Question

in the figure, line ae is perpendicular to ray bc and line df intersects line ae at point b. what is m\angle dbc? a 39° b 41° c 49° d 55°

Explanation:

Step1: Use the property of perpendicular lines

Since line \(AE\) is perpendicular to ray \(BC\), \(\angle EBC = 90^{\circ}\).

Step2: Use the angle - sum relationship

We know that \(\angle EBC=\angle EBF+\angle FBC\) (where \(\angle EBF = 49^{\circ}\) and \(\angle EBC = 90^{\circ}\)). Let \(\angle DBC=\angle FBC\) (vertical angles are equal).
We use the formula \(\angle FBC=\angle EBC-\angle EBF\).
Substitute \(\angle EBC = 90^{\circ}\) and \(\angle EBF = 49^{\circ}\) into the formula: \(\angle FBC=90^{\circ}- 49^{\circ}\).

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Since \(\angle DBC\) and \(\angle FBC\) are vertical angles, \(m\angle DBC = 41^{\circ}\)

Answer:

B. \(41^{\circ}\)