Question

I have three sides. One of my angles measures ${{15}^{\circ }}$. Another has a measure of ${{60}^{\circ }}$. What kind of polygon am I? If I am a triangle, then what kind of triangle am I?

Answer 1

Hint: We must first figure out the type of polygon. We know very well that a polygon having 3 sides is called a triangle. Then, we can calculate the measure of all of its interior angles to find the type of polygon.

Complete step by step solution:
We know that in any polygon of n sides, there are always n number of interior angles in that polygon. For example, a triangle has 3 sides and 3 angles, a parallelogram has 4 sides and 4 angles, a pentagon has 5 sides and 5 angles.
Also, we know that any polygon having 3 sides is called a triangle. Thus, this polygon will also have 3 angles.
We are given that one of the angles measures ${{15}^{\circ }}$, and the measure of another angle of this polygon is ${{60}^{\circ }}$. In the below figure, we have the schematic diagram of such a triangle.

Here the measurements of internal angle are $\angle A={{15}^{\circ }}$ and $\angle B={{60}^{\circ }}$.
We know that the sum of all internal angles of a triangle is ${{180}^{\circ }}$. Mathematically, we can write
$\angle A+\angle B+\angle C={{180}^{\circ }}$
So, by using the values of angles A and B, we can write.
${{15}^{\circ }}+{{60}^{\circ }}+\angle C={{180}^{\circ }}$
And thus, the value of angle C is,
$\angle C={{180}^{\circ }}-\left( {{15}^{\circ }}+{{60}^{\circ }} \right)$
$\Rightarrow \angle C={{105}^{\circ }}$
Here, we can see that the angle C is an obtuse angle. Thus, the triangle ABC is an obtuse angled triangle.
Also, we can see that all the three angles of triangle ABC are different, and no two angles are equal to one another. So, all the three sides of triangle ABC must also be of different lengths. Thus, the triangle ABC is also a scalene triangle.

Hence, we can say that the polygon is an obtuse angled scalene triangle.

Note:
We must be very clear that any angle that is greater than ${{90}^{\circ }}$ is called an obtuse angle, and a triangle that has an obtuse angle as one of its interior angles, is called an obtuse angled triangle. We can also calculate the measure of $\angle C$ using the exterior angle theorem.