An angle is the geometrical measure used to express the deviation between two lines or surfaces. Obtuse angles are observed wherever this deviation appears rather wide. Obtuse angle occur often along with the acute angle as these two supplement each other. Let us learn what an obtuse angle is how it is formed.

## Obtuse Angle Definition

An obtuse angle is an angle whose angle measure is greater than 90º and less than 180º.In general if angle A is obtuse, then its degree measure is greater than 90º, but less than 180º.

In the above diagram the angle BAC is greater than the right angle marked by a square sign.
m ∠BAC > 90º.

Indeed the obtuse angle is formed by adding an acute measure to the right angle.
Angle 2 which is obtuse makes a straight angle with the acute angle 1.
Thus 1 and 2 are supplementary angles.
In general the supplement of an obtuse angle is acute.
The complement for an acute angle does not exist, as its measure is greater than 90º.

## Obtuse Angle Examples

1.  Name the obtuse angles found in the following diagram.

We need to identify the angles whose measures are greater than 90º. Two angles are marked as right angles with ⌉ and ⌈ signs. We can get obtuse angles by annexing adjacent acute angles to these are angles.
m ∠TQR = m ∠TQP = 90º.                    Marked as right angles in the figure.
m ∠UQR = m ∠UQT + m ∠TQR             Angle addition Property.
= measure of an acute angle + measure of a right angle  > 90º.
Similarly,
m ∠PQS = m ∠PQT + m ∠TQS             Angle addition Property
= measure of a right angle + measure of an acute angle > 90º.Hence the two obtuse angles present in the given diagram are, angles UQR and PQS.

 In the adjoining diagram, the measures of two verticalangles are given as linear expressions. Using the factthat vertical angles are congruent,Find the value of x      2.  Determine whether the two angles vertical           angles marked are obtuse.

We can equate the two linear expressions and solve for x.
7x + 2 = 6x + 15                             Vertical angle are congruent. Hence their measures are equal.
- 2         - 2
--------------------
7x      = 6x + 13
-6x        -6x
--------------------
x      =     13
--------------------
Substituting x = 13 in one of the equations,
7x + 2 = 7(13) + 2 = 91 + 2 + 93º.  The angle measure is greater than 90º.
Hence the two vertical angles marked are obtuse angles.
Obtuse angle in Polygons:
A triangle can have at most one obtuse angle. If a triangle has an obtuse angle then it is called an obtuse angled triangle.

In triangle ABC, measure of angle A is marked as 130º. Thus ABC is an obtuse angled triangle.

Oblique parallelograms have one pair of opposite angles obtuse.

In general if two lines intersect, they form one pair of equal vertical angles obtuse, if they do not cut each other at right angles.

## Obtuse Angles in Real Life

1. When people work on lap tops, the the angle between the screen and the base is obtuse.

2. Boomerangs are flying toys which return to the thrower after moving some distance in the air. Some of the Boomerangs are designed in such a way, the angle between the wings is an obtuse angle ranging from 91º to 140º.

3. You can observe the obtuse angle at many roof tops, as the two roof surfaces slope down from it.

In general, obtuse angles are observed whenever two sides / arms or surfaces deviate widely.