Question

1. if $overline{ef}$ bisects $angle aed$, $mangle aef = (4x + 3)^circ$, and $mangle fed = (7x - 33)^circ$, find $mangle ceb$.

Explanation:

Step1: Use Angle Bisector Definition

Since \( \overline{EF} \) bisects \( \angle AED \), \( \angle AEF = \angle FED \). So, \( 4x + 3 = 7x - 33 \).

Step2: Solve for \( x \)

Subtract \( 4x \) from both sides: \( 3 = 3x - 33 \).
Add 33 to both sides: \( 36 = 3x \).
Divide by 3: \( x = 12 \).

Step3: Find \( m\angle AEF \) and \( m\angle FED \)

Substitute \( x = 12 \) into \( 4x + 3 \): \( 4(12) + 3 = 51^\circ \).
So, \( m\angle AED = 2 \times 51^\circ = 102^\circ \).

Step4: Use Vertical Angles

\( \angle CEB \) and \( \angle AED \) are vertical angles, so they are equal. Thus, \( m\angle CEB = 102^\circ \).

Answer:

\( 102^\circ \)