We are given a right-angled triangle $ABC$, where $\angle B = 90^\circ$, and triangle $DBC$ is an equilateral triangle with point $D$ lying on $AC$. The side $BC$ is provided as 5 cm. The tasks are:

a) Determine the measure of $\angle C$.

b) Calculate the area of triangle $ABC$.

c) Justify that triangle $ABD$ is isosceles.

Let's go through each part:

a) Determine $\angle C$

Since triangle $ABC$ is a right-angled triangle with $\angle B = 90^\circ$, the sum of angles in a triangle is always $180^\circ$. Therefore:

$\angle A + \angle B + \angle C = 180^\circ$

Given that $\angle B = 90^\circ$, we have:

$\angle A + \angle C = 90^\circ$

To solve for $\angle C$, we would need additional information like the length of side $AB$ or $AC$. Since that information isn't provided directly, we'll proceed based on the geometric properties or using trigonometric methods if the other side lengths are known.

b) Calculate the area of triangle $ABC$

The area of a right-angled triangle can be calculated using the formula:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Here, the base is $BC = 5 \, \text{cm}$, but we need the height, which would be the length of side $AB$.

Again, this requires knowing the length of side $AB$, which isn't provided in the problem, but if it's given later or if we deduce it, we can calculate the area easily.

c) Justify that triangle $ABD$ is isosceles

Since triangle $DBC$ is equilateral (as per the problem), all sides of $DBC$ are equal. This implies that $BD = BC = 5 \, \text{cm}$. If we can show that $AB = BD$, then triangle $ABD$ would be isosceles.

The key observation here is that in an equilateral triangle, the altitude from one vertex (in this case $D$) to the opposite side (here, $AC$) bisects that side. So, if we know the geometric properties of the triangle and the length of $AB$, we can conclude that triangle $ABD$ is isosceles if two sides are equal.

Let me know if you'd like more details on solving these parts! Here are some questions to explore the problem further:

1. What is the length of side $AB$ in triangle $ABC$?
2. Can the length of side $AC$ be deduced from the diagram?
3. How do the properties of an equilateral triangle help in solving this problem?
4. How does knowing $\angle C$ help in solving part (b) of the question?
5. Is there any way to confirm the isosceles property of triangle $ABD$ using coordinates?

Tip: The Pythagorean theorem is a powerful tool in solving for unknown sides in right-angled triangles.