Question

if ∠dfe is a right angle, m∠cfd = (4x)°, m∠cfe = (9x − 14)°, find each measurment below.
image with angle diagram
x=
$m\angle dfe = $
$m\angle cfd = $
$m\angle cfe = $

Explanation:

Step1: Analyze angle relationship

Since \( \angle DFE \) is a right angle, \( m\angle DFE = 90^\circ \). Also, \( \angle CFD + \angle CFE=\angle DFE \), so \( (4x)^\circ+(9x - 14)^\circ = 90^\circ \).

Step2: Solve for \( x \)

Combine like terms: \( 4x+9x-14 = 90 \)
\( 13x-14 = 90 \)
Add 14 to both sides: \( 13x=90 + 14=104 \)
Divide by 13: \( x=\frac{104}{13}=8 \)

Step3: Find \( m\angle DFE \)

Given \( \angle DFE \) is right angle, so \( m\angle DFE = 90^\circ \)

Step4: Find \( m\angle CFD \)

Substitute \( x = 8 \) into \( m\angle CFD=(4x)^\circ \): \( 4\times8 = 32^\circ \)

Step5: Find \( m\angle CFE \)

Substitute \( x = 8 \) into \( m\angle CFE=(9x - 14)^\circ \): \( 9\times8-14=72 - 14 = 58^\circ \)

Answer:

\( x = 8 \)
\( m\angle DFE = 90 \)
\( m\angle CFD = 32 \)
\( m\angle CFE = 58 \)