Question

read the proof.
given: \\(\overline{ab} \parallel \overline{de}\\)
prove: \\(\triangle abc \sim \triangle edc\\)
image of a geometric diagram with points a, b, c, d, e

statement	reason
2. \\(\angle acb\\) and \\(\angle ecd\\) are vert. \\(\angle\\)s	2. definition of vertical angles
3. \\(\angle acb \cong \angle dce\\)	3. vertical angles are congruent
4. \\(\angle bde\\) and \\(\angle dba\\) are alt. int. \\(\angle\\)s	4. definition of alternate interior angles
5. \\(\angle bde \cong \angle dba\\)	5. alternate interior angles are congruent
6. \\(\triangle abc \sim \triangle edc\\)	6.?

options:

• aa similarity theorem
• asa similarity theorem
• aas similarity theorem
• sas similarity theorem

Explanation:

To determine the reason for \(\triangle ABC \sim \triangle EDC\), we analyze the proven congruent angles. From steps 3 and 5, we have \(\angle ACB \cong \angle DCE\) (vertical angles) and \(\angle BDE \cong \angle DBA\) (alternate interior angles). The AA (Angle - Angle) similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Here, we have two pairs of congruent angles, so the AA similarity theorem applies.

Answer:

AA similarity theorem