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The problem states that \( \overrightarrow{EF} \) bisects \( \angle CEB \), and gives the following angle measures:

\[
m\angle CEF = (7x + 21)^\circ \quad \text{and} \quad m\angle FEB = (10x - 3)^\circ.
\]

Since \( \overrightarrow{EF} \) bisects \( \angle CEB \), it means that the two angles \( \angle CEF \) and \( \angle FEB \) are equal. Therefore, we can set up the equation:

\[
7x + 21 = 10x - 3
\]

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

\[
21 + 3 = 10x - 7x
\]

\[
24 = 3x
\]

\[
x = \frac{24}{3} = 8
\]

Now that we have the value of \( x \), we can find the measures of \( \angle CEF \) and \( \angle FEB \):

\[
m\angle CEF = 7x + 21 = 7(8) + 21 = 56 + 21 = 77^\circ
\]

\[
m\angle FEB = 10x - 3 = 10(8) - 3 = 80 - 3 = 77^\circ
\]

Since \( \overrightarrow{EF} \) bisects \( \angle CEB \), the entire angle \( \angle CEB \) is twice the measure of \( \angle CEF \) or \( \angle FEB \):

\[
m\angle CEB = 77^\circ + 77^\circ = 154^\circ
\]

Finally, the measure of \( \angle DEB \), which is the angle supplementary to \( \angle CEB \), can be found by subtracting from 180°:

\[
m\angle DEB = 180^\circ - m\angle CEB = 180^\circ - 154^\circ = 26^\circ
\]

So, the measure of \( \angle DEB \) is \( \boxed{26^\circ} \).

Would you like more details or have any questions? Here are some related questions for further practice:

1. If \( \overrightarrow{EF} \) trisects \( \angle CEB \), what would be the new measures of \( \angle CEF \) and \( \angle FEB \)?
2. If the angles \( \angle CEF \) and \( \angle FEB \) were complementary, how would that change the equations and result?
3. How would you solve for the angles if \( m\angle CEF = 2(7x + 21) \) instead?
4. What would be the measure of \( \angle DEB \) if \( m\angle FEB \) was \( (10x - 3)^\circ + 5^\circ \)?
5. What properties of angles and bisectors are used in solving this type of problem?

**Tip:** Always verify your solution by substituting the value of \( x \) back into the original equations to ensure consistency.

If EF bisects ∠CEB, m∠CEF = (7x + 21)° and m∠FEB = (10x − 3)°, find the measure of ∠DEB.

Learn how to find the measure of ∠DEB when EF bisects ∠CEB and the angle measures are given as algebraic expressions. This problem involves solving equations, using the angle bisector theorem, and applying the concept of supplementary angles. Suitable for students in Grades 8-10.

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