Use the image to answer the question. 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.(1 point)Option #

Answer 1

To help Easton prove the Base Angles Theorem, he should focus on the triangles that include the apex of the isosceles triangle \( \triangle ABC \) and the midpoint \( D \) of the base \( BC \).

The two triangles that he should prove are congruent are \( \triangle ABD \) and \( \triangle ACD \).

The reasoning for this congruence can be broken down as follows:

1. Side \( AB \) is congruent to side \( AC \) (given, since \( \triangle ABC \) is isosceles).
2. Side \( AD \) is common to both triangles \( \triangle ABD \) and \( \triangle ACD \) (common side).
3. Segment \( BD \) is congruent to segment \( CD \) (since \( D \) is the midpoint of \( BC \)).

With these three pieces of information, we can apply the SSS Triangle Congruence Theorem:

• \( AB \cong AC \)
• \( AD \cong AD \) (common side)
• \( BD \cong CD \)

Thus, we can conclude that \( \triangle ABD \cong \triangle ACD \).

Therefore, the appropriate option that Easton should choose is:

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