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The diagram provided shows a triangle \( \triangle ABC \), with point \( D \) on the extended line beyond \( C \), making \( \angle BCD \) an exterior angle. We are tasked with finding the measure of this exterior angle, \( \angle BCD \).

In the diagram:
- \( \angle A = (13x + 18)^\circ \)
- \( \angle B = 5x^\circ \)
- \( \angle BCD = (19x - 3)^\circ \)

### Approach:
Using the exterior angle theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In this case:

\[
\angle BCD = \angle A + \angle B
\]

Substituting the given values for \( \angle A \) and \( \angle B \):

\[
(19x - 3)^\circ = (13x + 18)^\circ + 5x^\circ
\]

### Solving:
1. Combine like terms on the right-hand side:
   \[
   19x - 3 = 13x + 5x + 18
   \]
   \[
   19x - 3 = 18x + 18
   \]

2. Subtract \( 18x \) from both sides:
   \[
   x - 3 = 18
   \]

3. Add 3 to both sides:
   \[
   x = 21
   \]

### Finding \( \angle BCD \):
Now that we know \( x = 21 \), substitute this back into the expression for \( \angle BCD \):

\[
\angle BCD = 19x - 3 = 19(21) - 3 = 399 - 3 = 396^\circ
\]

Thus, the measure of exterior angle \( \angle BCD \) is **396°**.

Would you like more details or explanations? Here are 5 related questions you can explore:

1. What is the interior angle sum theorem, and how does it relate to triangles?
2. How would the problem change if the exterior angle theorem wasn't known?
3. Can you verify the solution by calculating the other angles of the triangle?
4. Why is \( \angle BCD \) greater than \( 180^\circ \) in this problem?
5. What happens if the value of \( x \) leads to unrealistic angle measures?

### Tip:
Always verify your answers by checking if they satisfy all geometric constraints (such as the sum of interior angles).

Use the diagram to find the measure of exterior angle BCD.

Solve for the exterior angle BCD in a triangle using the exterior angle theorem, where the interior angles are given by expressions in terms of x. This problem is suitable for students in Grades 8-10 and involves geometric principles and algebra.

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