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Q.

In the given figure $2 b - a = 65^{\circ}$ and $∠ BOC = 90^{\circ},$ find the measure of

$∠ AOB, ∠ AOD$ and $∠ COD$ .

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a

∠ AOB = 135^{\circ}

,

∠ AOD = 120^{\circ}

and

∠ COD = 30^{\circ}

b

∠ AOB = 90^{\circ}

,

∠ AOD = 50^{\circ}

and

∠ COD = 145^{\circ}

c

∠ AOB = 35^{\circ}

,

∠ AOD = 100^{\circ}

and

∠ COD = 135^{\circ}

d

None of the above

answer is C.

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Detailed Solution

Given:

2 b - a = 65^{\circ}

…….(1)

∠ BOC = 90^{\circ}

From the given figure, we can say that,

∠ AOB = a^{\circ}

∠ AOD = 2 b^{\circ}

∠ COD = 2 a + b + 15

Since, lines are concurrent and the lines intersect at O. Thus,

∠ BOC + ∠ AOB + ∠ AOD + ∠ COD

= 360^{\circ}

90 + a + 2 b + 2 a + b + 15 = 360

3 a + 3 b = 255

a + b = 85

….(2)
Add equation 1 and 2 as,

2 b - a + a + b = 65 + 85

3 b = 150

b = 50^{\circ}

Now, put this value in equation 1 as,

2 b - a = 65^{\circ}

2 \times 50^{\circ} - a = 65^{\circ}

a = 100^{\circ} - 65^{\circ}

a = 35^{\circ}

Now, the measure of angles,

∠ AOB = a = 35^{\circ}

∠ AOD = 2 b = 100^{\circ}

∠ COD = 2 a + b + 15 = 70 + 50 + 15

\Rightarrow ∠ COD = 135^{\circ}

Thus,

∠ AOB = 35^{\circ}

,

∠ AOD = 100^{\circ}

and

∠ COD = 135^{\circ}

Hence, option 3 will be the correct option.

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