Question

given: \\(\overline{bc} \parallel \overline{ad}\\) and \\(\overline{ab} \parallel \overline{cd}\\).
prove: \\(\angle bcd \cong \angle bad\\).
note: quadrilateral properties are not permitted in this proof.
step 1: \\(\

$$\begin{aligned}\\overline{bc} \\parallel \\overline{ad}\\\\\\overline{ab} \\parallel \\overline{cd}\\end{aligned}$$

\\), reason: given
step 2: \\(\angle bac \cong \angle dca\\), reason: parallel lines cut by a transversal form congruent alternate interior angles
step 3: \\(\overline{ac} \cong \overline{ca}\\), reason: reflexive property
step 4: \\(\angle cad \cong \angle bca\\), reason: parallel lines cut by a transversal form congruent alternate interior angles
step 5: \\(\triangle abc \cong \triangle cda\\), reason: asa
(there is a dropdown menu labeled type of statement and a diagram of a quadrilateral with vertices a, b, c, d, showing parallel sides and angle markings.)

Explanation:

Step1: Add congruent angle pairs

$\angle BAC + \angle CAD \cong \angle DCA + \angle BCA$

Step2: Identify composite angles

$\angle BAC + \angle CAD = \angle BAD$, $\angle DCA + \angle BCA = \angle BCD$

Step3: Substitute to prove congruence

$\angle BCD \cong \angle BAD$

Answer:

$\angle BCD \cong \angle BAD$