Geometry Solution: Calculating Angle ACD in Trapezium ABCD

This problem involves finding a specific angle within a trapezium using its geometric properties and the given angle measures. We are given a trapezium ABCD where the sides BC and AD are parallel (BC || AD). We are also told that the sides AC and CD are equal in length (AC = CD), which makes triangle ACD an isosceles triangle. We know two angles: $\angle ABC = 24^\circ$ and $\angle BAC = 112^\circ$. Our goal is to find the measure of $\angle ACD$.

Properties of the Given Trapezium

• ABCD is a trapezium with BC || AD.
• Triangle ACD is an isosceles triangle because AC = CD.
• Given angles: $\angle ABC = 24^\circ$ and $\angle BAC = 112^\circ$.

Step-by-Step Angle Calculation

1. Calculate angles in Triangle ABC:

   The sum of the interior angles in any triangle is $180^\circ$. Applying this to triangle ABC, we have:

   $\angle BCA + \angle CAB + \angle ABC = 180^\circ$

   Substitute the known values:

   $\angle BCA + 112^\circ + 24^\circ = 180^\circ$

   Combine the known angles:

   $\angle BCA + 136^\circ = 180^\circ$

   Solve for $\angle BCA$:

   $\angle BCA = 180^\circ - 136^\circ = 44^\circ$.

2. Use the parallel lines property:

   Since BC is parallel to AD (BC || AD) and AC is a transversal line intersecting these parallel lines, the alternate interior angles are equal. Therefore:

   $\angle CAD = \angle BCA$

   From the previous step, we know $\angle BCA = 44^\circ$, so:

   $\angle CAD = 44^\circ$.

3. Analyze the isosceles Triangle ACD:

   We are given that AC = CD. In an isosceles triangle, the angles opposite the equal sides are equal. The angle opposite side CD is $\angle CAD$, and the angle opposite side AC is $\angle CDA$. Therefore:

   $\angle CDA = \angle CAD$

   Since $\angle CAD = 44^\circ$, we have:

   $\angle CDA = 44^\circ$.

4. Calculate the required angle $\angle ACD$:

   Now, we use the property that the sum of angles in triangle ACD is $180^\circ$.

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

   Substitute the known angles:

   $44^\circ + 44^\circ + \angle ACD = 180^\circ$

   Combine the known angles:

   $88^\circ + \angle ACD = 180^\circ$

   Solve for $\angle ACD$:

   $\angle ACD = 180^\circ - 88^\circ = 92^\circ$.

Conclusion

By applying the properties of triangles and parallel lines, we found that the measure of angle $\angle ACD$ is $92^\circ$.