Determine the value of angle x in the first figure.

Solution:

• The sum of angles in a triangle is 180 degrees. In the given triangle, two angles are 30° and 80°.
• Therefore, the third angle of the triangle is 180° - 30° - 80° = 70°.
• The angle marked as x is an inscribed angle subtended by an arc. The central angle subtended by the same arc would be twice the inscribed angle. Alternatively, the angle between the tangent and a chord is equal to the inscribed angle subtended by the chord in the alternate segment.
• The angle subtended by the arc at the circumference is 70°. The angle between the chord and the tangent (x) is equal to the angle subtended by the chord at the circumference in the alternate segment.
• Thus, x = 70°.

Answer: 70°