It is know that BD is an angle bisector with...

x = 10 or x = -6

Step-by-step explanation: If ray BD bisects <ABC, then <ABD ≅ <DBC.

So we can setup the equation x² - 4x = 60.

To solve this polynomial equation, we need to set it equal to 0.

So we subtract 60 from both sides to get x² - 4x - 60 = 0.

On the left, we have a trinomial in a special form that can

be factored as the product of two binomials.

In the first position of each binomial,

we have the factors of the x squared term, x and x.

In the second position, we're looking for the factors

of -60  that add to -4 which are -10 and +6.

So we have (x - 10)(x + 6) = 0.

Whenever two terms are multiplied together to equal 0,

this means that either one or the other must equal 0.

So if (x - 10)(x + 6) = 0, then either x - 10 = 0 or x + 6 = 0.

Solving each equation from here, we find that x = 10 or x = -6.

If you plug both values of x in for the measures of the angles,

you will find out that both of these give us true statements.