Question

ad || bc
ad = cb
prove:
ab || dc

1. ad || bc, ad = cb
2. ac = ac
3. ∠2 = ∠3
4. △acd = △cab
5. ∠1 = ∠4
6. ab || dc

reflexive property of equality
if alternate interior angles are congruent, then lines are parallel.
if lines are parallel, then alternate interior angles are equal.
cpcte (corresponding parts of congruent triangles are equal)
sas (side-angle-side)
given

Explanation:

Step1: Identify the reason for $\overline{AC}=\overline{AC}$

Reflexive Property of Equality states that a segment is equal to itself. So, $\overline{AC}=\overline{AC}$ is due to the Reflexive Property of Equality.

Step2: Identify the reason for $\angle2 = \angle3$

Since $\overline{AD}\parallel\overline{BC}$, by the property "If Lines are Parallel, then Alternate Interior Angles are Equal", we have $\angle2=\angle3$.

Step3: Prove $\triangle ACD\cong\triangle CAB$

We have $\overline{AD}=\overline{CB}$ (given), $\angle2 = \angle3$ (from step 2), and $\overline{AC}=\overline{AC}$ (from step 1). By the SAS (Side - Angle - Side) congruence criterion, $\triangle ACD\cong\triangle CAB$.

Step4: Identify the reason for $\angle1=\angle4$

Since $\triangle ACD\cong\triangle CAB$, by CPCTE (Corresponding Parts of Congruent Triangles are Equal), $\angle1=\angle4$.

Step5: Prove $\overline{AB}\parallel\overline{DC}$

Since $\angle1=\angle4$, by the property "If Alternate Interior Angles are Congruent, then Lines are Parallel", we can conclude that $\overline{AB}\parallel\overline{DC}$.

Answer:

1. Given
2. Reflexive Property of Equality
3. If Lines are Parallel, then Alternate Interior Angles are Equal
4. SAS (Side - Angle - Side)
5. CPCTE (Corresponding Parts of Congruent Triangles are Equal)
6. If Alternate Interior Angles are Congruent, then Lines are Parallel