Question

The angle bisectors of angle D and angle C intersect at O as shown in the figure. angle A = 75°
and angle B=60°. Find the measure of angle DOC.
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Answer 1

Answer:

Let $x$ = ∠DOC, $y$ = ∠ODC = ∠ODA, $z$ = ∠OCD = ∠OCB.

In ∆COD,

$x+y+z = 180$ (Sum of all interior angle of triangle is 180)

$z=180-x-y$ -------(i)

since,

$∠DAB + ∠ABC +∠BCD + ∠CDA = 360$ (Sum of all interior angles of quadrilateral is 360)

therefore,

$75 + 60 + z + z + y + y = 360 \\ 135 + 2z + 2y = 360 \\ 135 + 2(z + y) = 360$

from equation (i),

$135 + 2(180 - x - y + y) = 360 \\ 135 + 2(180 - x) = 360 \\ 135 + 360 - 2x = 360 \\ - 2x = 360 - 360 - 135 \\ - 2x = - 135 \\ 2x = 135 \\ x = \frac{135}{2} = 67.5$

since,

$∠DOC = x \\ ∠DOC = 67.5$