Question

1. if $overline{ef}$ bisects $angle ceb$, $mangle cef = (7x + 21)^circ$, and $mangle feb = (10x - 3)^circ$, find the measure of $angle deb$.

Explanation:

Step1: Use angle bisector property

Since \( \overline{EF} \) bisects \( \angle CEB \), we have \( m\angle CEF = m\angle FEB \).
So, \( 7x + 21 = 10x - 3 \).

Step2: Solve for \( x \)

Subtract \( 7x \) from both sides: \( 21 = 3x - 3 \).
Add 3 to both sides: \( 24 = 3x \).
Divide by 3: \( x = 8 \).

Step3: Find \( m\angle CEF \) and \( m\angle FEB \)

Substitute \( x = 8 \) into \( m\angle CEF = 7x + 21 \): \( 7(8) + 21 = 56 + 21 = 77^\circ \).
Since \( m\angle CEF = m\angle FEB \), \( m\angle FEB = 77^\circ \).

Step4: Find \( m\angle CEB \)

\( m\angle CEB = m\angle CEF + m\angle FEB = 77^\circ + 77^\circ = 154^\circ \).

Step5: Find \( m\angle DEB \)

Since \( \angle CEB \) and \( \angle DEB \) are supplementary (they form a linear pair), \( m\angle DEB = 180^\circ - m\angle CEB \).
Substitute \( m\angle CEB = 154^\circ \): \( 180 - 154 = 26^\circ \). Wait, no, wait. Wait, maybe I made a mistake here. Wait, looking at the diagram, \( \angle DEB \) and \( \angle CEB \)? Wait, no, maybe \( \angle DEB \) is related to vertical angles or adjacent angles. Wait, no, let's re-examine. Wait, the lines: \( A \) and \( B \) are opposite? Wait, no, the diagram: \( E \) is the intersection. So \( \angle AEC \) and \( \angle DEB \) are vertical angles? Wait, no, the problem is to find \( \angle DEB \). Wait, maybe I messed up the supplementary angle. Wait, let's check again.

Wait, \( EF \) bisects \( \angle CEB \), so \( \angle CEF = \angle FEB = 77^\circ \). Then, \( \angle DEB \): let's see, \( \angle DEB \) and \( \angle AEC \)? Wait, no, maybe \( \angle DEB \) is adjacent to \( \angle FEB \)? Wait, no, the diagram: points \( D, E, B \) and \( C, E, F \). Wait, maybe \( \angle DEB \) is equal to \( \angle AEC \), but no. Wait, maybe I made a mistake in step 5. Wait, let's re-express.

Wait, when \( EF \) bisects \( \angle CEB \), so \( \angle CEB = 2 \times \angle CEF \). Wait, no, \( \angle CEF = \angle FEB \), so \( \angle CEB = \angle CEF + \angle FEB = 2 \times \angle CEF \). Then, \( \angle DEB \): since \( \angle CEB \) and \( \angle AED \) are vertical angles? No, maybe \( \angle DEB \) is supplementary to \( \angle CEB \)? Wait, no, if \( \angle CEB \) is 154°, then \( \angle DEB \) would be 180 - 154 = 26°, but that seems small. Wait, maybe I made a mistake in solving for \( x \). Wait, let's check step 2 again.

Step 2: \( 7x + 21 = 10x - 3 \)
Subtract \( 7x \): \( 21 = 3x - 3 \)
Add 3: \( 24 = 3x \)
Divide by 3: \( x = 8 \). That's correct.

Then \( \angle CEF = 7(8) + 21 = 56 + 21 = 77 \), \( \angle FEB = 10(8) - 3 = 80 - 3 = 77 \). So \( \angle CEB = 77 + 77 = 154 \). Then, \( \angle DEB \): looking at the diagram, \( D, E, B \) and \( C, E, F \). Wait, maybe \( \angle DEB \) is equal to \( \angle AEC \), but \( \angle AEC \) and \( \angle DEB \) are vertical angles? No, \( \angle AEC \) and \( \angle DEB \) are vertical angles? Wait, no, \( A \) and \( B \) are opposite? Wait, the lines: \( A \) and \( B \) are opposite rays? \( C \) and \( D \) are opposite rays? Wait, maybe \( \angle AEC \) and \( \angle DEB \) are vertical angles, and \( \angle CEB \) and \( \angle AED \) are vertical angles. Wait, no, let's think again.

