CLASSIFYING TRIANGLES BY ANGLES

Triangles can be classified into three types by their angles.

1. Acute triangle

2. Right triangle

3. Obtuse triangle

Acute Triangle

An acute triangle is a triangle with all three angles are less than 90°.

Example :

A right triangle is a triangle in which one of the angles is 90°.

Example :

Right Triangle

Obtuse Triangle

An obtuse triangle is a triangle in which one of the angles is obtuse (greater than 90°).

Example :

Example 1 :

Identify the type of triangle whose diagram is given below.

Solution :

Let us consider the following two important points related to the given information.

(i) All the given three angles are different

(ii) Each of the three angles is less than 90°

So, the given triangle is a scalene and acute triangle.

Example 2 :

Identify the type of triangle whose diagram is given below.

Solution :

Let us consider the following two important points related to the given information.

(i) One of the angles is greater than 90°

(ii) Two sides are equal in length. The angles formed by the two congruent sides with the third side also must be equal

So, the given triangle is an isosceles and obtuse triangle. 

Example 3 :

Identify the type of triangle whose diagram is given below.

Solution :

In the triangle above, the lengths of all the three sides are same and all the three angles are congruent.

So, the given triangle is equiangular triangle.

Example 4 :

Identify the type of triangle whose diagram is given below.

Solution :

In the triangle above, two sides are congruent. The angles formed by the third side with two congruent sides will always be equal.

The diagram given below illustrates this.

By Triangle Sum theorem,

x° + x° + 40° = 180°

2x + 40 = 180

Subtract 40 from both sides.

2x = 140

Divide both sides by 2.

x = 70

The angles measures of the given triangle are 40°, 70° and 70°.

Let us consider the following two important points from the above calculation.

(i) Two of the angles are equal

(ii) Each of the three angles is less than 90°

So, the given triangle is an isosceles and acute triangle.

Example 5 :

If (2x + 15)°, (3x)° and (6x)° are the angles of a triangle, identify the type of triangle.

Solution :

The angle sum property of a triangle states that the angles of a triangle always add up to 180°.

(2x + 15)° + (3x)° + (6x)° = 180°

2x + 15 + 3x + 6x = 180

Simplify.

11x + 15 = 180

Subtract 15 from each side.

11x = 165

Divide each side by 11.

x = 15

Substitute 15 for x into the given expressions to find the angles of the triangle.

First angle = 2x + 15 = 2(15) + 15 = 45°

Second angle = 3x = 3(15) = 45°

Third angle = 6x = 6(15) = 90°

Let us consider the following two important points from the above calculation.

(i) Two of the angles are equal

(ii) One of the angles is 90°

So, the given triangle is an isosceles and right triangle.

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