Shapes and Space

Angles

Angles are everywhere, and we can visualize them in different aspects of daily lives.We will learn about different types of angles that include obtuse angles, acute angles, right angles, and straight angles.

What is an Angle?

Simply understood, an angle could be defined as a place where two lines intersect. Such lines could either be on a piece of paper in mathematics class.You can even measure angles between various things that don’t even have physical lines between them. Even for angles, we can categorize them in different categories such as right, straight, acute, and obtuse.

This table will provide a concise description of every important angle –

You will be able to better visualize different angles by referring to the below-given illustration.

Positive and Negative Angles

When measured from a specific line, a positive angle would go in counter-clockwise direction, i.e. opposite direction that a clock goes. However, a negative angle would be measured in a clockwise direction.

Thus, the direction in which a specific angle is measured would certainly lead to a difference in the overall result.

Parts of Angle

The corner points where two lines meet is known as the vertex. Two straight sides are termed as arms. Angle could be defined as the amount of turn between each arm.

Let’s learn about different angles in detail with the help of figures.

Acute Angle

An acute angle has value always less than 90°.

Right Angle

A right angle is where the internal angle is equivalent to 90°. Note that the value of the right angle will always remain 90° i.e. neither less nor more.

Obtuse Angle

An obtuse angle always has a value of more than 90° but always less than 180°

Reflex Angles

All angles greater than 180° are termed to be reflex in nature.

QUIZ

Based upon the discussion that has been made above regarding the concept of angles, think innovatively and try to answer the below-given questions optimally.

Q1. Which one of the following angles can not be termed as acute?

1. 180°
2. 80°
3. 18°
4. 1°

Answer – (a)

Solution –Acute angles have a value of less than 90° under all circumstances. According to this statement, option (b), (c), and (d) are acute, whereas option (a) is an obtuse angle.

Q2. When two acute angles are added together, which of the following cannot be a possible result of their sum:

1. Acute Angle
2. Right Angle
3. Obtuse Angle
4. Straight Angle

Answer – (d)

Solution – Simply understood, an acute angle is always less than 90°.

If we add up two acute angles, we can obtain the following results

• The resultant sum could be acute as 25° + 8° = 33° (an acute angle)
• The resultant sum could be a right angle as 45° + 45°= 90° (a right angle)
• The resultant sum could be an obtuse angle as 50°+70° = 120° (an obtuse angle)
• However, the resultant sum cannot be a straight angle because the acute angle is always less than 90°. A straight angle has a value equivalent to 180°. The largest acute angle is 89°.

If we add up 89° + 89° = 178°, the resultant would be a straight angle.

Hence, the addition of two acute angles can never result in a straight angle. Thus, option (d) is correct.

Q3. How many acute angles can you find out in the below-given diagram?

Think innovatively and think of all possible combinations.

1. 10
2. 8
3. 7
4. 9

Solution –An acute angle has to be always less than 90°, the below-given angles are acute –

∠AOB, ∠COD, ∠DOE, ∠AOD, ∠BOE, ∠COE, ∠BOC, ∠BOD, ∠AOC

Therefore, there isa total of 9 acute angles in the above-given figure. However, an obtuse angle is also present – ∠AOE. Hence, the correct option for the question is (d).

Q4. Which of the following angles will be an obtuse angle?

1. One half of a right-angle triangle
2. Two and a half right-angle triangles
3. Two right angles
4. One and a half right angle

Answer – (d)

Solution –To solve the given problem, we will have to understand what an obtuse angle is. An angle that has a value of more than 90° but less than 180° could be termed as obtuse.

• As per option (a), one half of a right-angle triangle equates to 45° (an acute angle)
• As per option (b), two and a half right-angle triangles result in 225°. (a reflex angle)
• As per option (c), two right angles will result in 180°. (a straight angle)
• As per option (d), one and a half right angle results in 135°. (an obtuse angle)

Q5. How many obtuse angles can you find out in the below-given diagram?

1. 2
2. 4
3. 8
4. 10

Answer – (c)

Solution – To find out different obtuse angles in the figure, we have to understand that every angle that has a value of more than 90° but less than 180° is obtuse.

By observing the above-given figure, we can find out that –

∠AOE, ∠AOD, ∠BOD, ∠AOF, ∠BOF, ∠BOE, ∠COFand ∠COE are all obtuse angles. Hence, there isa total of 8 obtuse angles in the figure.

Q6. In the below-given figure, how many obtuse angles can you find out?

1. 5
2. 10
3. 15
4. 20

Answer – (b)

Solution –For finding out different obtuse angles in the figure, we need to label the diagram as –

In the above-given figure, the following angles could be regarded as obtuse –

∠CDE, ∠ABC, ∠EFG, ∠IJA, ∠GHI, ∠JBD, ∠BDF,∠DFH, ∠HJB, and ∠FHJ

All angles that have been mentioned above are obtuse. Hence, there isa total of 10 obtuse angles in the figure.

Q7. In the below-given figure, how many pairs of right angles can you figure out?

1. 3
2. 8
3. 9
4. 16

Answer – (d)

Solution – In the given figure, we can visualize that one pair of a line is parallel and the other two lines are perpendicular to them. This defines that the shape formed inside must be a rectangle.

There are four right angles at each point where different lines are intersecting. Hence, there are 4 x 4 = 16 right angles present in the figure. Therefore, the correct answer to the problem is (d).

Q8. Which of the following angles in the below-given figure is straight?

1. ∠BOC
2. ∠DOC
3. ∠COA
4. BOD

Solution –Before finding out the straight angle, we have to understand that a straight angle always has a value of 180°. By visualizing the above-given figure, we can assess that.

• ∠AOB has a value of less than 90°. Hence, it is an acute angle.
• ∠AOC has a value greater than90° but less than 180°. Hence, it could be termed as obtuse.
• ∠BOCcan be observed to be a right-angle having value of 90°
• ∠BODprimarily comprises two right angles i.e. 2 x 90° that equates to 180°. Hence, it is a straight angle.

Q9. How many straight angles are comprised of 450°?

1. 5
2. 5
3. 3
4. 4

Answer – (b)

Solution – A straight angle has a value of 180°.

We can divide 450°/180° = 45/18 = 5/2 = 2.5. Hence, 450° comprises of 2.5 straight angles.

Q10. In the given figure angle AOB equals 67°. Find out the size of the reflex angle AOB.

1. 293°
2. 113°
3. 244°
4. 345°

Answer – (a)

Solution – If you add up a smaller angle (obtuse or acute) and reflex angle for the same shape, you will always be obtaining a value of 360°.

As per the given question, angle AOB = 63°.

Hence, the reflex angle AOB will be equal to 360° – 67° = 293°.

Thus, the correct answer to the problem will be an option (a).