Quadrilateral labeled with vertices upper A, upper B, upper C, and upper D. The segment between upper B and upper D is longer than the diagonal between upper A and upper C.

Question

Consider the following proof for quadrilateral ABCD .

Statements Reasons
1. AB¯¯¯¯¯¯¯¯≅CB¯¯¯¯¯¯¯¯ , ∠ABD≅∠CBD Quadrilateral ABCD

1. given
2. BD¯¯¯¯¯¯¯¯≅BD¯¯¯¯¯¯¯¯ 2. Reflexive Property of Congruence
3. ΔADB≅ΔCDB 3. ___________
4. AD¯¯¯¯¯¯¯¯≅CD¯¯¯¯¯¯¯¯ 4. CPCTC
5. ABCD is a kite. 5. definition of a kite (A quadrilateral with two pairs of adjacent congruent sides)
Which reason best completes the proof in step 3?

(1 point)
Responses

Side-Angle-Side Triangle Congruence Theorem
Side-Angle-Side Triangle Congruence Theorem

Definition Congruent Triangles
Definition Congruent Triangles

Angle-Angle-Side Triangle Congruence Theorem
Angle-Angle-Side Triangle Congruence Theorem

Side-Side-Side Triangle Congruence Theorem

Answer 1

To complete the proof in step 3, we need to establish that triangles ΔADB and ΔCDB are congruent based on the information given in the preceding statements.

The triangles share the segment BD (which is denoted as BD ≅ BD, using the Reflexive Property of Congruence). We also know that AB ≅ CB (stated in step 1) and ∠ABD ≅ ∠CBD (also from step 1).

With two sides and the included angle congruent between two triangles, we can apply the Side-Angle-Side (SAS) Triangle Congruence Theorem.

Thus, the best reason to complete the proof in step 3 is:

Side-Angle-Side Triangle Congruence Theorem