In the first diagram, find the value of x.

Solution:

• The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.
• In the given diagram, the arc AC subtends an angle of 63 degrees at point B on the circumference.
• Therefore, the angle AOC (at the center) is 2 * 63 = 126 degrees.
• The angle subtended by arc CD at point B is given as 130 degrees. This is an inscribed angle.
• The central angle subtended by arc CD would be 2 * 130 = 260 degrees. However, this forms a reflex angle. The other angle AOC is formed by the chord AC.
• Let's reconsider the information. We are given angle ABC = 63 degrees and arc CD corresponds to an inscribed angle which we don't directly have. The angle marked 130 degrees is an arc measure.
• The arc AD subtends angle ABD at the circumference. The angle subtended by arc AD at the center O is the angle AOD.
• The arc BC subtends angle BAC at the circumference. The angle subtended by arc BC at the center O is the angle BOC.
• The angle marked 130 degrees is the measure of arc AB.
• The central angle subtended by arc AB is angle AOB = 130 degrees.
• Angle x is angle AOC. We need to find the measure of arc AC.
• The sum of arcs in a circle is 360 degrees. Arc AB + Arc BC + Arc CA = 360 degrees.
• We are given that the inscribed angle ABC = 63 degrees. This angle subtends arc AC.
• Therefore, the measure of arc AC = 2 * angle ABC = 2 * 63 = 126 degrees.
• The angle marked 'x' is the central angle subtended by arc AC. So, x = angle AOC = measure of arc AC.
• Therefore, x = 126 degrees.

Answer: x = 126°