In the given isosceles triangle ABC, we know the following:

• $\angle B = (3x + 32)^\circ$
• $\angle C = (87 - 2x)^\circ$
• $AB = AC$, so $\angle B = \angle C$.

Since $\angle B = \angle C$, we can set up an equation:

$3x + 32 = 87 - 2x$

Now, let's solve for $x$.

1. Add $2x$ to both sides:

$3x + 2x + 32 = 87$

$5x + 32 = 87$

2. Subtract 32 from both sides:

$5x = 87 - 32$

$5x = 55$

3. Divide both sides by 5:

$x = \frac{55}{5}$

$x = 11$

So, the value of $x$ is $11$.

Would you like a more detailed breakdown or have additional questions? Here are five related questions:

1. How can you verify if the value of $x$ is correct by substituting back into the angles?
2. What happens if the triangle is not isosceles? How would this change the approach?
3. What is the sum of the interior angles of a triangle?
4. How would you solve this problem if the triangle were scalene instead of isosceles?
5. Can the same method be used to solve for unknowns in other geometric shapes like quadrilaterals?

Tip: Always double-check your angle calculations when working with geometric shapes.