Question

Write true or false. Give reason to your answer: An angle ${52.5^0}$ can be constructed.

Answer 1

Angles can be constructed using a compass and ruler, if the angles are multiples of ${7.5^0}$, for example, angles like ${30^0},{45^0},{60^0}$. But certain angles ${5^0}$, ${10^0}$, ${20^0}$, ${40^0}$, ${50^0}$cannot be constructed using a ruler and compass.

Complete step-by-step answer:
Now let’s check how the angle ${52.5^0}$can be constructed using a compass and ruler.
The given angle is ${52.5^0}$we have to divide this angle to a simpler form so that we can check if this given angle can be written as a product ${7.5^0}$. Then only we can say this angle can be constructed.

We can write ${52.5^0}$ as $\left( {{{45}^0} + {{7.5}^0}} \right)$ Or we can further expand it as \left\\{ {\left( {{{30}^0} + {{15}^0}} \right) + {{7.5}^0}} \right\\}we can see ${30^0},{15^0}$and ${7.5^0}$are multiples of${7.5^0}$. This satisfies our condition of construction using a ruler and compass.

So, the angle ${52.5^0}$can be constructed. Now by using a ruler and compass we can construct the given angle. This method is for checking whether the given angle can be constructed or not.

Now to construct the angle ${52.5^0}$we have to follow another step. First, construct an angle ${90^0}$. Then construct an angle ${120^0}$Now plot angle bisectors of ${90^0}$and ${120^0}$, we will get the angle${105^0}$. Now construct the angle bisector ${105^0}$ we will get${52.5^0}$.
The answer is true. Hence the given statement is true.

Note: We have to divide this angle to a simpler form so that we can check if this given angle can be written as a product ${7.5^0}$. Always check whether the given angle can be divided into angles and check whether they are the multiples of${7.5^0}$when it is needed to construct them using a compass and ruler.
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