Question

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

Answer 1

Hint: 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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