The problem states that $ABCD$ is a rhombus. We need to find $AB$ and the measure of angle $ABC$. The given information includes: $\angle A = 12y$ $\angle B = 4x+15$ $\angle C = 4y-1$ $\angle D = 7x+2$

1. Problem Description

The problem states that $ABCD$ is a rhombus. We need to find $AB$ and the measure of angle $ABC$.

The given information includes:

$\angle A = 12y$

$\angle B = 4x+15$

$\angle C = 4y-1$

$\angle D = 7x+2$

2. Solution Steps

Since $ABCD$ is a rhombus, opposite angles are equal, and consecutive angles are supplementary (add up to 180 degrees).

Therefore:

$\angle A = \angle C$ and $\angle B = \angle D$

$12y = 4y - 1$ implies $8y = -1$, so $y = -1/8$.

Since an angle cannot be negative, there might be an error in the given expression for angle $C$. We will use the supplementary property of adjacent angles instead. Also we can't find length of $AB$ without more information.

We also know that consecutive angles in a rhombus are supplementary, so:

$\angle A + \angle B = 180$

$12y + 4x + 15 = 180$

And

$\angle B = \angle D$, so $4x + 15 = 7x + 2$, which implies $3x = 13$, so $x = 13/3$.

Substituting $x = 13/3$ into the equation $12y + 4x + 15 = 180$, we get:

$12y + 4(13/3) + 15 = 180$

$12y + 52/3 + 45/3 = 540/3$

$12y = (540 - 52 - 45)/3$

$12y = 443/3$

$y = 443/36$

Now, we can find the measure of angles A and B:

$\angle A = 12y = 12(443/36) = 443/3 = 147.67$ degrees (approx)

$\angle B = 4x + 15 = 4(13/3) + 15 = 52/3 + 45/3 = 97/3 = 32.33$ degrees (approx)

$m\angle ABC = 4x+15 = \frac{97}{3}$

We check if the values are reasonable with $A+B = 180$:

$\frac{443}{3} + \frac{97}{3} = \frac{540}{3} = 180$

$m\angle ABC = \frac{97}{3}$ degrees.

3. Final Answer

m∠ABC = 97/3 degrees. We can't determine the length of AB with the information provided.