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In the provided diagram, we are tasked with finding the measure of \(\angle ABC\). Here's how we can approach the solution:

1. The angle at point \(C\), labeled as \(\angle BCA\), is given as \(150^\circ\).
2. The interior angles of a triangle add up to \(180^\circ\).
3. From the diagram, the two other angles are \(\angle ABC\) (which we are solving for) and \(\angle BAC\), labeled as \(x\).

We will use the angle sum property of a triangle, which states that:

\[
\angle ABC + \angle BAC + \angle BCA = 180^\circ
\]

Substituting the known values:

\[
\angle ABC + x + 150^\circ = 180^\circ
\]

Simplifying this equation:

\[
\angle ABC + x = 30^\circ
\]

The diagram also shows that the length of side \(AB\) is labeled as \(x - 20\). This may suggest a relationship between the length and the angles, but for the specific question asking for \(\angle ABC\), we conclude from the above equation that:

\[
\angle ABC = 30^\circ - x
\]

At this point, we do not have enough information to directly calculate \(x\) without additional data about the side lengths or a further relationship between the angles. Therefore, the expression for \(\angle ABC\) in terms of \(x\) is \(30^\circ - x\).

Would you like a further explanation or additional details on solving this triangle?

### Related Questions:
1. How do we calculate the exact value of \(x\) in the triangle?
2. What are the general steps for solving triangles when given angle-side relationships?
3. How does the exterior angle theorem apply to this triangle?
4. How can the Law of Sines or Cosines be applied to this triangle?
5. What are the properties of an obtuse triangle like the one in this figure?

**Tip:** When solving for angles in a triangle, always remember that the sum of the internal angles must equal 180°.

What is the measure of ∠ABC in the given triangle, where ∠BCA = 150° and one side is labeled x - 20, with another angle labeled as x?

Learn how to find the measure of angle ABC in a triangle where ∠BCA is given as 150°. This problem applies the angle sum property of triangles and explores the relationship between side lengths and angles. Suitable for students in Grades 8-10.

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