Wait, the problem is to find \( m\angle DEB \). Let's see, \( EF \) bisects \( \angle CEB \), so \( \angle CEF = \angle FEB = 77^\circ \). Then, \( \angle DEB \): if \( \angle DEB \) is adjacent to \( \angle FEB \), but no, \( D, E, B \): \( D \) and \( B \) are on different rays. Wait, maybe \( \angle DEB \) is equal to \( \angle AEC \), but \( \angle AEC \) is vertical to \( \angle DEB \). Wait, no, maybe \( \angle DEB \) is supplementary to \( \angle CEB \)? Wait, 180 - 154 = 26, but that seems wrong. Wait, maybe I misread the problem. Wait, the problem says "find the measure of \( \angle DEB \)". Wait, maybe \( \angle DEB \) is equal to \( \angle CEF \) or \( \angle FEB \)? No, that doesn't make sense. Wait, maybe the diagram is different. Wait, maybe \( \angle DEB \) is a right angle? No, the problem doesn't say that. Wait, maybe I made a mistake in the angle bisector. Wait, angle bisector means it splits the angle into two equal parts, so \( \angle CEF = \angle FEB \), so 7x + 21 = 10x - 3, which gives x = 8, so each angle is 77, so \( \angle CEB = 154 \), then \( \angle DEB \) is 180 - 154 = 26? But that seems small. Wait, maybe the diagram has \( \angle DEB \) as the angle adjacent to \( \angle FEB \), but no, \( D \) is on a different ray. Wait, maybe the lines are \( AB \) and \( CD \) intersecting at \( E \), and \( EF \) bisecting \( \angle CEB \). So \( AB \) and \( CD \) are straight lines, so \( \angle AEC + \angle CEB = 180 \), but \( \angle DEB \) is vertical to \( \angle AEC \). Wait, no, \( \angle DEB \) is vertical to \( \angle AEC \), so \( \angle DEB = \angle AEC \). And \( \angle AEC = 180 - \angle CEB = 180 - 154 = 26 \). So \( \angle DEB = 26^\circ \). Wait, but that seems correct. Wait, maybe I was overcomplicating. So the steps are:

1. Set \( 7x + 21 = 10x - 3 \) (angle bisector)
2. Solve for \( x \): \( x = 8 \)
3. Find \( \angle CEF = 77^\circ \), \( \angle FEB = 77^\circ \), so \( \angle CEB = 154^\circ \)
4. \( \angle DEB \) is supplementary to \( \angle CEB \) (since \( CD \) is a straight line), so \( 180 - 154 = 26^\circ \)

Wait, but maybe the diagram is different. Wait, the problem says "find the measure of \( \angle DEB \)". So according to the calculations, it's 26 degrees. But let me check again.

Wait, \( x = 8 \), so \( \angle CEF = 7*8 +21 = 56 +21=77 \), \( \angle FEB=10*8 -3=80-3=77 \). So \( \angle CEB=77+77=154 \). Then, since \( \angle DEB \) and \( \angle CEB \) are adjacent and form a linear pair (if \( D, E, C \) are colinear? Wait, no, \( D \) and \( C \) are on opposite rays? Wait, the diagram: \( D \) and \( C \) are opposite? So \( CD \) is a straight line, so \( \angle DEB + \angle CEB = 180 \)? No, \( \angle DEB \) and \( \angle CEB \): if \( E \) is the intersection, and \( CD \) is a straight line, then \( \angle DEB + \angle BEC + \angle CEA = 180 \)? No, maybe \( AB \) and \( CD \) are intersecting at \( E \), so \( \angle AEC = \angle DEB \) (vertical angles), and \( \angle AEC + \angle CEB = 180 \) (linear pair). So \( \angle DEB = 180 - \angle CEB = 180 - 154 = 26 \). Yes, that makes sense.

Answer:

\( 26^\circ \)