Question

Given that CD¯¯¯¯¯¯¯¯ is the perpendicular bisector of AB¯¯¯¯¯¯¯¯, which triangle congruence criteria can be used to prove that △ADC and △BDC are congruent?

Answer 1

The triangles are drawn in the picture attached.

Let's call M the intersection point of AB and CD. Since CD is the perpendicular bisector, we know that:
AM ≡ MB (bisector = divides into two equal pieces)
∠AMD ≡ ∠AMC ≡ ∠BMC ≡ ∠BMD (perpendicular = forms 4 angles of 90°)

Considering ΔAMD and ΔBMD, they have also MD in common, therefore, we can use the SAS (side - angle - side) congruency criterium to prove that they are congruent.

Similarly for ΔAMC and ΔBMC.

Therefore, ΔADC ≡ ΔBCD because they are made of congruent triangles.

Hence, with the SAS congruency theorem, we can demonstrate that ΔADC ≡ ΔBCD