Determine the value of angle x in the first diagram.

Solution:

• The sum of angles in a triangle is 180 degrees.
• In the triangle shown, two angles are 30 degrees and 80 degrees.
• Therefore, the third angle in this triangle is $$180 - 30 - 80 = 70$$ degrees.
• The angle marked with double arcs is the angle of the inscribed triangle. This angle is equal to the angle formed by the tangent and the chord, which is angle x.
• The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
• The angle subtended by the arc at the circumference is half the angle subtended at the center.
• The angle marked with double arcs is 70 degrees.
• The angle x is the angle formed by a tangent and a chord. This angle is equal to the angle subtended by the chord in the alternate segment.
• The angle subtended by the chord at the circumference is 70 degrees.
• Therefore, angle x is also 70 degrees.

Ответ: 70°