**Construction Of An Angle Using Compass And Ruler**

**To draw an angle equal to a given angle**

In this section, we will learn how to construct angles of 60º, 30º, 90º, 45º and 120º with the help of ruler and compasses only.

### Construction Of Some Standard Angles

**Construction of an Angle of 60º**

In order to construct an angle of 60º with the help of ruler and compasses only, we follow the following steps :

**Steps of Construction**

**Step I:** Draw a ray OA.

**Step II:** With centre O and any radius draw an arc PQ with the help of compasses, cutting the ray OA at P.

**Step III:** With centre P and the same radius draw an arc cutting the arc PQ at R.

**Step IV:** Join OR and produce it to obtain ray OB.

The angle ∠AOB so obtained is the angle of measure 60º.

**Justification:** In above figure, join PR.

In ΔOPR, we have

OP = OR = PR

⇒ ΔOPR is an equilateral triangle.

⇒ ∠POR = 60º

⇒ ∠AOB = 60º [∵ ∠POR = ∠AOB]

**(ii) Construction of An Angle of 30º**

**Steps of Construction:**

**Step I:** Draw ∠AOB = 60º by using the steps mentioned above.

**Step II:** With centre O and any convenient radius draw an arc cutting OA and OB at P and Q respectively.

**Step III:** With centre P and radius more than \(\frac { 1 }{ 2 } \)(PQ), draw an arc in the interior of ∠AOB.

**Step IV:** With centre Q and the same radius, as in step III, draw another arc intersecting the arc in step III at R.

**Step V:** Join OR and product it to any point C.

**Step VI:** The angle ∠AOC is the angle of measure 30º.

**(iii) Construction of An Angle of 90º**

**Steps of Construction:**

**Step I:** Draw a ray OA.

**Step II:** With O as centre and any convenient radius, draw an arc, cutting OA at P.

**Step III:** With P as centre and the same radius, an arc cutting the arc drawn in step II at Q.

**Step IV:** With Q as centre and the same radius as in steps II and III, draw an arc, cutting the arc drawn in step II at R.

**Step V:** With Q as centre and the same radius, draw an arc.

**Step VI:** With R as centre and the same radius, draw an arc, cutting the arc drawn in step V at B.

**Step VII:** Draw OB and produce it to C. ∠AOC is the angle of measure 90º.

**(iv) Construction of An Angle of 45º**

**Steps of Construction:**

**Step I:** Draw ∠AOB = 90º by following the steps given above.

**Step II:** Draw OC, the bisector of ∠AOB.

The angle ∠AOC so obtained is the required angle of measure 45º.

**(v) Construction of An Angle of 120º**

**Steps of Construction:**

**Step I:** Draw a ray OA.

**Step II:** With O as centre and any convenient radius, draw an arc cutting OA at P.

**Step III:** With P as centre and the same radius draw an arc, cutting the first arc at Q.

**Step IV:** With Q as centre and the same radius, draw an arc, cutting the arc drawn in step II at R.

**Step V:** Join OR and produce it to any point C. ∠AOC so obtained is the angle of measure 120º.

**Read More:**

- Construction of an Equilateral Triangle
- Construction Of Similar Triangle As Per Given Scale Factor
- Construction Of A Line Segment
- Construction Of The Bisector Of A Given Angle
- Construction Of Perpendicular Bisector Of A Line Segment