https://file.aiquickdraw.com/mathbot/file/1726596548806wdlbwwgo.jpg

Let's analyze the diagram and solve for the value of \(\angle ACB\).

The diagram shows two parallel lines \(a\) and \(b\), which are intersected by two transversal lines \(c\) and \(d\). The angles given in the diagram are:
- \(\angle CAB = (3x + 20)^\circ\)
- \(\angle ABC = 5x^\circ\)

Since \(a \parallel b\) and lines \(c\) and \(d\) are transversals, we can apply the **alternate interior angle theorem**. According to this theorem, alternate interior angles formed by a transversal with two parallel lines are equal.

Therefore, the two angles at points \(C\) and \(B\) are equal:
\[
(3x + 20)^\circ = 5x^\circ
\]

Now, we solve for \(x\):
\[
3x + 20 = 5x
\]
Subtract \(3x\) from both sides:
\[
20 = 2x
\]
Divide by 2:
\[
x = 10
\]

Now that we know \(x = 10\), we can find the value of \(\angle ACB\):
\[
\angle ACB = 5x = 5(10) = 50^\circ
\]

Thus, the value of \(\angle ACB\) is \(50^\circ\).

Would you like further clarification on any step or more details?

Here are 5 related questions to explore:
1. How can we use the alternate exterior angle theorem in a similar diagram?
2. What happens if the lines are not parallel?
3. How do we handle more than two transversals intersecting parallel lines?
4. What other properties of angles can we use in parallel line configurations?
5. How do vertical angles relate to alternate interior angles?

**Tip**: Always check for angle theorems like alternate interior, corresponding, and co-interior angles when dealing with parallel lines and transversals. They can simplify many geometric problems.

This geometry problem involves finding the value of ∠ACB using the alternate interior angle theorem. Parallel lines are intersected by transversals, and algebraic expressions are used to solve for the angle. Suitable for students in Grades 8-10.

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