Question

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

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

\\)
\\( \bigcirc \\) aa similarity theorem
\\( \bigcirc \\) asa similarity theorem
\\( \bigcirc \\) aas similarity theorem
\\( \bigcirc \\) sas similarity theorem

Explanation:

Step1: Recall similarity theorems

There are AA (Angle - Angle), ASA (Angle - Side - Angle), AAS (Angle - Angle - Side), and SAS (Side - Angle - Side) similarity theorems.

Step2: Check angles in the triangles

From the proof:

• We have \(\angle ACB\cong\angle DCE\) (from step 3)
• We have \(\angle BDE\cong\angle DBA\) (from step 5)

So, two pairs of corresponding angles in \(\triangle ABC\) and \(\triangle EDC\) are congruent.

Step3: Apply AA similarity theorem

By the 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. Since we have two pairs of congruent angles (\(\angle ACB\cong\angle DCE\) and \(\angle BDE\cong\angle DBA\)) in \(\triangle ABC\) and \(\triangle EDC\), \(\triangle ABC\sim\triangle EDC\) by the AA similarity theorem.

Answer:

AA similarity theorem