Question

given abcd is a trapezoid,
(overline{ba} cong overline{cd})
prove: (overline{bd} cong overline{ca})
assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Identify Trapezoid Properties

ABCD is a trapezoid, so \( AB \parallel CD \) (definition of trapezoid: a quadrilateral with at least one pair of parallel sides). Also, given \( \overline{BA} \cong \overline{CD} \).

Step2: Identify Congruent Angles

Since \( AB \parallel CD \) and \( AD \) is a transversal, \( \angle BAD \cong \angle CDA \) (alternate interior angles theorem).

Step3: Identify Common Side

\( \overline{AD} \cong \overline{AD} \) (reflexive property of congruence).

Step4: Prove Triangles Congruent

In \( \triangle BAD \) and \( \triangle CDA \):

• \( \overline{BA} \cong \overline{CD} \) (given)
• \( \angle BAD \cong \angle CDA \) (alternate interior angles)
• \( \overline{AD} \cong \overline{AD} \) (reflexive property)

By SAS (Side - Angle - Side) congruence criterion, \( \triangle BAD \cong \triangle CDA \).

Step5: Conclude Segment Congruence

Since \( \triangle BAD \cong \triangle CDA \), corresponding parts \( \overline{BD} \cong \overline{CA} \) (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Statements and Reasons Table:

Statements	Reasons
2. \( \angle BAD \cong \angle CDA \)	2. Alternate Interior Angles Theorem (since \( AB \parallel CD \), \( AD \) transversal)
3. \( \overline{AD} \cong \overline{AD} \)	3. Reflexive Property of Congruence
4. \( \triangle BAD \cong \triangle CDA \)	4. SAS Congruence Criterion ( \( \overline{BA} \cong \overline{CD} \), \( \angle BAD \cong \angle CDA \), \( \overline{AD} \cong \overline{AD} \))
5. \( \overline{BD} \cong \overline{CA} \)	5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

To assemble the proof by dragging tiles, the order of statements (and corresponding reasons) should follow the above logical flow. The key is to first establish the parallel sides and given congruent side, then the congruent angles, then the common side, prove triangle congruence, and finally conclude the congruence of \( \overline{BD} \) and \( \overline{CA} \).

Answer:

The proof is assembled by showing \( \triangle BAD \cong \triangle CDA \) using SAS (with \( \overline{BA} \cong \overline{CD} \), \( \angle BAD \cong \angle CDA \), \( \overline{AD} \cong \overline{AD} \)) and then using CPCTC to conclude \( \overline{BD} \cong \overline{CA} \). The statements and reasons follow the order: 1. Given trapezoid and \( \overline{BA} \cong \overline{CD} \); 2. \( \angle BAD \cong \angle CDA \) (alternate interior angles); 3. \( \overline{AD} \cong \overline{AD} \) (reflexive); 4. \( \triangle BAD \cong \triangle CDA \) (SAS); 5. \( \overline{BD} \cong \overline{CA} \) (CPCTC).