Question

the secret beams on a bridge intersect in the pattern shown in the accompanying diagram where ab is parallel to ef. if ae || bf, m∠aef=(2x + 20.6)°, and m∠bfe=(x - 11.8)°, find m∠bfe.
note: the diagram is not drawn to scale.
20.6°
45.8°
134.8°

Explanation:

Step1: Use the property of parallel lines

Since \(AB\parallel EF\), \(\angle ABE=\angle BEF\) (alternate - interior angles). And \(AE\parallel BF\), so \(ABFE\) is a parallelogram. Then \(\angle AEF+\angle BFE = 180^{\circ}\) (consecutive - interior angles of parallel lines \(AE\parallel BF\)). Also, \(\angle ABE=(2x + 20.6)^{\circ}\) and \(\angle BFE=(x-11.8)^{\circ}\).

Step2: Set up the equation

Because \(AB\parallel EF\), \(\angle ABE=\angle BEF\) (alternate - interior angles). And \(AE\parallel BF\), so \(\angle AEB=\angle EBF\) (alternate - interior angles). Then \(ABFE\) is a parallelogram (a quadrilateral with both pairs of opposite sides parallel). Using the property of consecutive - interior angles (\(AE\parallel BF\)), we have \((2x + 20.6)+(x-11.8)=180\).
Simplify the left - hand side of the equation: \(2x+x+20.6 - 11.8=180\), \(3x+8.8 = 180\).
Subtract \(8.8\) from both sides: \(3x=180 - 8.8=171.2\).
Divide both sides by \(3\): \(x=\frac{171.2}{3}\approx57.07\).

Step3: Calculate \(\angle BFE\)

Substitute \(x = 57.07\) into \(\angle BFE=(x - 11.8)^{\circ}\). \(\angle BFE=(57.07-11.8)^{\circ}=45.27^{\circ}\approx45.3^{\circ}\)

Answer:

\(45.3^{\circ}\)