Question

two-column proofs (continued)

1. fill-in the statements and reasons for the following proof.

given: $overline{de} \parallel \overline{av}$
$\triangle dav \cong \triangle eva$
prove: dave is an isosceles trapezoid
diagram of quadrilateral dave with vertices d, e, v, a; diagonals intersecting at f
\

$$\begin{tabular}{|c|c|} \\hline statement & reason \\\\ \\hline 1. $\\overline{de} \\parallel \\overline{av}$ & 1. given \\\\ \\hline 2. $\\triangle dav \\cong \\triangle eva$ & 2. given \\\\ \\hline 3. $\\overline{da} \\cong \\overline{ev}$ & 3. definition of congruent triangles (partial) \\\\ \\hline 4. & 4. definition of trapezoid (partial) \\\\ \\hline 5. & 5. definition of isosceles trapezoid (partial) \\\\ \\hline \\end{tabular}$$

possible statements:
dave is a trapezoid
$\overline{da} \cong \overline{ev}$
dave is an isosceles trapezoid
$\triangle dav \cong \triangle eva$
$\overline{de} \parallel \overline{av}$

possible reasons:
given
definition of isosceles trapezoid
given
definition of trapezoid
definition of congruent triangles

Explanation:

Step 1: Analyze Statement 4

We know that a trapezoid is defined as a quadrilateral with at least one pair of parallel sides. We are given that \(\overline{DE} \parallel \overline{AV}\), so to conclude that DAVE is a trapezoid, we use the definition of a trapezoid. The statement for step 4 should be "DAVE is a trapezoid" and the reason is "Definition of trapezoid".

Step 2: Analyze Statement 5

An isosceles trapezoid is a trapezoid with the non - parallel sides (legs) being congruent. We found in step 3 that \(\overline{DA} \cong \overline{EV}\) (from the congruent triangles \(\triangle DAV\cong\triangle EVA\)) and in step 4 that DAVE is a trapezoid. So to conclude that DAVE is an isosceles trapezoid, we use the definition of an isosceles trapezoid. The statement for step 5 should be "DAVE is an isosceles trapezoid" and the reason is "Definition of isosceles trapezoid".

Statement	Reason
2. \(\triangle DAV\cong\triangle EVA\)	2. Given
3. \(\overline{DA} \cong \overline{EV}\)	3. Definition of congruent triangles
4. DAVE is a trapezoid	4. Definition of trapezoid
5. DAVE is an isosceles trapezoid	5. Definition of isosceles trapezoid

Answer:

The completed two - column proof is as shown in the above table.