Question

OA,OB,OC,OD,OE and OF are six roads and all join at a point 'O'. These roads make angles (∠A ∠B and ∠C ∠ D and ∠E ) according to the figure. Roads OD and OE are perpendicular to each other AD & CF are straight lines and intersect each other at 'O' if ∠A:∠B:∠C are in the ratio 2:3:4. A teacher showed this figure to all the students of maths and asked the following​

Answer 1

1. What is the angle between OB and OC roads ?

Road OB and road OC makes the ∠b

We are given,

∠A: ∠B : ∠C are in the ratio of 2:3:4 respectively.

∴Let , their common multiple be 'x'

$\angle COD + \angle BOC + \angle BOA = 180 \degree \\i.e \: \: \angle a + \angle b + \angle c = 180\degree$

- ( Angles forming linear pair )

$\longmapsto 2x + 3x + 4x = 180 \degree \\ \longmapsto \: 9x = 180 \degree \\ \longmapsto \: x = \frac{180}{9} \\ \longmapsto \: { \blue{x = 20 \degree}}$

Measure of ∠b = 3x

= 3 (20)

= 60°

Option ( C ) is Correct .

2. What is the value of ∠d ?

We are given,

AD and CF are straight lines.

which means,

$\angle AOF = \angle COD$

- ( Vertically opposite angles )

∠COD = 4x

= 4 (20)

= 80°

$\therefore AOF = 80 \degree \\ i.e \: \: \boxed { \blue{ ∠d = 80}}$

Option ( D ) is Correct .

3. What is the complementary angles out of the four option ?

We know that,

Complementary angles are the pair of angle with the sum of 90 ° .

We are given ,

AD and CF are straight lines and OE is perpendicular

Conclusion - There's only one complementary angle in the figure

$i.e \: \: \: \: { \blue {∠d + ∠e = 90° }}$

Option ( D ) is Correct.

Answer 2

Answers are:

a) (option c) 60 degrees

b) (option b) 40 degrees

c) (option d) ∠D and ∠E

Step-by-step explanation:

a) Since ∠A:∠B:∠C is in the ratio 2:3:4, let them be 2x, 3x, and 4x.

Thus, the sum of these angles is 180 degrees (angles form a straight line)

$2x + 3x + 4x = 180$

$9x =180$

$x =20$

Thus, ∠A = 2x = 40, ∠B = 3x = 60, and ∠C = 4x = 80

• So, the angle between OB and OC is b = 60 degrees. (option c)

b) ∠ D = ∠ A ( vertically opposite angle)

• So, ∠ D = 40 degrees(option b)

c) Complementary angles add up to give the sum 90 degrees.

• From the figure, the sum of ∠D and ∠E will give 90 degrees. (option d)