Easton is working to prove the Base Angles Theorem. He starts with an isosceles triangle △ABC with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ . He then constructs point D , the midpoint of BC¯¯¯¯¯¯¯¯ . He knows he is supposed to prove that two triangles are congruent in order to prove the Base Angles Theorem. Help him to construct a viable argument. Which two triangles should he prove are congruent and by what reason are they congruent? Option #1: △ABC≅△ADC by the SSS Triangle Congruence Theorem.Option #2: △ABD≅△ACD by the SSS Triangle Congruence Theorem.Option #3: △ABC≅△ADC by the SAS Triangle Congruence Theorem.Option #4: △ABD≅△ACD by the SAS Triangle Congruence Theorem.

Answer 1

To help Easton prove the Base Angles Theorem using triangle congruence, we can analyze the options and identify the most appropriate triangles to prove congruent.

Given triangle \( \triangle ABC \) where \( AB \cong AC \) since it's isosceles, and point \( D \) is the midpoint of \( BC \), we can look at triangles \( \triangle ABD \) and \( \triangle ACD \):

1. Sides:

   • \( AB \cong AC \) (given, since \( \triangle ABC \) is isosceles)
   • \( AD \cong AD \) (the common side, reflexive property)
   • \( BD \cong CD \) (since \( D \) is the midpoint of \( BC \))

2. Reason for Congruence: By the SSS (Side-Side-Side) Triangle Congruence Theorem, since all three corresponding sides of \( \triangle ABD \) and \( \triangle ACD \) are congruent, we can conclude that:

\[ \triangle ABD \cong \triangle ACD \text{ by SSS Triangle Congruence Theorem.} \]

Thus, the correct option is:

Option #2: \( \triangle ABD \cong \triangle ACD \) by the SSS Triangle Congruence Theorem.

After establishing that \( \triangle ABD \cong \triangle ACD \), we can conclude that the angles opposite the congruent sides are equal, namely \( \angle ABD \cong \angle ACD \), thereby proving the Base Angles Theorem.