Introduction

Any angle that measures greater than 90° but less than 180° is called an obtuse angle

An obtuse angle is larger than an acute angle and a right angle, but smaller than a straight angle.

Acute Angle	Right Angle	Straight Angle
Angle measure less than 90o	Angle measure 90o	Angle measure 180o

An obtuse angle can be observed in 3-D and 2-D polygons. It is a crucial measure for describing angles. 

Properties of Obtuse Angles

• Magnitude of Obtuse Angles: The measurement of an obtuse angle always lies between 90° and 180°.
• Sum of Angles: In triangles, the sum of all angles is always 180°, and if one angle is obtuse, the other two must be acute.
• Supplementary Angles: An obtuse angle and an acute angle form a supplementary pair whose sum equals 180°).

• An obtuse angle is more than a quarter of a circle but less than half of a circle. So, if we divide a circle into four parts, an obtuse angle will always occupy between 1/4 and 1/2 of a circle.
• Obtuse angles are often mistaken for reflex angles. The value of obtuse angles lies between 90^o and 180^o whereas the reflex angle values lie between 180^o and 360^o.

Examples of Obtuse Angles in the world

Obtuse angles can be observed in clock timings such as 7.15, 1.30, 6.10, 10.15, etc.

When the door is ajar

Reclining chairs

Wings of the Windmill

Clothes hangar

Geometric Shapes with Obtuse Angle

Obtuse angle triangles

The obtuse angle can be observed on the obtuse angle triangle – isosceles triangle and scalene triangle.

Rhombus – a rhombus will have at least two obtuse angles.

Trapezoid – trapezoid will have at least one obtuse angle.

Pentagons - Can have five obtuse angles, each measuring 108°

Hexagon – all its angles are obtuse angles.

Heptagon – it has at least one obtuse angle.

Octagon – all its interior angles are obtuse.

Parallelogram – it has two obtuse angles.

Obtuse angle Triangle

The sum of the interior angles of a triangle will be 180^o. Hence, in an obtuse angle triangle, if one angle is an obtuse angle, the sum of the other two angles must be less than 90°. There are two types of obtuse angle triangle –

Isosceles triangle – it’s one angle is obtuse angle the other two angles are acute angles of equal measure.

Scalene triangle – all its angles are of unequal measures and there is one obtuse angle.

Example Calculation

In the above diagram of ?ABC, find the value of x.

The above triangle is a scalene triangle with one obtuse angle. Since the total value of all angles of a triangle is 180^o, 

130^o + 20^o + x = 180^o

150^o + x = 180^o

x = 180^o – 150^o

   = 30^o

Measuring an Obtuse Angle

Obtuse angles are measured using a protractor or calculated based on other given angles.

When you place the centre point of the protractor at the vertex, lying on the base, the point on the protractor at which the perpendicular rests will show the angle. On a protractor, the 90-angle point will always be perpendicular to the vertex. Thus, if the perpendicular arm of the angle arches beyond the 90^o point then it forms an obtuse angle. 

In the above figure, the perpendicular arm forms a 115^o angle. 

Example: In the below figure, if ∠DCA is 30^o, then what is the value of its adjacent angle?

The line ACB forms a straight angle of value 180^o, if ∠DCA, which is formed by the ray DC that is perpendicular to AB is 30^o, then the value of ∠DCB, which is its adjacent angle is given as - 

 ∠DCA + ∠DCB = 180^o

30^o + ∠DCB = 180^o

∠DCB = 180^o – 30^o

= 150^o

∠DCB is an obtuse angle of value 150^o.

Difference between obtuse angle and other angles

Obtuse angle	Measures <90o and >180o
Right Angle	Measures 90o
Acute Angle	Measures <0oand >90o
Straight Angle	Measures 180o
Reflex Angle	Measures <180o and >360o
Complete Angle	Measures 360o

Practical examples of Obtuse Angles

• Engineering and Construction: Obtuse angles are used in construction for structural integrity. It can be observed in the arches of bridges, roofs, road intersections etc.

• Art and Design: Obtuse angles are used to create aesthetic designs in art, logos, and product design.

• Sports: How obtuse angles appear in various sports, such as the angles formed in throwing and hitting movements.