Question

given: $overline{ab}||overline{dc}$ and $overline{ad}||overline{cb}$
prove: $overline{ab}congoverline{cb}$
1 $overline{ab}||overline{dc}$ given
2 $overline{ad}||overline{cb}$ given
3
4
5
arrange the cards to complete the proof.
mark sets of congruent sides and angles if the corresponding congruence statement is justified in the proof.
$\triangle adbcong\triangle cdb$
alternate interior angles are congruent
alternate interior angles are congruent
$overline{bd}congoverline{bd}$
$angle adbcongangle cbd$
asa triangle congruence theorem
$angle dbacongangle bdc$
cpctc

Explanation:

Step1: Alternate Interior angles

Since \( \overline{AB}\parallel\overline{DC}\), by the Alternate Interior Angles Theorem, \( \angle DBA\cong\angle BDC\).

Step2: Alternate Interior angles

Since \( \overline{AD}\parallel\overline{CB}\), by the Alternate Interior Angles Theorem, \( \angle ADB\cong\angle CBD\).

Step3: Reflexive Property

\( \overline{BD}\cong\overline{BD}\) (Reflexive Property of Congruence).

Step4: ASA Congruence

By the ASA (Angle - Side - Angle) Triangle Congruence Theorem, \( \triangle ADB\cong\triangle CDB\).

Step5: CPCTC

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \( \overline{AB}\cong\overline{CB}\).

Answer:

1. \( \angle DBA\cong\angle BDC\) (Alternate Interior angles are congruent)
2. \( \angle ADB\cong\angle CBD\) (Alternate Interior angles are congruent)
3. \( \overline{BD}\cong\overline{BD}\)
4. \( \triangle ADB\cong\triangle CDB\) (ASA triangle congruence theorem)
5. \( \overline{AB}\cong\overline{CB}\) (CPCTC)