Question

Given: ABCD is a rhombus and △ACB ≅ △DBC

Prove: ABCD is a square

Answer 1

A B C D is a rhombus so if you turn it sideways you get a square

Answer 2

Given: ABCD is a rhombus and △ACB ≅ △DBC

Prove: ABCD is a square

Proof:

1. ABCD is a rhombus, so AB = BC = CD = DA.
2. △ACB ≅ △DBC, so AC = DB and ∠ACB = ∠DBC.
3. Since ABCD is a rhombus, opposite angles are congruent, so ∠ABC = ∠CDA and ∠BCD = ∠DAB.
4. Since ∠ACB = ∠DBC, then ∠ACB + ∠ABC + ∠BCD = ∠DBC + ∠DAB + ∠CDA.
5. Substituting the congruent angles from step 3, we get ∠ABC + ∠CDA + ∠BCD + ∠DAB = 360°.
6. Since the sum of the angles of a quadrilateral is 360°, then ABCD is a quadrilateral.
7. Since all sides and angles of ABCD are congruent, then ABCD is a square.