Question

Find $a + b + c + d + e + f$ in the given figure if AOD, BOE and FOC are straight lines.

Answer 1

Hint: An angle is a figure formed by two rays, called sides of the angle sharing a common endpoint called the vertex of the angle and it is represented by the symbol $\angle $. In this we use angle sum property of a triangle, vertically opposite angle and complete angle.
$a + b + c = 180^\circ \,$ (Angle sum property)
Here, in the question we need to determine the sum of the angles of the figure except the central angle. For this we need to use the properties of the angles of the triangles. Some of the properties that we have to use here are:
->Sum of the angles in a triangle is 180 degrees.
->Alternative and the corresponding angles are equal.
->Sum of the angles on a straight line is always 180 degrees.


Complete step by step solution:
Following the figure given in the question,

In $\Delta AOB$, $a + b + \angle AOB = 180^\circ ...(1)$
Similarly, in$\Delta EOF$, $f + e + \angle EOF = 180^\circ ...(2)$(angle sum property)
Again, in$\Delta COD$, $c + d + \angle COD = {180^ \circ }$…………….(3)

Now, $\angle AOF + \angle EOF + \angle EOD + \angle COD + \angle BOC + \angle AOB = 360^\circ $(complete angle)…(4)

From (1), (2) and (3)
$
  \angle EOF = 180^\circ - f - e \\
  \angle COD = 180^\circ - c - d \\
  \angle AOB = 180^\circ - a - b \\
 $
Put the value of $\angle EOF,\angle COD\,and\,\angle AOB$in equation (4)
$\angle AOF + (180^\circ - f - e) + \angle EOD + (180^\circ - c - d) + \angle BOC + (180^\circ - a - b) = 360^\circ $

Now, $\angle BOC = \angle EOF$(vertically opposite angle)
$\angle AOF + \angle EOD + \angle EOF = 180^\circ $[Straight line (given)]
\[
  180^\circ + 180^\circ + 180^\circ + 180^\circ - (a + b + c + d + e + f) = 360^\circ \\
  720^\circ - 360^\circ = a + b + c + d + e + f \\
  a + b + c + d + e + f = 360^\circ \\
 \]

Note: Students must have knowledge of different types of angles. They would need to know some commonly used angle properties and practice a lot of such problems, so it becomes easier to “see” which properties need to be applied.