Question

finding an angle measure
what is the measure of \\( \angle cbe \\)?
\\( 36 ^ { \circ } \\)
\\( 72 ^ { \circ } \\)
\\( 108 ^ { \circ } \\)
\\( 144 ^ { \circ } \\)

Explanation:

Step1: Use the property of parallelogram

In parallelogram \(CBED\), \(CB\parallel DE\). So, \(\angle CBA=\angle DEF\) (corresponding angles). Also, \(\angle ABC + \angle CBE+\angle BED+\angle DEF = 180^{\circ}\) (linear pair). Since \(CB\parallel DE\), \(3x = \angle DEF\) (corresponding angles). And \(3x+2x= 180^{\circ}\) (consecutive interior angles on the same side of a transversal \(AEF\) for parallel lines \(CB\) and \(DE\)).

Step2: Solve for \(x\)

We have the equation \(3x + 2x=180^{\circ}\), which simplifies to \(5x = 180^{\circ}\). Then \(x=\frac{180^{\circ}}{5}=36^{\circ}\).

Step3: Find \(\angle CBE\)

\(\angle CBE = 180^{\circ}-3x\). Substitute \(x = 36^{\circ}\), we get \(\angle CBE=180^{\circ}-3\times36^{\circ}=180^{\circ}- 108^{\circ}=72^{\circ}\)

Answer:

\(72^{\circ}\)