Question

n the diagram below, \\( \overline { b e } \perp \overline { e d } , \overline { a c } \cong \overline { b c } \\), and \\( \mathrm { m } \angle d = 48 ^ { \circ } \\). find \\( \mathrm { m } \angle a \\).
answer attempt 1 out of 2
\\( \mathrm { m } \angle a = \\) submit answer

Explanation:

Step1: Find \(m\angle DCE\)

In \(\triangle DCE\), since \(\angle E = 90^{\circ}\) and \(\angle D=48^{\circ}\), using the angle - sum property of a triangle (\(\angle D+\angle E+\angle DCE = 180^{\circ}\)), we have \(\angle DCE=180^{\circ}-\angle D - \angle E\).
Substitute \(\angle D = 48^{\circ}\) and \(\angle E = 90^{\circ}\) into the formula: \(\angle DCE=180^{\circ}-48^{\circ}-90^{\circ}=42^{\circ}\).

Step2: Use vertical angles

\(\angle BCA\) and \(\angle DCE\) are vertical angles. Vertical angles are equal. So \(m\angle BCA=m\angle DCE = 42^{\circ}\).

Step3: Use the property of isosceles triangle

Since \(\overline{AC}\cong\overline{BC}\), \(\triangle ABC\) is an isosceles triangle with \(\angle A=\angle B\). Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle BCA = 180^{\circ}\)) and \(\angle A=\angle B\), we can write \(2\angle A+\angle BCA=180^{\circ}\).
Substitute \(\angle BCA = 42^{\circ}\) into the formula: \(2\angle A+42^{\circ}=180^{\circ}\).
Then \(2\angle A=180^{\circ}- 42^{\circ}=138^{\circ}\), and \(\angle A=\frac{138^{\circ}}{2}=69^{\circ}\).

Answer:

\(69\)