Beatrix is working to prove the Base Angles Theorem. They start with an isosceles triangle △ABC with AB¯¯¯¯¯¯¯¯≅AC¯¯¯¯¯¯¯¯ . Then they construct AD¯¯¯¯¯¯¯¯ , the angle bisector of point A . They know they are supposed to prove that two triangles are congruent in order to prove the Base Angles Theorem. They show how △ABD≅△ACD by the SAS Congruence Theorem. They aren’t sure how to finish the proof. Help them construct a viable argument. What should Beatrix do next? Option #1: Show that ∠BAD≅∠CAD by the definition of an angle bisector.Option #2: Show that ∠C≅∠A by the CPCTC Theorem.Option #3: Show that ∠B≅∠C by the CPCTC Theorem.Option #4: Show that BD¯¯¯¯¯¯¯¯≅DC¯¯¯¯¯¯¯¯ by the definition of a midpoint.

Answer 1

To finish the proof for the Base Angles Theorem, Beatrix should proceed with the following argument:

Option #1: Show that ∠BAD ≅ ∠CAD by the definition of an angle bisector.

Explanation: Since \( AD \) is the angle bisector of \( \angle A \), by definition, it divides the angle \( \angle A \) into two equal parts. Therefore, it follows that \( \angle BAD \) is equal to \( \angle CAD \). This is critical because it allows Beatrix to then show that the two angles at the base of triangle \( ABC \) (i.e., \( \angle B \) and \( \angle C \)) are equal due to the congruence of triangles \( ABD \) and \( ACD \) established by the SAS Congruence Theorem along with the corresponding parts of congruent triangles (CPCTC).

Thus, the steps would be:

1. Since \( \triangle ABD \cong \triangle ACD \) by the SAS Congruence Theorem, we can state that corresponding angles are congruent by CPCTC.
2. Therefore, it follows that \( \angle B \) (from triangle \( ABD \)) is equal to \( \angle C \) (from triangle \( ACD \)), proving the Base Angles Theorem.

By doing this, Beatrix can complete her proof effectively.