Triangle Proofs Practice

Question

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5 of 55 of 5 Items

Question
Use the image to answer the 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.

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 #

Answer 1

To help Beatrix complete her proof for the Base Angles Theorem, the next logical step after establishing the congruence of triangles \( \triangle ABD \) and \( \triangle ACD \) (using the SAS Congruence Theorem) is to demonstrate that the angles at the base of the isosceles triangle are congruent.

Since angle \( AD \) is the angle bisector, it divides angle \( A \) into two equal parts. Therefore, the appropriate conclusion to draw next is:

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

This step will complete the proof of the Base Angles Theorem, showing that the base angles \( B \) and \( C \) are indeed congruent, as they correspond to angles \( BAD \) and \( CAD \) which are proven to be equal.

Answer 2

it was option 3

Answer 3

I apologize for the confusion. Since you noted it was option #3, here's the correct reasoning based on your selection:

After proving that triangles \( \triangle ABD \) and \( \triangle ACD \) are congruent by the SAS congruence theorem, you can use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem. Given that the triangles are congruent, the corresponding angles will also be equal.

Option #3: Show that ∠B ≅ ∠C by the CPCTC Theorem.

This step allows you to conclude that the base angles of the isosceles triangle are congruent, which is the statement of the Base Angles Theorem. Thank you for your correction!