Question

in the diagram below, $overline{be}perpoverline{ed},overline{ac}congoverline{bc}$, and $mangle a = 74^{circ}$. find $mangle d$. you may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

Explanation:

Step1: Find angle B in triangle ABC

Since \(AC\cong BC\) in \(\triangle ABC\), \(\triangle ABC\) is isosceles. So \(\angle A=\angle B = 74^{\circ}\).

Step2: Find angle ACB in triangle ABC

The sum of interior - angles of a triangle is \(180^{\circ}\). In \(\triangle ABC\), \(\angle ACB=180^{\circ}-\angle A - \angle B\). Substituting \(\angle A = 74^{\circ}\) and \(\angle B = 74^{\circ}\), we get \(\angle ACB=180^{\circ}-74^{\circ}-74^{\circ}=32^{\circ}\).

Step3: Find angle DCE

\(\angle ACB\) and \(\angle DCE\) are vertical angles. Vertical angles are equal. So \(\angle DCE=\angle ACB = 32^{\circ}\).

Step4: Find angle D in right - triangle CDE

In right - triangle \(CDE\) with \(\angle E = 90^{\circ}\) (because \(BE\perp ED\)), and using the angle - sum property of a triangle (\(\angle D+\angle E+\angle DCE = 180^{\circ}\)). Substituting \(\angle E = 90^{\circ}\) and \(\angle DCE = 32^{\circ}\), we get \(\angle D=180^{\circ}-\angle E-\angle DCE=180^{\circ}-90^{\circ}-32^{\circ}=58^{\circ}\).

Answer:

\(58^{\circ}\)