drag each reason to the correct location on the flow chart. not all reasons will be used given: $\overline{ab}\parallel\overline{cd}$ and $\overline{ad}\parallel\overline{bc}$ prove: $\overline{ab}\cong\overline{cd}$ and $\overline{ad}\cong\overline{bc}$ complete the flow chart proof. flow chart and reasons (sas, alternate exterior angles theorem, alternate interior angles theorem, cpctc, reflexive property of congruence, asa) are shown with geometric proof steps including constructing $\overline{ac}$, angle congruences, triangle congruence $\triangle acd\cong\triangle cab$, and final segment congruences

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Accepted Answer

# Explanation:
## Step1: Analyze ∠DAC ≅ ∠BCA
Since $ \overline{AD} \parallel \overline{BC} $ and $ \overline{AC} $ is a transversal, by the **alternate interior angles theorem**, $ \angle DAC \cong \angle BCA $.

## Step2: Analyze ∠ACD ≅ ∠BAC
Since $ \overline{AB} \parallel \overline{CD} $ and $ \overline{AC} $ is a transversal, by the **alternate interior angles theorem**, $ \angle ACD \cong \angle BAC $.

## Step3: Analyze $ \overline{AC} \cong \overline{AC} $
By the **reflexive property of congruence**, a segment is congruent to itself, so $ \overline{AC} \cong \overline{AC} $.

## Step4: Prove $ \triangle ACD \cong \triangle CAB $
We have two pairs of congruent angles ($ \angle DAC \cong \angle BCA $, $ \angle ACD \cong \angle BAC $) and a common side ($ \overline{AC} \cong \overline{AC} $). By the **ASA (Angle - Side - Angle)** congruence criterion, $ \triangle ACD \cong \triangle CAB $.

## Step5: Prove $ \overline{AB} \cong \overline{CD} $ and $ \overline{AD} \cong \overline{BC} $
Since $ \triangle ACD \cong \triangle CAB $, by the **CPCTC (Corresponding Parts of Congruent Triangles are Congruent)** theorem, the corresponding sides $ \overline{AB} \cong \overline{CD} $ and $ \overline{AD} \cong \overline{BC} $.

# Answer:
- For $ \angle DAC \cong \angle BCA $: alternate interior angles theorem
- For $ \angle ACD \cong \angle BAC $: alternate interior angles theorem
- For $ \overline{AC} \cong \overline{AC} $: reflexive property of congruence
- For $ \triangle ACD \cong \triangle CAB $: ASA
- For $ \overline{AB} \cong \overline{CD} $ and $ \overline{AD} \cong \overline{BC} $: CPCTC

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze ∠DAC ≅ ∠BCA

Step2: Analyze ∠ACD ≅ ∠BAC

Step3: Analyze \( \overline{AC} \cong \overline{AC} \)

Step4: Prove \( \triangle ACD \cong \triangle CAB \)

Step5: Prove \( \overline{AB} \cong \overline{CD} \) and \( \overline{AD} \cong \overline{BC} \)

Answer: