Question

question
in the diagram below, ( overline{be} cong overline{ec} ), ( mangle bec = 48^{circ} ) and ( mangle a = 40^{circ} ). find ( mangle aeb ).
you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.
answer attempt 2 out of 5
( mangle aeb = )

Explanation:

Step1: Find angles in isosceles triangle \( \triangle BEC \)

Since \( \overline{BE}\cong\overline{EC} \), \( \triangle BEC \) is isosceles. Using the angle - sum property of a triangle (\( \angle EBC+\angle ECB+\angle BEC = 180^{\circ} \)) and \( \angle EBC=\angle ECB \), we have \( \angle EBC=\frac{180^{\circ}-\angle BEC}{2} \). Substituting \( \angle BEC = 48^{\circ} \), we get \( \angle EBC=\frac{180 - 48}{2}=66^{\circ} \).

Step2: Use exterior - angle property

\( \angle EBC \) is an exterior angle of \( \triangle ABE \). By the exterior - angle property (\( \angle EBC=\angle A+\angle AEB \)). Given \( \angle A = 40^{\circ} \) and \( \angle EBC = 66^{\circ} \), we can solve for \( \angle AEB \) as \( \angle AEB=\angle EBC-\angle A \).

Step3: Calculate \( \angle AEB \)

Substitute \( \angle EBC = 66^{\circ} \) and \( \angle A = 40^{\circ} \) into \( \angle AEB=\angle EBC-\angle A \), so \( \angle AEB=66 - 40=26^{\circ} \).

Answer:

\( 26 \)