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 $180^\circ$.