Question

refer to the figure. if ( mangle bdc = (8x + 12)^circ ) and ( mangle fdb = (12x - 32)^circ ), find ( mangle fde ). assume the figure is not drawn to scale.

Explanation:

Step1: Identify angle relationship

Angles \( \angle BDC \) and \( \angle FDB \) are supplementary (they form a linear pair along line \( FC \)), so their sum is \( 180^\circ \).
\[
(8x + 12) + (12x - 32) = 180
\]

Step2: Solve for \( x \)

Combine like terms:
\[
20x - 20 = 180
\]
Add 20 to both sides:
\[
20x = 200
\]
Divide by 20:
\[
x = 10
\]

Step3: Find \( m\angle BDC \)

Substitute \( x = 10 \) into \( m\angle BDC = (8x + 12)^\circ \):
\[
m\angle BDC = 8(10) + 12 = 92^\circ
\]

Step4: Determine \( m\angle FDE \)

\( \angle FDE \) and \( \angle BDC \) are vertical angles (or alternate angles, since \( BE \) and \( FC \) intersect at \( D \)), so they are equal. Thus, \( m\angle FDE = m\angle BDC = 92^\circ \).

Answer:

\( 92 \)