Notice that ∠ ABC is opposite to ∠ CDA. Since a rhombus is a parallelogram, its opposite angles are congruent. This means that their measures are equal.
m ∠ ABC = m ∠ CDA
Let's write the measures of these angles, ∠ ABC and ∠ CDA, as the sum of two angle measures.
m ∠ ABC = m ∠ CDA
⇓
m ∠ ABD + m ∠ DBC = m ∠ CDB + m ∠ BDA
We know that if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.
m ∠ DBC = m ∠ ABD
and
m ∠ CDB = m ∠ BDA
Now, let's substitute these values in the equation m ∠ ABD + m ∠ DBC = m ∠ CDB + m ∠ BDA.
We already know that m ∠ ABD = 55. Therefore, by the Transitive Property of Equality, the measure of ∠ BDA is also 55.
m ∠ ABD = m ∠ BDA m ∠ ABD=55 ⇓ m ∠ BDA = 55
