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Theory:

In this lesson, let us construct some special angles by without using the protractor.
 
i) Construction of \(60°\) angle:
 
Step 1: First draw a straight line, and mark a point \(A\) on it.
 
pic 18.svg
 
Step 2: Take the point \(A\) as centre, draw an arc of convenient radius to the line to meet at a point \(B\).
 
pic 19.svg
 
Step 3: With the same radius, take \(B\) as centre, draw an arc to cut the previous arc at \(C\).
 
pic 20.svg
 
Step 4: Join \(AC\). Then \(∠BAC\) is the required angle with the measure \(60°\).
 
pic 21.svg
 
 
ii) Construction of \(135°\) angle:
 
Step 1: First draw a straight line, and mark a point \(A\) on it.
 
pic 22.svg
 
Step 2: Take the point \(A\) as centre, draw an arc as below which meet at a point \(B\).
 
pic 23.svg
 
Step 3: Take radius, and draw an arc for \(60°\), as shown above. This cuts an arc at point \(C\).
 
pic 24.svg
 
Step 4: Now take \(C\) as centre, draw an arc to with the same radius, point it as \(D\). Here the point \(D\) gives the angles of \(120°\).
 
pic 25.svg
 
Step 5: With the same radius, draw an arc from the point \(D\) that cuts the \(E\) which gives the \(180°\).
 
pic 26.svg
 
Step 6: Finally, draw the angle bisector between the two points \(D\) \((120°)\) and \(E\) \((180)\). The angle bisector separates the two points \(D\) and \(E\) and gives the angle \(150°\). Now again find the angle bisector between \(150°\) and \(120°\), the resultant gives the required angle \(135°\).
 
pic 27.svg
 
The measure of \(135°\) angle, can be drawn in many ways, without using the protractor. Refer the below video for further information and clarification to construct the \(135°\) angle.
 
Construct \(135°\) degree using ruler and compass: