Question

select the correct answer.
given: \\(\overline{ad} \parallel \overline{bc}\\)
prove: \\(dc = 6\\) units
diagram of quadrilateral with triangle and sides labeled

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?
a. \\(\triangle dca \cong \triangle bca\\) by sas
b. \\(\triangle dac \cong \triangle bca\\) by asa
c. \\(\triangle dca \cong \triangle bca\\) by asa
d. \\(\triangle dac \cong \triangle bca\\) by sas

Explanation:

Step1: Analyze known congruent parts

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

Step2: Identify triangle congruence criterion

The ASA (Angle - Side - Angle) 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, then the triangles are congruent. Here, in \(\triangle DAC\) and \(\triangle BCA\), \(\angle DAC \cong \angle BCA\), \(\overline{AC}\) is the included side, and \(\angle DCA \cong \angle BAC\). So, \(\triangle DAC \cong \triangle BCA\) by ASA.

Step3: Match with options

Looking at the options, option B states \(\triangle DAC \cong \triangle BCA\) by ASA, which is the missing step (the step before using CPCTC to show \(\overline{DC} \cong \overline{BA}\)).

Answer:

B. \(\triangle DAC \cong \triangle BCA\) by ASA