Triangles are classified according to the angle measures or according to side lengths. Triangles are classified based on angle measures as acute, obtuse and right triangles. Any triangle should have at least two acute angles to comply with angles sum property of a triangle.

An obtuse triangle hence has one obtuse angle, the other two angles being the required acute angles. We can define an obtuse triangle as a triangle one of whose angle measure is greater than 90º and the sum of the measures of other two angles is less than 90º.

If Δ ABC is a obtuse triangle with C as the obtuse angle then the measure of $\angle$C > 90º   and measure of $\angle$A + measure of $\angle$B < 90º.

## Obtuse Triangle Properties

1. An obtuse triangle will have one and only one obtuse angle. The other two angles are acute angles.
2. The sum of the two angles other than the obtuse angle is less than 90º.
3. The side opposite to the obtuse angle is the longest side of the triangle.
4. The points of concurrency, the Circumcenter and the Orthocenter lie outside of an obtuse triangle, while Centroid and Incenter lie inside the triangle.

## Obtuse Scalene Triangle

A triangle is called scalene if no two sides of the triangle are congruent. In addition to this property, an obtuse scalene triangle will have one of its angle obtuse. If Δ ABC is an obtuse scalene triangle with obtuse angle at C, then we can say,

AB $\neq$ BC $\neq$ CA and m $\angle$A $\neq$ m $\angle$B $\neq$ m $\angle$C.

## Obtuse Isosceles Triangle

An isosceles triangle has two of its sides are congruent. In an obtuse isosceles triangle, the sides containing the obtuse angle are congruent. The two non obtuse angles serve as the base angles and they are also congruent.

In $\triangle$ PQR: R is an obtuse angle with measure 130º and the bases angles P and Q are acute angles with measure 25º each. The congruent sides PR and RQ form the legs of the isosceles triangle. Base PQ, the side opposite to the obtuse angle is the longest.

## Obtuse Equilateral Triangle

An equilateral triangle is also equiangular. An obtuse triangle can never be equiangular as the obtuse angle in the triangle has to be necessarily greater than the other two angles which are acute. Hence it is impossible to find an Obtuse equilateral triangle.

## Right Obtuse Triangle

When we say Right obtuse triangle, we mean that the triangle has one right angle and also an obtuse angle. Any triangle need to have two acute angles and the third angle can either be acute, obtuse or right. The classification acute, obtuse and right triangle is done on this fact.

Hence, if the third angle is obtuse it cannot be a right angle, and if it is a right angle it cannot be obtuse. Hence, a right obtuse triangle is non existent.

## Obtuse Triangle Formula

If any two angles of a triangle are given, it can be easily determined whether the triangle is an obtuse triangle or not. But how to determine this, when the three sides of the triangle are known? We have an inequality in the lines of Pythagorean identity to test this.

The triangle is an obtuse triangle if the sum of the squares of the smaller sides is less than the square of the largest side.

Let a, b and c are the lengths of the sides of triangle ABC and c is the largest side, then the triangle is obtuse if

a2 + b2 < c2

### Solved Example

Question: Janet needs to draw an obtuse triangle for her Geometry Project. She chooses the lengths of the sides as 8 cm, 9 cm and 13 cm. Will this give an obtuse triangle when constructed?
Solution:

The lengths of the smaller sides are 8 cm and 9 cm and the largest side is 13 cm
a2 + b2 = 82 + 92 = 64 + 81 = 145  and c2 = 132 = 169
We find a2 + b2 < c2.  Hence Janet is right in her choice of the sides to construct an obtuse triangle.

## Obtuse Triangle in Real Life

The triangles formed by the longer diagonal of a parallelogram are obtuse triangles.

Obtuse triangle can be visualized in some flight formations. The structures seen in certain roof constructions use obtuse triangular braces.

## Obtuse Triangle Examples

### Solved Examples

Question 1: Determine whether the following are always, sometimes or never true.
a. Obtuse triangles are isosceles triangles.
b. Obtuse triangles are equilateral triangles.
c. Obtuse triangles are right triangles.
Solution:

The first statement is sometimes true.
The second and the third statements are never true.

Question 2: The measures of two angles of a triangle are given as 48º and 52º. Determine whether the triangle is obtuse.
Solution:

The sum of the two angles = 48o + 52o = 100o > 90o.
Hence the triangle cannot be obtuse. Indeed the third angle measures = 180o - 100o = 80º, which is an acute angle.

Question 3: Using the inequality test for obtuse triangles, test whether the three numbers 24, 36 and 48 can refer to the sides of an obtuse triangle.
Solution:

Sum of the squares of the smaller numbers = a2 + b2 = 242 + 362 = 576 + 1296 = 1872
Square of the largest number = 482 = 2304
Since a2 + b2 < c2, the numbers can represent the sides of an obtuse triangle.