Question

read the proof.
given: \\( \overline { a b } \parallel \overline { d e } \\)
prove: \\( \triangle a b c \sim \triangle e d c \\)
\\( \

$$\begin{array} { | l | l | } \\hline \\text { statement } & \\text { reason } \\\\ \\hline 1. \\overline { a b } \\parallel \\overline { d e } & 1. \\text { given } \\\\ \\hline 2. \\angle a c b \\text { and } \\angle e c d \\text { are } & 2. \\text { definition of vertical angles } \\\\ \\text { vert. } \\angle s & \\\\ \\hline 3. \\angle a c b \\cong \\angle d c e & 3. \\text { vertical angles are } \\\\ & \\text { congruent } \\\\ \\hline 4. \\angle b d e \\text { and } \\angle d b a \\text { are } & 4. \\text { definition of alternate } \\\\ \\text { alt. int. } \\angle s & \\text { interior angles } \\\\ \\hline 5. \\angle b d e \\cong \\angle d b a & 5. \\text { alternate interior angles } \\\\ & \\text { are congruent } \\\\ \\hline 6. \\triangle a b c \\sim \\triangle e d c & 6. ? \\\\ \\hline \\end{array}$$

\\)
aa simllanty theorem
asa similarity theorem
aas simllarity theorem
sas similarity theorem

Explanation:

Step1: Recall similarity theorems

• AA (Angle - Angle) similarity theorem: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.
• ASA (Angle - Side - Angle) is a congruence theorem (not a similarity theorem in the traditional sense for similarity as we focus on angle - angle relationships for similarity here).
• AAS (Angle - Angle - Side) is a congruence theorem.
• SAS (Side - Angle - Side) similarity theorem: If the ratio of two sides of one triangle is equal to the ratio of two sides of another triangle and the included angles are congruent, then the two triangles are similar.

Step2: Identify the angles in the triangles

• In \(\triangle ABC\) and \(\triangle EDC\), we have \(\angle ACB=\angle DCE\) (from step 3 of the proof, vertical angles are congruent) and \(\angle BDE = \angle DBA\) (from step 5 of the proof, alternate interior angles are congruent). So, two pairs of angles are congruent.

Answer:

AA similarity theorem