# Obtuse Angle

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 1 Introduction To Angles 2 What is an Obtuse Angle? 3 Examples of Obtuse Angles 4 Obtuse Angle of a Triangle 5 Obtuse Angles of a Rhombus 6 Obtuse Angles of a Parallelogram 7 Solved Examples on Obtuse Angles 8 Thinking Out of the box! 9 Practice Questions on Obtuse Angles 10 Important Notes on Obtuse Angles 11 Maths Olympiad Sample Papers 12 Frequently Asked Questions (FAQs)

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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 $$^\circ$$ ## 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. 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? ## 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.  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. 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 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 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. 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. 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? 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? 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.

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.

## 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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