Find the measure of ∠BDC.
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Given:

∠A = 120°

∠DBC = 2 ∠ABD and ∠DCB = 2 ∠ACD.

To do:

We have to find the measure of ∠BDC.

Solution:

Let ∠ABC = 3x and ∠ACB = 3y

This implies,

∠ABD = x and ∠CBD = 2x

∠ACD = y and ∠DCB = 2y

We know that,

Sum of the angles in a triangle is 180°.

Therefore,

In triangle ABC,

$∠BAC + ∠ACB + ∠CBA = 180°$

$120° + ∠ACB + ∠CBA = 180°$

$∠ACB + ∠CBA = 180°-120°$

$3x + 3y = 60°$

$3(x+y) = 60°$

$x+y = \frac{60°}{3}$

$x+y=20°$

Let it be equation (1).

In triangle BDC,

$∠BDC + ∠CBD + ∠DBC = 180°$

 $∠BDC + 2x + 2y = 180°$

$∠BDC + 2(x+y) = 180°$     (From equation 1)

$∠BDC + 2(20°) = 180°$ 

$∠BDC + 40° = 180°$ 

$∠BDC = 180°-40°$

$∠BDC = 140°$

The measure of ∠BDC is 140°.

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Updated on: 10-Oct-2022

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