Final answer:
To prove ∆ACD ≅ ∆BCD, we use the SAS congruence theorem, as D is the midpoint of AB and AC = BC, giving us congruent sides AD = DB and AC = BC, as well as congruent angles ∠ACD = ∠BCD between those sides. This satisfies the SAS criteria.
Step-by-step explanation:
To prove that ∆ACD ≅ ∆BCD given D is the midpoint of AB and AC = BC, we can apply the Side-Angle-Side (SAS) congruence theorem. Since D is the midpoint of AB, it implies AD = DB. It's given that AC = BC which gives us two pairs of congruent sides. Because AC = BC, the base angles of the isosceles triangle ABC must also be congruent, therefore ∠ACD = ∠BCD. Combining the congruent angles with the congruent sides from AD = DB and AC = BC, we satisfy the conditions of the SAS congruence theorem which states that two triangles are congruent if they have two sides and the included angle of one triangle equal respectively to the two sides and the included angle of another triangle.
Therefore, ∆ACD ≅ ∆BCD because they have two congruent sides and one congruent angle between those sides, satisfying the criteria for the SAS postulate for triangle congruence.