Question

quadrilateral abcd is a parallelogram. complete the statements to prove that $overline{ab}congoverline{cd}$ and $overline{bc}congoverline{ad}$. statement: quadrilateral abcd is a parallelogram. reason: given $overline{ab}paralleloverline{cd}$ and $overline{bc}paralleloverline{ad}$. definition of a parallelogram. draw $overline{ac}$. $overline{ac}$ is a transversal that intersects two pairs of parallel lines $overline{ab}$ and $overline{cd}$ and $overline{ad}$ and $overline{bc}$. drawing a line segment. $angle cabcongangle acd$ and $angle bcacongangle cad$. $mangle cab = mangle acd$ and $mangle bca = mangle cad$. $overline{ac}=overline{ac}$. $\triangle abccong\triangle cda$. reflexive property of equality. $overline{ab}congoverline{cd}$ and $overline{bc}congoverline{ad}$. asa criterion for congruence. corresponding sides of congruent triangles are congruent.

Explanation:

Step1: Recall parallel - line properties

When two parallel lines are intersected by a transversal, alternate - interior angles are congruent. Since $AB\parallel CD$ and $AD\parallel BC$, and $AC$ is a transversal, $\angle CAB\cong\angle ACD$ and $\angle BCA\cong\angle CAD$ because of the alternate - interior angles theorem. So the reason for $m\angle CAB = m\angle ACD$ and $m\angle BCA=m\angle CAD$ is "alternate - interior angles theorem".

Step2: Identify common side

The line segment $AC$ is common to both $\triangle ABC$ and $\triangle CDA$. The reason for $AC = AC$ is the "Reflexive Property of Equality".

Step3: Prove triangle congruence

In $\triangle ABC$ and $\triangle CDA$, we have $\angle CAB\cong\angle ACD$, $AC = AC$, and $\angle BCA\cong\angle CAD$. By the Angle - Side - Angle (ASA) criterion for congruence, $\triangle ABC\cong\triangle CDA$.

Step4: Use congruent - triangle property

Since $\triangle ABC\cong\triangle CDA$, by the property that corresponding sides of congruent triangles are congruent, we have $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{AD}$.

Answer:

The reason for $m\angle CAB = m\angle ACD$ and $m\angle BCA=m\angle CAD$ is "alternate - interior angles theorem"; the reason for $AC = AC$ is "Reflexive Property of Equality"; $\triangle ABC\cong\triangle CDA$ by "ASA criterion for congruence"; $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{AD}$ because "corresponding sides of congruent triangles are congruent".