This is a geometry problem involving parallel lines and angles in a quadrilateral. Based on the information in the image:

• $AB \parallel CD$ (AB is parallel to CD)
• $AC = BD$ (AC is equal to BD)
• $\angle ADB = 80^\circ$
• $\angle CAD = 20^\circ$

The goal is to find $\angle ADC$.

Step-by-step solution:

1. Use the property of parallel lines: Since $AB \parallel CD$, $\angle CAD$ and $\angle ADB$ form a pair of alternate angles. Thus, $\angle ADB = 80^\circ$ implies that the angle adjacent to $\angle CAD$ at point $C$, denoted as $\angle DCA$, is also $80^\circ$ (corresponding angle property).

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