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
