Exercise 13.1

1. Construct the following angles at the initial point of a given ray and justify the construction.

(a) $$90^{\circ}$$

(b) $$45^{\circ}$$

Sol:

(a) $$90^{\circ}$$

Fig

1. Let AB be the given ray.

2. produce BA to D

3. Taking A as center draw a semi-circle with some radius.

4. With X and Y as center draw two intersecting arcs of the same radius.

(b) $$45^{\circ}$$

Fig

1. Construct $$90^{\circ}$$ with the given ray AB.

2. Bisect it from ∠BAD = $$45^{\circ}$$

2. Construct the following angles using ruler and compass and verify by measuring them by a protractor.

(a) $$30^{\circ}$$

(b) 22$$\frac{1}{2}^{\circ}$$

(c) $$15^{\circ}$$

(d) $$75^{\circ}$$

(e) $$105^{\circ}$$

f) $$135^{\circ}$$

Sol:

(a) $$30^{\circ}$$

Fig

Construct ∠ABY = $$60^{\circ}$$

Bisect ∠ABY, such that ∠ABC = ∠CBY  = $$30^{\circ}$$

(b) 22$$\frac{1}{2}^{\circ}$$

Fig

Construct ∠ABD = $$90^{\circ}$$

Bisect ∠ABD, such that ∠ABC = ∠CBD  = $$45^{\circ}$$

Bisect ∠ABC such that ∠ABE = ∠EBC = 22$$\frac{1}{2}^{\circ}$$.

(c) $$15^{\circ}$$

Fig

Construct ∠BAE = $$60^{\circ}$$

Bisect ∠BAE such that ∠BAC = ∠CAE = $$30^{\circ}$$.

Bisect ∠BAC such that ∠BAF = ∠FAC = $$15^{\circ}$$.

(d) $$75^{\circ}$$

Fig

Construct ∠BAC = $$60^{\circ}$$

Construct ∠CAD = $$60^{\circ}$$

Bisect ∠CAD such that ∠BAE = $$90^{\circ}$$.

Bisect ∠CAE such that ∠BAF = $$75^{\circ}$$.

(e) $$105^{\circ}$$

Fig

Construct ∠ABC = $$90^{\circ}$$

Construct ∠CBE = $$30^{\circ}$$

Bisect ∠CBE such that the angle formed ∠ABD = $$105^{\circ}$$.

f) $$135^{\circ}$$

Fig

Construct ∠ABC = $$120^{\circ}$$

Construct ∠CBD = $$30^{\circ}$$

Bisect ∠CBE such that the angle formed ∠ABE = $$135^{\circ}$$.

3. Construct an equilateral triangle, given its side of length of 4.5 cm and justify the construction.

Sol:

Fig

Steps:

1. Draw a line segment AB = 4.5 cm

2. With B and A as centers draw two arcs of radius 4.5 cm meeting at C.

3. Join C to A and B

4. ΔABC is the required triangle. 

4. Construct an isosceles triangle, given its base and base angle and justify the construction.

[Hint: You can take any measure of side and angle]

Sol:

Fig

Steps:

1. Draw a line segment AB of any given length.

2. Construct ∠BAX and  ∠ABY  at A and B such that ∠A = ∠B.

3. Ax and BY will intersect at C.

4. ΔABC is the required triangle.