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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.

Trijstūris ABC ir taisnleņķa trijstūris (∠B = 90°), trijstūris DBC (D ∈ AC) ir vienādmalu trijstūris. BC = 5 cm.

a) Nosaki ∠C lielumu.

b) Aprēķini trijstūra ABC laukumu.

c) Pamato, ka trijstūris ABD ir vienādsānu.

Solve a problem involving a right-angled triangle ABC with ∠B = 90° and an equilateral triangle DBC. Learn to determine angles, calculate areas, and prove that triangle ABD is isosceles. Suitable for students in Grades 8-10.

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