Question

in the diagram below, \\(\overline{ad} \cong \overline{ae}\\), \\(\overline{ba} \parallel \overline{ce}\\), \\(\overline{cb} \parallel \overline{da}\\) and \\(\text{m}\angle dae = 54^{\circ}\\). find \\(\text{m}\angle bad\\).

Explanation:

Step1: Find base angles of $\triangle ADE$

Since $\overline{AD} \cong \overline{AE}$, $\triangle ADE$ is isosceles. The base angles are equal:
$$\text{m}\angle ADE = \text{m}\angle AED = \frac{180^\circ - 54^\circ}{2} = 63^\circ$$

Step2: Use parallel lines property

$\overline{CB} \parallel \overline{DA}$ and $\overline{BA} \parallel \overline{CE}$, so $ABCD$ is a parallelogram. $\angle ADE$ and $\angle BAD$ are same-side interior angles (supplementary) because $\overline{BA} \parallel \overline{CE}$:
$$\text{m}\angle BAD = 180^\circ - \text{m}\angle ADE = 180^\circ - 63^\circ = 117^\circ$$

Answer:

$\text{m}\angle BAD = 117^\circ$