Question

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 \qquad statement \qquad reason
1 \qquad $\overline{bc} \parallel \overline{ad}$ \qquad given
\qquad $\overline{ab} \parallel \overline{cd}$ \qquad
\boxed{} type of statement

Explanation:

Step1: Identify alternate interior angles

Since $\overline{BC} \parallel \overline{AD}$ and $\overline{AC}$ is a transversal, $\angle BCA = \angle DAC$.
Since $\overline{AB} \parallel \overline{CD}$ and $\overline{AC}$ is a transversal, $\angle BAC = \angle DCA$.

Step2: Identify common side

$\overline{AC} = \overline{CA}$ (Reflexive Property of Congruence)

Step3: Prove triangle congruence

By the ASA (Angle-Side-Angle) Congruence Postulate, $\triangle ABC \cong \triangle CDA$.

Step4: Corresponding parts of congruent triangles

Corresponding angles of congruent triangles are congruent, so $\angle ABC = \angle CDA$.

Step5: Use angle addition postulate

$\angle BCD = \angle BCA + \angle DCA$, $\angle BAD = \angle BAC + \angle DAC$.
Substitute congruent angles: $\angle BCD = \angle DAC + \angle BAC = \angle BAD$.

Answer:

$\angle BCD \cong \angle BAD$ is proven.