A rhombus A B C D. From the top, clockwise, the vertices are A, D, C, and B.
If you wanted to explain why △ABD≅CBD by the SAS Congruence Theorem, which of the following would be helpful to know?
(1 point)
Responses
AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BAD.
Modifying above upper A upper C with bar is the angle bisector of angle upper B upper A upper D .
AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.
Modifying above upper A upper C with bar is the angle bisector of angle upper B upper C upper D .
BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.
Modifying above upper B upper D with bar is the angle bisector of angle upper B upper C upper D .
BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠ABC.
Modifying above upper B upper D with bar is the angle bisector of angle upper A upper B upper C .
To explain why △ABD≅CBD by the SAS Congruence Theorem, it would be helpful to know that BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.
The helpful information to know would be: BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD.
To prove that △ABD≅CBD by the SAS Congruence Theorem, we need to know two sides and the included angle of both triangles.
In a rhombus, all sides are congruent, so we know that AB¯¯¯¯¯¯¯¯=BD¯¯¯¯¯¯¯¯.
To use the SAS Congruence Theorem, we need to find the included angle. From the given information, we know that the rhombus vertices are ordered clockwise as A, D, C, and B. Therefore, angle ABD is the included angle in triangle ABD, and angle CBD is the included angle in triangle CBD. We need to ensure that these angles are congruent.
From the options provided, the helpful information to know is that AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BAD (option 1) or that BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD. (option 3).
If AC¯¯¯¯¯¯¯¯ is the angle bisector of ∠BAD, it means that angle CAB is congruent to angle DAB. Therefore, angle ABD is congruent to angle DBC. Hence, triangle ABD is congruent to triangle CBD by the SAS Congruence Theorem.
If BD¯¯¯¯¯¯¯¯ is the angle bisector of ∠BCD, it means that angle CBD is congruent to angle DBC. Therefore, angle BDA is congruent to angle DBA. Hence, triangle ABD is congruent to triangle CBD by the SAS Congruence Theorem.
It is important to note that option 2 and option 4 do not provide the necessary information to prove the congruence of the triangles using the SAS Congruence Theorem.