Question

1. in the figure below points a, e, and d are on the same line. what is the measure of \\( \angle bec \\)?

a) \\( 35 ^ { \circ } \\)
b) \\( 45 ^ { \circ } \\)
c) \\( 55 ^ { \circ } \\)
d) \\( 65 ^ { \circ } \\)
e) \\( 85 ^ { \circ } \\)

Explanation:

Step1: Recall the property of a straight - line angle

Since points \(A\), \(E\), and \(D\) are on the same line, \(\angle AEB+\angle BEC+\angle CED = 180^{\circ}\) (a straight - line angle measures \(180^{\circ}\)).

Step2: Substitute the known angles

We know that \(\angle AEB = 45^{\circ}\) and \(\angle CED=80^{\circ}\). Let \(\angle BEC=x\). Then the equation becomes \(45^{\circ}+x + 80^{\circ}=180^{\circ}\).

Step3: Solve for \(x\)

Combine like terms: \(x+125^{\circ}=180^{\circ}\). Subtract \(125^{\circ}\) from both sides of the equation: \(x=\angle BEC=180^{\circ}-125^{\circ}\).

Answer:

\(\angle BEC = 55^{\circ}\), so the answer is C.