Question

How many angles are there?

a.11
b.10
c.8
d.5

Answer 1

Hint:Any angles in geometry are formed with the two rays having a common endpoint and other proceeding to infinite. Thus any two rays as given is the figure from one angle ,solving this problem will need a formula of combinations but before that I would like to give a brief introduction about an angle.
Formula used- An angle is defined as when the tails of the two rays coincide, the common point is known as the angle. The angle is represented as $ \angle $

Complete step-by-step answer:
Here $ \angle AOB $ is the angle in the shown figure.
Given in the figure

There are 5 lines
For making an angle the tales of the two lines must intersect.
Therefore we need to select any two lines from the five lines and proceeding in the same manner we will see all the cases.
Here we can use combination to see all possible combinations
Therefore total numbers of angles are
 $
   = {}^5{C_2} \\
   = \dfrac{{5!}}{{2! \times 3!}} \\
   = \dfrac{{5 \times 4 \times 3!}}{{2 \times 1 \times 3!}} \\
   = \dfrac{{5 \times 2}}{1} \\
   = 10 \\
 $
Therefore, number of angles possible are 10
Hence, the correct option is B.

Note:The angles are measured in degrees or radians. We can measure angles using a protractor. The two rays whose tails start from a common point to form an angle is called arms of the angle. In the above figure OA and OB are the arms of the angle. Also the common end point where the tails of the rays meet is called the vertex of the angle. In the above figure the point O is the vertex of the angle.