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Draw using a compass and ruler $ 135^{\circ} $ at $ C $ in the line segment CD
Drawing an angle of 135 degrees at C of CD line segment given in the figure.
Steps of Construction:
1). Use a ruler and draw a Line segment CD of any convenient length.
2). Now use the compass and open it to any convenient radius. And with C as a center, draw an arc which cuts line segment CD at X.
3). Again use the compass and opened to the same radius and with X as a center, draw an arc which cuts the first arc at A.
4). Again use the compass and opened to the same radius and with A as a center, draw another arc which cuts the first arc at B.
5). Again use the compass and opened to the same radius and with A & B as centers, draw two arcs which cut each other at E.
6). Join CE and extend it to F.
7). Above formed angle, FCD = 90 Degrees.
8). Extend DC to Z.
9). Since ZD is a straight line, so formed Angle FCZ = 90 Degrees.
10). Again use the compass and open it to any convenient radius and with C as the center, draw an arc which cuts line segment CD at P and CF at Q.
11). Again use the compass and opened to the same radius (as of step 10) and with P & Q as centers and draw two arcs that cut each other at a point G.
12). Join CG and extend it to H.
13). HC is the bisector of angle FCD.
Therefore, Angle FCH = Angle HCD = ½ of Angle FCD = 45 Degree each
14). Now observe that:
Angle ZCF = 90
Angle FCH = 45
Add both the angles and we get
Angle ZCF + Angle FCH = 9O + 45 = Angle ZCH
So Angle ZCH = 135 degrees
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