Obtuse Angle

Table of Contents 


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Introduction To Angles

When two rays meet at a vertex, an angle is formed.

An angle is represented by the symbol \(\angle\) and is measured in degrees \(^\circ\)

An angle is formed when 2 rays meet


What is an Obtuse Angle?

The definition of an obtuse angle in Geometry states that an angle whose measure is greater than \({90^\circ}\) and less than \({180^\circ}\) is called an obtuse angle.

Obtuse angle definition in Math: angles measuring more than 90 degrees and less than 180 degrees

Here is an activity for you to try.

Drag the point B to the left and observe all the obtuse angle measures.


Examples of Obtuse Angles

Let's recall the definition of obtuse angles in Math.

We know that angles measuring greater than \(90^\circ \) and less than \(180^\circ \) are called obtuse angles.

Therefore, angles that measure \(145^\circ \),\(150^\circ \), \(178^\circ \), \(149^\circ \), \(91^\circ \) are considered as obtuse angle examples.

Here are some real-life examples of obtuse angles.

Can you observe the obtuse angles in all these images?

Can you think of more objects in real life that include obtuse angles? 

Real life obtuse angle examples: A sofa, a staircase, a roof, a wall clock, a hand fan, a cloth hanger


Obtuse Angle of a Triangle

When one of the vertex angles of a triangle is greater than \(90^\circ \), it is called an obtuse triangle.

Examples of obtuse triangles

Examples of obtuse triangles

The triangles above have one angle greater than \(90^\circ \)

Hence, they are called obtuse-angled triangles or simply obtuse triangles.

In an obtuse triangle, the sum of the squares of the two sides is less than the square of the longest side.

An obtuse triangle with 3 sides a, b, and c

In \(\Delta ABC \), the sides measure \(a,b,c \) such that \(c\) is the largest side, thus:

\(a^2 + b^2  < c^2\)

Conversely, if in a triangle, if \(a^2 + b^2  < c^2\), then the triangle is an obtuse triangle.


Obtuse Angles of a Rhombus

A rhombus is a special type of quadrilateral which includes:

  • four equal sides
  • two pairs of parallel sides
  • equal opposite angles 

Obtuse angles of a rhombus

The sum of the interior angles of any quadrilateral is \(180^\circ \) and in a rhombus, consecutive angles are supplementary and opposite angles are equal.

Thus, at any given time, a  rhombus has two obtuse angles that are equal and the other two angles are acute and they are also  equal.


Obtuse Angles of a Parallelogram

 A Parallelogram is a special type of quadrilateral which includes:

  • two pairs of parallel sides
  • opposite sides of equal lengths
  • equal opposite angles

Obtuse Angles of a parallelogram

The sum of the interior angles of any quadrilateral is \(180^\circ \) and in a parallelogram, consecutive angles are supplementary and opposite angles are equal.

Thus, at any given time, a  parallelogram has two obtuse angles that are equal and the other two angles are acute and they are also  equal.  

 
Thinking out of the box
Think Tank
  1. Can a triangle have more than one obtuse angles?
  2. Which polygon has all its internal angles as obtuse angles?
  3. Can there be a parallelogram without an obtuse angle? If so, what shape can it be?


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Solved Examples

Example 1

 

 

Here is a small interactive activity; few random angles are generated.

Measure the angles using the protractor and identify the obtuse angles.

Solution:

All the angles measure more than \(90^\circ \) and less than \(180^\circ \)

They are all obtuse angles.

All the given angles are obtuse angles.
Example 2

 

 

Choose all the obtuse angles from the following figures.

Obtuse angles in math and their measures

Solution:

Option (b) and option (c) are more than \(90^\circ \) and less than \(180^\circ \)

Hence, they are obtuse angles.

Option (b) and option (c) are obtuse angles.
Example 3

 

 

 At what times, in the clocks shown below, an acute angle is formed?

angle formed on a wall clock

Solution:

 We can observe that in all instances, an obtuse angle is formed between the hour's hand and minutes hand of the clock.

\(therefore\) An obtuse angle is formed at 5:00, 10:15, 2:40 and 8:00.
Example 4

 

 

 Can an obtuse triangle have sides measuring 4 cm, 5 cm and 8 cm?

Solution:

In an obtuse triangle, the sum of the square of two sides should be less than the square of the greatest side.

i.e   \(a^2 + b^2 < c^2\) where c is the largest side.

Let a = 4 cm

b = 5 cm

c = 8 cm (largest side)

\(a^2 = 16\)

\(b^2 = 25\)

\(c^2 = 64\)

\(a^2 +b^2= 16 +25 = 41\)

Since  \( 41 <  64 \)

\( \implies  a^2 + b^2 \)  is less than \(c^2\)

Hence, the given measures form an obtuse triangle.

4 cm, 5 cm and 8 cm forms an obtuse triangle.
Example 5

 

 

Which vertices of the parallelogram \(ABCD\) have obtuse angles?

What is their measure?

Parallelogram

Solution:

In a parallelogram, consecutive angles are supplementary and opposite angles are equal.

Since \(\angle A \) = \(60^\circ \), the opposite angle \(\angle C\) is also \(60^\circ \)

\(\angle A \) + \(\angle D \) = \(180^\circ \) (consecutive angles are supplementary)

Therefore, \(\angle D \) =  \(120^\circ \) and \(\angle B \) = \(120^\circ \) (opposite angles are equal)

\(\angle B \) and \(\angle D \) are obtuse angles and they measure 120° each

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Practice Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result

 
 
 
 
 
 

 

 
important notes to remember
Important Notes
  1.  All angles measuring more than \(90^\circ \) and less than \(180^\circ \) are called obtuse angles.
  2.  In an obtuse triangle, the sum of the squares of the two sides is less than the square of the longest side.

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. How do you create an obtuse angle?

The definition of an obtuse angle in Geometry states that an angle larger than \(90^\circ \) but less than \(180^\circ \) is called as an obtuse angle.

We can use a protractor and mark any angle between \(90^\circ \) and \(180^\circ \) to make an obtuse angle.

2. What are some examples of obtuse angles?

\(145^\circ \),\(150^\circ \), \(178^\circ \), \(149^\circ \), \(91^\circ \) are all examples of obtuse angles as they are more than \(90^\circ \) and less than \(180^\circ \).

3. How do you determine an obtuse angle?

The definition of an obtuse angle states that if an angle measures more than \(90^\circ \) and less than \(180^\circ \), it is an obtuse angle.

 

We can use a protractor and verify the same.

  
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