Calculate the unknown angles in the image.

Okay, let's analyze each diagram and calculate the unknown angles. Diagram 1: In the first diagram, we have a circle with points A, M, and B on the circumference. The angle \(\angle AMB\) is given as 80°. We need to find the angle \(\angle AOB\), where O is the center of the circle. According to the inscribed angle theorem, the angle subtended by an arc at the center of a circle is twice the angle subtended by it at any point on the remaining part of the circle. Therefore: $$\angle AOB = 2 \times \angle AMB = 2 \times 80° = 160°$$ Answer: 160° Diagram 4: In the fourth diagram, we have points A, B, and C on the circumference of a circle with center O. \(\angle OAC\) is given as 35°. We need to find \(\angle ABC\). Since OA and OC are radii of the circle, triangle OAC is an isosceles triangle. Therefore, \(\angle OCA = \angle OAC = 35°\). Now, we can find \(\angle AOC\): $$\angle AOC = 180° - \angle OAC - \angle OCA = 180° - 35° - 35° = 110°$$ \(\angle ABC\) is an inscribed angle that subtends the same arc AC as the central angle \(\angle AOC\). Therefore: $$\angle ABC = \frac{1}{2} \times \angle AOC = \frac{1}{2} \times 110° = 55°$$ Answer: 55° Diagram 5: In the fifth diagram, points A, D, and C are on the circumference, and O is the center. \(\angle OCA\) is given as 64°. We need to find \(\angle ADC\). Triangle OAC is an isosceles triangle (OA = OC). So \(\angle OAC = \angle OCA = 64°\). Then, \(\angle AOC = 180° - 64° - 64° = 52°\). \(\angle ADC\) is an inscribed angle subtending the same arc as \(\angle AOC\) but on the opposite side. Therefore, \(\angle ADC = \frac{1}{2} (360 - \angle AOC)\), as \(\angle ADC\) is a reflex angle. However, since A, D, and C are on the circumference, \(\angle ADC\) subtends the arc AC, which is the reflex angle of \(\angle AOC\) at the center. So, the reflex \(\angle AOC = 360° - 52° = 308°\). Thus, \(\angle ADC = \frac{1}{2} \times 308° = 154°\). Answer: 154° Diagram 6: In the sixth diagram, points K, A, T, and C lie on the circumference of the circle. \(\angle AKT\) is 42°. We need to find \(\angle ACK\). Since angles \(\angle AKT\) and \(\angle ACK\) subtend the same arc AK, they are equal. Thus, \(\angle ACK = \angle AKT = 42°\). Answer: 42° Diagram 7: In the seventh diagram, we have points D, N, E, and C on the circumference of a circle. \(\angle DNE\) is 20°. We need to find \(\angle DCE\) and \(\angle CDE\). Since \(\angle DCE\) and \(\angle DNE\) subtend the same arc DE, they are equal. So, \(\angle DCE = \angle DNE = 20°\). Without additional information or constraints, it is impossible to determine \(\angle CDE\). Answer: \(\angle DCE = 20°\) Diagram 8: In the eighth diagram, points A, B, C, and D lie on the circumference of a circle, and \(\angle BCD\) is given as 22°. We need to find \(\angle BAD\). Since angles \(\angle BCD\) and \(\angle BAD\) are opposite angles in a cyclic quadrilateral, their sum is 180°. Therefore, \(\angle BAD = 180° - \angle BCD = 180° - 22° = 158°\). Answer: 158° Diagram 3: In diagram 3, point K, N, and M lie on the circumference of the circle, and \(\angle NKM\) is 24°. The unknown angle \(\angle NOM\) is the central angle subtended by the same arc as \(\angle NKM\). The central angle is twice the inscribed angle. Therefore, \(\angle NOM = 2 \times 24° = 48°\). Answer: 48°