Question

in the figure, ed and ea are opposite rays, and eb bisects ∠aec. if ( mangle ced = 72^circ ) and ( mangle aeb = (7x - 2)^circ ), then what is the ( mangle bec )? ( mangle bec = )

Explanation:

Step1: Recall straight angle property

A straight angle (∠AED) is \(180^\circ\). So, \(m\angle AEB + m\angle BEC + m\angle CED = 180^\circ\). Since \(EB\) bisects \(\angle AEC\), \(m\angle AEB = m\angle BEC=(7x - 2)^\circ\).

Step2: Set up the equation

Substitute the known values: \((7x - 2)+(7x - 2)+72 = 180\).
Simplify the left - hand side: \(14x+68 = 180\).
Subtract 68 from both sides: \(14x=180 - 68=112\).
Divide both sides by 14: \(x=\frac{112}{14} = 8\).

Step3: Find \(m\angle BEC\)

Since \(m\angle BEC=(7x - 2)^\circ\), substitute \(x = 8\): \(7\times8-2=56 - 2=54^\circ\).

Answer:

\(54^\circ\)