Question

select the correct answer.
given: \\(\overline{ad} \parallel \overline{bc}\\)
prove: \\(dc = 6\\) units
image of a quadrilateral with points a, b, c, d; ab is 6 units, ad and bc have arrows indicating parallel, and ac is a diagonal

statements	reasons
\\(\angle dac \cong \angle bca\\)	alternate interior angles theorem
\\(\overline{ac} \cong \overline{ac}\\)	reflexive property of congruence
\\(\angle dca \cong \angle bac\\)	alternate interior angles theorem
?	?
\\(\overline{dc} \cong \overline{ba}\\)	cpctc
\\(dc = ba\\)	definition of congruent sides
\\(dc = 6\\) units	substitution property of equality

which step is missing?
\\(\circ\\) a. \\(\triangle dca \cong \triangle bca\\) by sas
\\(\circ\\) b. \\(\triangle dac \cong \triangle bca\\) by asa
\\(\circ\\) c. \\(\triangle dac \cong \triangle bca\\) by sas
\\(\circ\\) d. \\(\triangle dca \cong \triangle bca\\) by asa

Explanation:

Step1: Analyze Known Angles/Sides

We have \(\angle DAC \cong \angle BCA\), \(\overline{AC} \cong \overline{AC}\), and \(\angle DCA \cong \angle BAC\).

Step2: Identify Triangle Congruence

The ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. Here, for \(\triangle DCA\) and \(\triangle BCA\): \(\angle DCA \cong \angle BAC\) (angle), \(\overline{AC} \cong \overline{AC}\) (side, included between the angles), and \(\angle DAC \cong \angle BCA\) (angle). So, \(\triangle DCA \cong \triangle BCA\) by ASA.

Answer:

D. \(\triangle DCA \cong \triangle BCA\) by ASA