https://file.aiquickdraw.com/mathbot/file/1724907242089596odigq.jpg

To solve problem #39 from the image, let's break it down:

### Problem Statement:
Given \( m \angle ACB = 9x + 5 \) and \( m \angle DCB = 4x + 35 \), and that \( \angle ACB \) and \( \angle DCB \) are opposite rays, find \( m \angle DCE \).

### Solution:
Since \( \angle ACB \) and \( \angle DCB \) are opposite rays, they form a straight line. The sum of the angles on a straight line is \( 180^\circ \). Therefore, we can set up the equation:

\[
m \angle ACB + m \angle DCB = 180^\circ
\]

Substitute the given expressions:

\[
(9x + 5) + (4x + 35) = 180^\circ
\]

Combine like terms:

\[
13x + 40 = 180^\circ
\]

Solve for \( x \):

\[
13x = 180^\circ - 40 = 140^\circ
\]

\[
x = \frac{140^\circ}{13} \approx 10.77^\circ
\]

Now, we can find \( m \angle DCB \):

\[
m \angle DCB = 4x + 35 = 4(10.77) + 35 = 43.08 + 35 \approx 78.08^\circ
\]

Finally, since \( \angle DCE \) and \( \angle DCB \) are on the same straight line, \( \angle DCE \) is the supplement of \( m \angle DCB \):

\[
m \angle DCE = 180^\circ - m \angle DCB = 180^\circ - 78.08^\circ = 101.92^\circ
\]

### Final Answer:
\[
m \angle DCE \approx 101.92^\circ
\]

Would you like more details on any part of this solution?

### Relative Questions:
1. How do opposite rays form a straight angle?
2. What is the significance of supplementary angles in geometry?
3. Can you explain how to solve linear equations with one variable?
4. How does the calculation change if the angles given are not opposite rays?
5. What are other methods to solve for \( x \) in similar problems?

### Tip:
When dealing with angles formed by opposite rays, always remember that they add up to \( 180^\circ \), which can be used to set up a simple linear equation.

Learn how to solve problem #39 involving angles formed by opposite rays. This detailed solution uses the Angle Sum Theorem and Supplementary Angles Theorem to find angle DCE. Suitable for high school students.

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