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

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

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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.

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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? 

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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.

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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.

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 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.  

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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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Here are a few activities for you to practice.

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

 

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