Question

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

Explanation:

Step1: Use angle bisector property

Since \( \overrightarrow{EF} \) bisects \( \angle CEB \), \( m\angle CEF = m\angle FEB \). So we set the two expressions equal:
\( 7x + 21 = 10x - 3 \)

Step2: Solve for \( x \)

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

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

Substitute \( x = 8 \) into \( m\angle CEF = 7x + 21 \):
\( m\angle CEF = 7(8) + 21 = 56 + 21 = 77^\circ \)
Since \( \angle CEF = \angle FEB \), \( m\angle FEB = 77^\circ \), so \( m\angle CEB = 77^\circ + 77^\circ = 154^\circ \)

Step4: Find \( m\angle DEB \)

From the diagram, \( \angle CEB \) and \( \angle DEB \) are supplementary? Wait, no, looking at the diagram, \( \angle CEA \) and \( \angle DEB \) might be vertical angles? Wait, actually, \( \angle CEB \) and \( \angle AED \) are vertical angles? Wait, no, let's re-examine. Wait, the lines \( AB \) and \( CD \) intersect at \( E \), so \( \angle CEB \) and \( \angle AED \) are vertical angles, but \( \angle DEB \) and \( \angle CEA \) are vertical angles? Wait, no, maybe \( \angle CEB \) and \( \angle DEB \) are adjacent and form a linear pair? Wait, no, the diagram: \( A \) and \( B \) are on a line, \( C \) and \( D \) are on another line, intersecting at \( E \). So \( \angle CEB \) and \( \angle DEB \): Wait, no, \( \overrightarrow{EF} \) bisects \( \angle CEB \), and we need \( \angle DEB \). Wait, actually, \( \angle CEB \) and \( \angle DEB \): Wait, maybe \( \angle CEB \) and \( \angle DEB \) are such that \( \angle CED \) is a straight line? Wait, no, let's think again. Wait, when two lines intersect, vertical angles are equal, and linear pairs are supplementary. Wait, maybe \( \angle CEB \) and \( \angle DEB \): Wait, no, the key is that after finding \( \angle CEB = 154^\circ \), then \( \angle DEB \) is supplementary to \( \angle CEB \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, no, let's check the diagram again. The lines \( AB \) (with points \( A, E, B \)) and \( CD \) (with points \( C, E, D \)) intersect at \( E \). So \( \angle CEB \) and \( \angle AED \) are vertical angles, \( \angle CEA \) and \( \angle DEB \) are vertical angles. Also, \( \angle CEB + \angle DEB = 180^\circ \)? Wait, no, \( \angle CED \) is a straight line, so \( \angle CEB + \angle DEB = 180^\circ \)? Wait, no, \( \angle CED \) is a straight line, so \( \angle CEB + \angle DEB = 180^\circ \)? Wait, no, \( \angle CEB \) and \( \angle DEB \) share the side \( EB \), and \( C - E - D \) is a straight line? Wait, yes! So \( \angle CEB \) and \( \angle DEB \) are supplementary, meaning their measures add up to \( 180^\circ \). Wait, no, that would mean \( \angle CEB + \angle DEB = 180^\circ \), so \( \angle DEB = 180^\circ - \angle CEB \). Wait, but we found \( \angle CEB = 154^\circ \), so \( \angle DEB = 180 - 154 = 26^\circ \)? Wait, no, that doesn't make sense. Wait, maybe I messed up the angle bisector. Wait, let's recheck Step 3. \( m\angle CEF = 7x + 21 \), \( x = 8 \), so \( 7*8=56 \), 56+21=77. \( m\angle FEB = 10x - 3 = 80 - 3 = 77 \). So \( \angle CEB = 77 + 77 = 154^\circ \). Now, looking at the diagram, \( \angle DEB \) and \( \angle CEA \) are vertical angles, and \( \angle CEA \) is supplementary to \( \angle CEB \)? Wait, no, \( \angle CEA + \angle CEB = 180^\circ \) because \( A - E - B \) is a straight line. So \( \angle CEA = 180 - 154 = 26^\circ \), so \( \angle DEB = \angle CEA = 26^\circ \)? Wait, no, that can't be. Wait, maybe the diagram is different. Wait, the problem says "find the measure of \( \angle DEB \)". Wait, maybe \( \angle DEB \) is equal to \( \angle CEF \) or something? No, let's re-examine the diagram. The lines: \( A \) and \( B \) are on a horizontal line? \( C \) is up, \( D \) is down, \( F \) is to the right. So \( EF \) bisects \( \angle CEB \), so \( \angle CEF = \angle FEB \). Then, \( \angle DEB \): since \( CD \) is a straight line? Wait, no, \( CD \) is a line with \( C \) and \( D \), so \( \angle CEB + \angle DEB = 180^\circ \)? Wait, no, \( \angle CED \) is a straight line, so \( \angle CEB + \angle DEB = 180^\circ \). So \( \angle DEB = 180 - 154 = 26^\circ \)? Wait, but that seems small. Wait, maybe I made a mistake in the angle relationship. Wait, alternatively, maybe \( \angle DEB \) is equal to \( \angle CEF \)? No, that doesn't make sense. Wait, let's check the calculation again. Step 1: \( 7x + 21 = 10x - 3 \). Subtract 7x: 21 = 3x - 3. Add 3: 24 = 3x. x = 8. Correct. Then \( \angle CEF = 7*8 +21=56+21=77 \), \( \angle FEB=10*8 -3=80-3=77 \). So \( \angle CEB=77+77=154 \). Now, looking at the diagram, \( \angle DEB \): since \( AB \) and \( CD \) intersect at \( E \), \( \angle DEB \) and \( \angle CEA \) are vertical angles. \( \angle CEA \) is adjacent to \( \angle CEB \), so \( \angle CEA + \angle CEB = 180^\circ \) (linear pair). So \( \angle CEA = 180 - 154 = 26^\circ \), so \( \angle DEB = 26^\circ \). Wait, but that seems correct? Wait, no, maybe the diagram is such that \( \angle DEB \) is equal to \( \angle CEF \)? No, that would be 77, but that would mean \( 154 + 77 = 231 \), which is more than 180. So no. So the correct measure is 26? Wait, no, wait, maybe I got the angle bisector wrong. Wait, the problem says \( \overrightarrow{EF} \) bisects \( \angle CEB \), so \( \angle CEF = \angle FEB \), which we used. Then, \( \angle DEB \): let's see, the lines \( AB \) and \( CD \) intersect at \( E \), so \( \angle AEC = \angle DEB \) (vertical angles). \( \angle AEC + \angle CEB = 180^\circ \) (linear pair), so \( \angle AEC = 180 - 154 = 26^\circ \), so \( \angle DEB = 26^\circ \). Yes, that makes sense.

Answer:

\( 26^\circ \)