The diagram shows an isosceles triangle $\triangle ABC$, where $AB = AC$. The measure of angle $A$, denoted as $m\angle A$, is given by $(4x + 28)^\circ$. Since $\triangle ABC$ is isosceles, the two base angles, $\angle B$ and $\angle C$, are congruent.

We know that the sum of the angles in any triangle is $180^\circ$. Let the measure of angles $B$ and $C$ each be $\theta$. Therefore, we have the equation:

$m\angle A + m\angle B + m\angle C = 180^\circ$

Since $m\angle B = m\angle C = \theta$, the equation becomes:

$(4x + 28) + 2\theta = 180$

Now, solve for $\theta$:

$2\theta = 180 - (4x + 28)$ $2\theta = 152 - 4x$ $\theta = \frac{152 - 4x}{2} = 76 - 2x$

Since $\theta$ represents the base angles $B$ and $C$, we now have an expression for these angles in terms of $x$.

Let me know if you'd like me to continue solving for $x$ or if you need clarification on any steps so far.

Here are 5 follow-up questions related to this problem:

1. How do we derive the sum of interior angles of any triangle?
2. What property of isosceles triangles makes $\angle B$ and $\angle C$ equal?
3. How can we verify that the equation setup for solving $x$ is correct?
4. What happens to the values of the angles when $x$ is increased?
5. How would the problem change if it were not an isosceles triangle?

Tip: In any triangle, identifying congruent angles can simplify the calculation of unknown values using symmetry properties.