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Question

An isosceles triangle with a sideways orientation is marked counter-clockwise from the right as upper A upper B upper C. Sides upper A upper B and upper A upper C are marked with a single congruent tick mark. A line bisects the triangle.

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 prove the Base Angles Theorem using the isosceles triangle \( \triangle ABC \), Easton should focus on the two smaller triangles formed by the midpoint \( D \) of \( BC \).

The two triangles he should consider are \( \triangle ABD \) and \( \triangle ACD \). Both of these triangles have the following characteristics:

1. \( AB \cong AC \) (given that \( \triangle ABC \) is isosceles)
2. \( AD \) is common to both triangles.
3. \( BD \cong CD \) because \( D \) is the midpoint of \( BC \).

Thus, the criteria for the SSS (Side-Side-Side) Triangle Congruence Theorem is satisfied.

Therefore, the correct option is:

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