Given that is the angle bisector of

Notice that AD also creates two supplementary angles, BDA and ADE. Let y be the measure of the smaller angle, ADE; then the measure of the larger angle, BDA, is 180° - y.

Since AD bisects angle A, we have by the law of sines

sin(A/2)/x = sin(y)/8

and

sin(A/2)/6 = sin(180° - y)/12

Now,

sin(180° - y) = sin(180°) cos(y) - cos(180°) sin(y) = sin(y)

so that

sin(A/2) = x/8 sin(y)

and

sin(A/2) = 6/12 sin(y) = 1/2 sin(y)

Solve for x :

x/8 sin(y) = 1/2 sin(y)

x/8 = 1/2

x = 8/2

x = 4

i believe the answer is the polygon, heptagon.

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