Anna wrote an informal proof to prove ∠1≅∠2

Question

in △ABC
. She is given that AB≅AC
and that AD
bisects ∠BAC
.

1. Since AD
bisects ∠BAC
, then ∠BAD≅∠CAD
.
2. Additionally, AD≅AD
because of the reflexive property.
3. Therefore, we can say △BAD≅△CAD
by ____________.
4. Since △BAD≅△CAD
, we can say the corresponding parts are congruent, therefore, BD≅CD

5. Hence, ∠1≅∠2
because ______________________.


What statements would complete Anna's proof?

SSA; △BDC
is isosceles

SAS; △BDC
is isosceles

SAS; △BDC
is equilateral

SSA; △BDC
is equilateral

Answer 1

To complete Anna's proof, we need to identify the correct reasoning to identify the congruence of the triangles and the congruence of the angles. Here's how the proof can be completed with the correct statements:

1. Since AD bisects ∠BAC, then ∠BAD≅∠CAD.
2. Additionally, AD≅AD because of the reflexive property.
3. Therefore, we can say △BAD≅△CAD by SAS (Side-Angle-Side) because two sides and the included angle of triangle BAD are congruent to two sides and the included angle of triangle CAD.
4. Since △BAD≅△CAD, we can say the corresponding parts are congruent, therefore, BD≅CD.
5. Hence, ∠1≅∠2 because △BDC is isosceles (since BD≅CD, the angles opposite those sides are also congruent).

So, the correct statements to complete Anna's proof would be:

3. SAS; △BDC is isosceles.

5. Hence, ∠1≅∠2 because △BDC is isosceles.