Triangle Proofs PracticeComplete this assessment to review what you’ve learned. It will not count toward your grade.5 of 5 5 of 5 ItemsQuestionUse the image to answer the question. 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.(1 point)Option #

Question

Triangle Proofs PracticeComplete this assessment to review what you’ve learned. It will not count toward your grade.5 of 5 5 of 5 ItemsQuestionUse the image to answer the question. 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.(1 point)Option #

GPT-4o mini · Answer

To finish the proof of the Base Angles Theorem, Beatrix should proceed with Option #1: Show that ∠BAD ≅ ∠CAD by the definition of an angle bisector.

Here's why:

Since AD is the angle bisector of ∠A, by definition, it divides ∠A into two equal parts, which means ∠BAD is congruent to ∠CAD.
Once Beatrix establishes that ∠BAD ≅ ∠CAD, she can then use the congruence of triangles ΔABD and ΔACD (which she proved using SAS) to show that the base angles B and C are congruent as well (using the CPCTC theorem).

This way, Beatrix can conclude that the angles opposite the equal sides of the isosceles triangle (the base angles) are congruent, successfully proving the Base Angles Theorem.