7 th maths chapter 5 || 5.5

7th Standard Mathematics — Chapter 5: Geometry

Exercise 5.5 Solutions (Construction of Angles using Ruler and Compass Only)

1. Construct the following angles using ruler and compass only.

(i) Angle = 60°

Steps of Construction:

1. Draw a ray OA using a ruler.
2. With center O and any suitable radius, draw a semicircular arc cutting ray OA at point P.
3. With center P and the same radius, draw an arc intersecting the first arc at point Q.
4. Draw ray OB passing through point Q.
5. ∠AOB = 60° is the required angle.

(ii) Angle = 120°

Steps of Construction:

1. Draw ray OA with vertex O.
2. With center O and a fixed radius, draw a long arc cutting ray OA at P.
3. With center P and the same radius, draw an arc cutting the main arc at Q (60°).
4. With center Q and the same radius, cut the main arc further at point R (120°).
5. Draw ray OB passing through R to form ∠AOB = 120°.

(iii) Angle = 30°

Steps of Construction:

1. Construct a 60° angle (∠AOC) using the compass as in sub-question (i).
2. With centers at P (0° mark) and Q (60° mark), draw two intersecting arcs with equal radius.
3. Let the arcs intersect at point R.
4. Draw ray OB through R. Ray OB bisects 60° to give ∠AOB = 30°.

(iv) Angle = 90°

Steps of Construction:

1. Draw ray OA. Mark the 60° point Q and 120° point R on the main arc.
2. With centers Q and R and equal radius, draw two arcs intersecting above the main arc at point S.
3. Draw ray OB through S.
4. ∠AOB = 90° is the required right angle (bisector of the arc between 60° and 120°).

(v) Angle = 45°

Steps of Construction:

1. Construct a 90° angle (∠AOC) as described in sub-question (iv).
2. With centers at point P (0°) and point T (90° on the main arc), draw two intersecting arcs in the interior of the angle.
3. Let the arcs intersect at point E.
4. Draw ray OB through E to get ∠AOB = 45°.

(vi) Angle = 150°

Steps of Construction:

1. Extend ray OA backwards to form a straight line XOA (180°).
2. Construct the 120° mark R on the arc.
3. Bisect the region between 120° (point R) and 180° (straight line mark X) using two arcs intersecting at S.
4. Draw ray OB through S.
5. ∠AOB = 120° + 30° = 150°.

(vii) Angle = 135°

Steps of Construction:

1. Construct a straight line XOA (180°) and construct a perpendicular 90° ray.
2. The angle between the 90° mark and the 180° straight line is 90°.
3. Bisect this 90° region on the left side with two arcs intersecting at point E.
4. Draw ray OB through E.
5. ∠AOB = 90° + 45° = 135°.