Question

In the figure, show that $\angle A + \angle B + \angle C + \angle D + \angle E + \angle F = {360^ \circ }$.

Answer 1

Hint: We will use the fact that the sum of angles in a triangle is ${180^ \circ }$. Using this in the two triangles which combine to form the given figure of a star and then adding those equations, we will get the required answer.

Step-By-Step answer:
We can clearly see that we are given the figure of a star with vertices A, B, C, D, E and F. Now, we can also very easily observe that it is formed of two different triangles.
We can observe that the triangles AEC and the triangle BDF form two different triangles and then when they are put together in an inverted form, we get the given star.
Now, we also know that we have a property named Triangle Property which states that the sum of interior angles of a triangle is ${180^ \circ }$.
Using this property in $\vartriangle AEC$:-
$ \Rightarrow \angle A + \angle C + \angle E = {180^ \circ }$ ………………….(1)
Now, using the same property in $\vartriangle BFD$:-
$ \Rightarrow \angle B + \angle F + \angle D = {180^ \circ }$ ………………….(2)
Now, let us add both the equations (1) and (2), we will then get:-
$ \Rightarrow \left( {\angle A + \angle C + \angle E} \right) + \left( {\angle B + \angle F + \angle D} \right) = {180^ \circ } + {180^ \circ }$
Adding up the right hand side and writing the left hand side without the parenthesis, we will obtain the following expression:-
$ \Rightarrow \angle A + \angle B + \angle C + \angle D + \angle E + \angle F = {360^ \circ }$

Hence, proved.

Note: The students must note that trying to prove it without using will be a difficult task since, we are not even given about the length of the sides and thus we cannot divide the interior angles of the hexagon (if we join the vertices of the figure given to us, we will get a hexagon).
The students must note that they can easily prove that the sum of all the angles in a triangle is ${180^ \circ }$ by using the property of alternate angles on the straight line.