Question

proving trapezoid theorems given: abcd is a trapezoid. ba = cd prove: bd = ca assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Identify trapezoid type

Since $\overline{BA}=\overline{CD}$, $ABCD$ is an isosceles trapezoid.

Step2: Recall property of isosceles trapezoid

In an isosceles trapezoid $ABCD$ with $AB\parallel CD$, $\angle BAD=\angle CDA$ (base - angles of an isosceles trapezoid are equal).

Step3: Consider triangles

In $\triangle BAD$ and $\triangle CDA$, we have $\overline{BA}=\overline{CD}$ (given), $\angle BAD = \angle CDA$ (proven above), and $\overline{AD}=\overline{DA}$ (common side).

Step4: Apply congruence criterion

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

Step5: Use congruent - triangle property

Since $\triangle BAD\cong\triangle CDA$, corresponding parts of congruent triangles are equal. So, $\overline{BD}=\overline{CA}$.

Answer:

1. Statements: $ABCD$ is an isosceles trapezoid; Reason: Given $\overline{BA}=\overline{CD}$
2. Statements: $\angle BAD=\angle CDA$; Reason: Base - angles of an isosceles trapezoid are equal
3. Statements: $\overline{AD}=\overline{DA}$; Reason: Common side
4. Statements: $\triangle BAD\cong\triangle CDA$; Reason: SAS congruence criterion
5. Statements: $\overline{BD}=\overline{CA}$; Reason: Corresponding parts of congruent triangles are equal