Question

find the lettered angles.

Explanation:

Step1: Recall circle - angle property

In a circle, angles in the same segment are equal. Also, the angle subtended by an arc at the center is twice the angle subtended by the same arc at the circumference.

Step2: Analyze the first circle

In the first circle, if we assume the center is \(O\), the angle subtended by arc \(BC\) at the center is \(2\times50^{\circ}=100^{\circ}\) (angle - at - the - center property). Since \(OA = OB\) (radii of the same circle), \(\triangle OAB\) is isosceles. Let's assume we want to find \(\angle BAC\). The angle \(\angle BAC\) and the given \(50^{\circ}\) angle are angles in the same segment. So \(\angle BAC = 50^{\circ}\).

Step3: Analyze the second circle

In the second circle, if the center is \(O\), and we know one angle at the circumference is \(30^{\circ}\) and another is \(50^{\circ}\). The sum of angles in a triangle inscribed in a circle is \(180^{\circ}\). Let the third angle be \(x\). Then \(x=180^{\circ}-(30^{\circ} + 50^{\circ})=100^{\circ}\).

Answer:

For the first circle, if the lettered angle is the one in the same segment as the \(50^{\circ}\) angle, it is \(50^{\circ}\). For the second circle, if the lettered angle is the third - angle of the inscribed triangle, it is \(100^{\circ}\).