In a rhombus ABCD, opposite sides are parallel and equal. This means AB is parallel and equal to CD, and BC is parallel and equal to AD. Since AB is parallel to CD, angle ABC is equal to angle CDA. Corresponding angles are equal in parallel lines. Therefore, in rhombus ABCD, angle ABC is equal to angle CDA, proving that ABC = CDA.
Since opposite sides are equal in a rhombus, AB = CD and BC = AD. This means that triangle ABC is congruent to triangle CDA by the Side-Angle-Side congruence theorem.
Therefore, angle A = angle C and angle B = angle D. Since all angles in a quadrilateral add up to 360 degrees, we can rewrite angle ABC as (angle A + angle B) and angle CDA as (angle C + angle D).
Therefore, (angle A + angle B) = (angle C + angle D). Since angle A = angle C and angle B = angle D, we can rewrite this as 2(angle A) = 2(angle C).
This simplifies to angle A = angle C, proving that angles ABC and CDA are indeed equal in the rhombus ABCD.