Determine the value of angle x in the third figure.

Solution:

• The green shaded angle is an inscribed angle.
• The inscribed angle is subtended by an arc. The angle x is also an angle formed by a tangent and a chord.
• The angle between the tangent and a chord is equal to the angle in the alternate segment.
• The green shaded angle is in the alternate segment.
• Therefore, x is equal to the green shaded angle. However, the value of the green shaded angle is not given.
• Let's reconsider the diagram. The line passing through the center is a diameter or a radius. The angle marked x and the green angle are adjacent angles that form a larger angle.
• If we assume the green angle is given and we need to find x, it is not directly possible without more information.
• Let's assume the green angle is a given value, say 'y'. Then x = y because they subtend the same arc or are alternate segment angles. Without a specific value for the green angle, x cannot be determined numerically.
• Looking closely at the diagram, the green angle is shown as 90 degrees. This is suggested by the right-angle symbol within the green shaded area.
• If the green angle is 90°, and x is adjacent to it, and together they form an angle subtended by some arc, we need to find that relationship.
• Let's interpret the green area as the angle subtended by an arc. The angle between the tangent and the chord is equal to the angle in the alternate segment. So, x is equal to the angle in the alternate segment. The green angle is given as 90 degrees. This implies that the arc subtends an angle of 90 degrees at the circumference.
• Therefore, x = 90°.

Answer: 90°