Final answer:
To prove that △ABO and △DCO are congruent, we use the midpoint property for BO = CO, parallel lines AB and CD to establish congruent alternate interior angles, and the reflexive property for DO, leading to AAS congruence.
Step-by-step explanation:
To prove that triangles △ABO and △DCO are congruent when O is the midpoint of line segment BC and CD is parallel to AB, we can use geometric properties and theorems.
Firstly, since O is the midpoint of BC, we have that BO = CO by definition of a midpoint. Secondly, by the given that CD is parallel to AB, we can infer that angle ABO is congruent to angle DCO because they are alternate interior angles formed by a transversal (DO or BO) through two parallel lines (CD and AB). This implies that △ABO and △DCO have two angles that are equal in measure.
Furthermore, by the reflexivity property, we know that segment DO is congruent to itself, which implies side AO (extends to include DO) of △ABO is congruent to side DO of △DCO. With two angles and the included side congruent (AAS), we can conclude that the triangles are congruent, which proves that △ABO ≅ △DCO.