.

In figure, AB = AC, so ΔABC is an isosceles triangle.

**Equilateral Triangle:**

A triangle which has all the three sides equal in length, is called an equilateral triangle.

In figure, PQ = QR = PR, so ΔPQR is an equilateral triangle.

**Classification of Triangles by Angles**

**Acute-angled Triangle:**

A triangle whose all angles are acute (less than 90°), is called an acute-angled triangle or simply acute triangle.

In figure, ∠A, ∠B, and ∠C are all less than 90°, hence ΔABC is an acute-angled triangle.

**Right-angled Triangle:**

A triangle whose one of the angles is a right angle, i.e., 90°, is called a right-angled triangle or simply right triangle.

In figure, ∠Q = 90°, hence ΔPQR is a right triangle.

**Obtuse-angled Triangle:**

A triangle whose one angle is obtuse, is called an obtuse-angled triangle or simply obtuse triangle.

In figure, ∠Y is obtuse, hence ΔXYZ is an obtuse-angled triangle or simply obtuse triangle.

**Equiangular Triangle:**

When all the three angles of a triangle are equal, then it is known as an equiangular triangle.

An equiangular triangle is also known as equilateral triangle because all the three sides are equal.

In figure, ∠P = ∠Q = ∠R = 60°and PQ = QR = PR, hence ΔPQR is an equiangular triangle.

**Example 1:** One of the equal angles of an isosceles triangle is 50°. Find its third angle.

**Solution:** Let third angle = x

∴ x + 50° + 50° = 180°

(Angle sum property of a triangle)

or, x + 100° = 180°

or, x = 180° – 100°

= 80°

∴ Third angle = 80°

**Example 2:** Each ofthe two equal angles of a triangle is four times the third angle. Find all the angles of the triangle.

**Solution:** Let the smaller angle = x

∴ other two angles = 4x and 4x

Thus, x + 4x + 4x = 180°

(Angle sum property of a triangle) or, 9x = 180°

∴ 4x = 4 x 20°

= 80°

Thus, the angles of the triangle are 20°, 80°, and 80°.

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