Question

in the diagram below, ( overline{be} cong overline{ec} ), ( mangle aeb = 21^{circ} ) and ( mangle ecd = 118^{circ} ). find ( mangle a ).

Explanation:

Step1: Find \(m\angle ECB\)

Since \(\angle ECD = 118^{\circ}\) and \(\angle ECB+\angle ECD = 180^{\circ}\) (linear - pair of angles), then \(m\angle ECB=180^{\circ}- 118^{\circ}=62^{\circ}\).

Step2: Use the property of isosceles triangle \(\triangle BEC\)

Because \(BE\cong EC\), \(\triangle BEC\) is isosceles. So \(m\angle EBC=m\angle ECB = 62^{\circ}\).

Step3: Find \(m\angle ABE\)

Since \(\angle ABE+\angle EBC = 180^{\circ}\) (linear - pair of angles), then \(m\angle ABE=180^{\circ}-62^{\circ}=118^{\circ}\).

Step4: Use the angle - sum property of \(\triangle ABE\)

In \(\triangle ABE\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). Let \(m\angle A=x\). Then \(x + 21^{\circ}+118^{\circ}=180^{\circ}\).

$$x=180^{\circ}-(21^{\circ}+118^{\circ})$$

$$x = 41^{\circ}$$

Answer:

\(41^{\circ}\)