Question

in the figure shown below, $overline{bc} parallel overline{de}$, $overline{af}$ is a transversal, $mangle cpq = 2x$ and $mangle eqf = 5x - 51$. find the measure of $angle fqd$. $angle fqd = square$ degrees

Explanation:

Step1: Identify Alternate Interior Angles

Since \( \overline{BC} \parallel \overline{DE} \) and \( \overline{AF} \) is a transversal, \( \angle CPQ \) and \( \angle EQ F \) are alternate interior angles. So, \( m\angle CPQ = m\angle EQF \).
\[ 2x = 5x - 51 \]

Step2: Solve for \( x \)

Subtract \( 2x \) from both sides:
\[ 0 = 3x - 51 \]
Add 51 to both sides:
\[ 3x = 51 \]
Divide by 3:
\[ x = 17 \]

Step3: Find \( m\angle EQF \)

Substitute \( x = 17 \) into \( 5x - 51 \):
\[ m\angle EQF = 5(17) - 51 = 85 - 51 = 34^\circ \]

Step4: Find \( m\angle FQD \)

\( \angle FQD \) and \( \angle EQF \) are supplementary (they form a linear pair), so:
\[ m\angle FQD + m\angle EQF = 180^\circ \]
\[ m\angle FQD = 180^\circ - 34^\circ = 146^\circ \]

Answer:

146