Understanding the Trapezium Geometry Problem

The problem asks us to find the measure of angle $\angle ACD$ in a trapezium ABCD. We are given that the side BC is parallel to the side AD (BC || AD) and that the lengths of sides AC and CD are equal (AC = CD). We are also given two angles: $\angle ABC = 20^{\circ}$ and $\angle BAC = 131^{\circ}$.

Step-by-Step Calculation of $\angle ACD$

We can solve this problem by applying the properties of angles in a triangle and the properties of parallel lines.

1. Calculate $\angle BCA$ in triangle ABC:

   The sum of angles in any triangle is $180^{\circ}$. In triangle ABC, we have:

   $ \angle BAC + \angle ABC + \angle BCA = 180^{\circ} $

   Substitute the given values:

   $ 131^{\circ} + 20^{\circ} + \angle BCA = 180^{\circ} $

   $ 151^{\circ} + \angle BCA = 180^{\circ} $

   Solving for $\angle BCA$:

   $ \angle BCA = 180^{\circ} - 151^{\circ} = 29^{\circ} $

2. Use the property of parallel lines (BC || AD):

   When a transversal line intersects two parallel lines, the alternate interior angles are equal. Here, AC is the transversal intersecting parallel lines BC and AD.

   Therefore, the alternate interior angles $\angle BCA$ and $\angle CAD$ are equal:

   $ \angle CAD = \angle BCA $

   $ \angle CAD = 29^{\circ} $

3. Use the property of isosceles triangle ACD:

   We are given that AC = CD. This means that triangle ACD is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal.

   The angle opposite side CD is $\angle CAD$.

   The angle opposite side AC is $\angle CDA$.

   So, we have:

   $ \angle CDA = \angle CAD $

   Since $\angle CAD = 29^{\circ}$, we get:

   $ \angle CDA = 29^{\circ} $

4. Calculate $\angle ACD$ in triangle ACD:

   Now, consider triangle ACD. The sum of its angles must be $180^{\circ}$.

   $ \angle ACD + \angle CAD + \angle CDA = 180^{\circ} $

   Substitute the values we found:

   $ \angle ACD + 29^{\circ} + 29^{\circ} = 180^{\circ} $

   $ \angle ACD + 58^{\circ} = 180^{\circ} $

   Solving for $\angle ACD$:

   $ \angle ACD = 180^{\circ} - 58^{\circ} $

   $ \angle ACD = 122^{\circ} $

Final Answer Verification

The calculated value for $\angle ACD$ is $122^{\circ}$. This matches one of the provided options.