Question

question in the diagram below, $overline{be}congoverline{ec}$, $mangle bec = 54^{circ}$ and $mangle a=42^{circ}$. find $mangle aeb$. step angle reason 1 $mangle bec = 54^{circ}$ given 2 $mangle a = 42^{circ}$ given try $mangle aeb=21^{circ}$ congruent angles

Explanation:

Step1: Find base - angles of isosceles triangle BEC

Since $\overline{BE}\cong\overline{EC}$, in $\triangle BEC$, $\angle EBC=\angle ECB$. Using the angle - sum property of a triangle ($\angle EBC+\angle ECB+\angle BEC = 180^{\circ}$), and $\angle BEC = 54^{\circ}$, we have $\angle EBC=\angle ECB=\frac{180 - 54}{2}=63^{\circ}$.

Step2: Use the exterior - angle property of triangle ABE

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. In $\triangle ABE$, $\angle EBC$ is an exterior angle. Let $\angle AEB=x$. Then $\angle EBC=\angle A+\angle AEB$. We know $\angle A = 42^{\circ}$ and $\angle EBC = 63^{\circ}$. So, $63^{\circ}=42^{\circ}+x$.

Step3: Solve for $\angle AEB$

Subtract $42^{\circ}$ from both sides of the equation $63^{\circ}=42^{\circ}+x$. We get $x=\angle AEB=63 - 42=21^{\circ}$.

Answer:

$21^{\circ}$