The image you uploaded shows a geometrical problem involving a circle and triangle with given conditions.

Let's break it down:

• $\triangle ABC$ is an isosceles triangle because $AB = AC$.
• The given angle $\angle ABC = 50^\circ$.
• The task is to find the measure of $\angle BDC$.

Step-by-step approach:

1. Since $\triangle ABC$ is isosceles, $\angle ACB = \angle ABC = 50^\circ$.
2. The angle $\angle ABC$ is subtended by the arc $AC$, and by the inscribed angle theorem, the angle subtended by the same arc on the circle will be half of $\angle ABC$, which leads to the relation for $\angle BDC$.

I will now calculate $\angle BDC$.

Let's calculate it:We are given the following information:

• $AB = AC$ (isoceles triangle),
• $\angle ABC = 50^\circ$.

Since $AB = AC$, the angles $\angle ABC$ and $\angle ACB$ are equal, so: $\angle ACB = 50^\circ.$

Next, the total sum of angles in any triangle is $180^\circ$. Therefore, in $\triangle ABC$: $\angle BAC + \angle ABC + \angle ACB = 180^\circ.$ Substituting the known values: $\angle BAC + 50^\circ + 50^\circ = 180^\circ,$ $\angle BAC + 100^\circ = 180^\circ,$ $\angle BAC = 180^\circ - 100^\circ = 80^\circ.$

Now, let's focus on $\angle BDC$, which is the angle subtended by arc $BC$. By the inscribed angle theorem, the angle subtended by an arc at the circumference is half of the central angle subtended by the same arc.

The central angle subtended by arc $BC$ is $2 \times \angle BAC = 2 \times 80^\circ = 160^\circ$.

Thus, the inscribed angle $\angle BDC$, which subtends the same arc, is: $\angle BDC = \frac{160^\circ}{2} = 80^\circ.$

So, the measure of $\angle BDC$ is $80^\circ$.

Let me know if you need further clarification or details!

Here are 5 related questions:

1. What is the inscribed angle theorem, and how does it apply in this case?
2. How can you prove that a triangle inscribed in a circle has a total of 180°?
3. What other properties of isosceles triangles can help solve geometric problems like this one?
4. How does the central angle relate to the inscribed angle in a circle?
5. What are some real-world examples of inscribed angles in geometry?

Tip: Always remember that the sum of angles in any triangle is 180°, and use the properties of circles to relate central and inscribed angles!