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The image shows a geometric problem where we are tasked with finding the measure of angle \( \angle ABD \). The given expressions for the angles are:

- \( \angle ABC = 2x \) (for the straight line segment \( BC \)),
- \( \angle ABD = 3x + 10 \).

Since \( \angle ABC \) and \( \angle ABD \) form a straight line, their sum must be 180°. This is because the angles along a straight line are supplementary.

We can set up the equation:
\[
(3x + 10) + 2x = 180
\]

Now, let's solve for \( x \):

1. Combine like terms:
\[
5x + 10 = 180
\]

2. Subtract 10 from both sides:
\[
5x = 170
\]

3. Divide by 5:
\[
x = 34
\]

Now, substitute \( x = 34 \) into the expression for \( \angle ABD \):
\[
\angle ABD = 3(34) + 10 = 102 + 10 = 112^\circ
\]

Thus, the measure of \( \angle ABD \) is \( 112^\circ \).

Would you like more details, or have any other questions?

Here are some related questions for further exploration:

1. How do you determine when angles are supplementary?
2. What is the definition of a straight angle in geometry?
3. If angle \( ABD \) were changed, how would the calculation adjust?
4. How are angle relationships used in real-world applications?
5. Can you calculate the angle \( \angle ABC \) based on this solution?

**Tip:** When dealing with straight-line angles, remember that their sum is always \( 180^\circ \).

This problem involves solving for the measure of angle ABD using algebra and the supplementary angles theorem. Given two expressions for angles along a straight line, learn how to set up an equation and solve for the unknown variable x.

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