https://file.aiquickdraw.com/mathbot/file/1729634668305dutille9.jpg

In the diagram, we see that there is an angle at point \( C \). The angles \( \angle BCD \) (a straight angle) and \( \angle ECD \) (labeled as \( 10x + 5^\circ \)) add up to a straight line, which means their sum must be \( 180^\circ \).

The angle \( \angle BCD = 75^\circ \) (as given), and the other angle \( \angle ECD = 10x + 5^\circ \).

Using the fact that these angles form a straight line:
\[
75^\circ + (10x + 5^\circ) = 180^\circ
\]

Simplify the equation:
\[
75 + 10x + 5 = 180
\]
\[
10x + 80 = 180
\]

So the correct equation to solve for \( x \) is **\( 10x + 80 = 180 \)**.

Now, let's solve for \( x \):
\[
10x + 80 = 180
\]
\[
10x = 180 - 80
\]
\[
10x = 100
\]
\[
x = \frac{100}{10} = 10
\]

Thus, \( x = 10 \).

Let me know if you'd like more details or explanations! Here are 5 related questions to extend this concept:

1. What are supplementary angles, and how do they relate to straight lines?
2. How would the equation change if the angle was \( 12x + 5^\circ \) instead?
3. What is the general method to solve for an unknown in a linear equation?
4. Can you derive \( x \) if the angle was \( (15x + 10)^\circ \) instead of \( 10x + 5^\circ \)?
5. How do you verify that your solution for \( x \) is correct in this type of problem?

**Tip**: When working with angles on a straight line, always remember their sum is \( 180^\circ \). This helps in setting up equations easily.

Solve for x in the angle diagram where the angles form a straight line, including the expression (10x + 5)° and the given angle of 75°.

Learn how to solve for x using supplementary angles in a straight line geometry problem. The problem involves the equation 10x + 80 = 180 and is suitable for students in Grades 6-8.

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