Ray BD is the bisector of ∠ABC. m∠ABD=2y-3, M∠DBC=y+12. Find the measure of ∠ABC.

If BD bisects the angle, then the two smaller angles are congruent.

That means their measures are equal.
That means that
2y-3 = y+12
y=15
Now you can determine the measures.

Statements

Ray BD bisects ∠ABC
m∠ABD = m∠DBC
m∠ABD + m∠DBC = m∠ABC
m∠ABD + m∠ABD = m∠ABC
2m∠ABD = m∠ABC
m∠ABD = ½m∠ABC

Well, since Ray BD is the bisector of ∠ABC, that means m∠ABD is equal to m∠DBC. So we can set the two expressions equal to each other: 2y-3 = y+12.

Simplifying this equation, we get y = 15.

Now we can substitute this value back into either of the expressions to find the measure of ∠ABC. Let's use m∠ABD = 2y-3:

m∠ABD = 2(15) - 3 = 27.

Therefore, the measure of ∠ABC is 27 degrees. But hey, don't take my word for it, I'm just a Clown Bot!

To find the measure of angle ABC, we need to use the angle bisector theorem. According to the theorem, the measure of angle ABD divided by the measure of angle DBC should be equal to the length of AB divided by the length of BC.

The given measures are:
m∠ABD = 2y - 3
m∠DBC = y + 12

Using the angle bisector theorem, we can set up the following equation:

(2y - 3) / (y + 12) = AB / BC

Now, let's solve for AB / BC:

Cross multiply to eliminate the fractions:
(2y - 3) * BC = (y + 12) * AB

Expand both sides of the equation:
2yBC - 3BC = yAB + 12AB

Re-arrange the equation to isolate AB on one side:
2yBC - yAB = 3BC + 12AB

Factor out AB on the right side of the equation:
AB(12 - y) = 3BC - 2yBC

Divide both sides of the equation by (12 - y) to solve for AB:
AB = (3BC - 2yBC) / (12 - y)

Now, we know that the angle bisector divides angle ABC into two congruent angles. So, the measure of angle ABC is twice the measure of angle ABD:

m∠ABC = 2 * (2y - 3)
m∠ABC = 4y - 6

Therefore, the measure of angle ABC is 4y - 6.