Let’s solve each problem step by step:

1. To find angle ATC, we need to apply the properties of tangents and the fact that the angle between a tangent and a radius at the point of contact is 90 degrees.

Given:

• Circle with center O, where O is (0, 0) since it’s the center of the circle.
• Tangents TA and TB from point T touch the circle at points A and B, respectively.
• Angle AOT is given as 67 degrees.

To find angle ATC, we need to determine the measure of angle ATO first. Since TA and TB are tangents to the circle, angle ATO is a right angle (90 degrees) because it is formed between the radius OT and the tangent TA.
Therefore, angle ATC is the exterior angle of triangle ATO. The exterior angle of a triangle is equal to the sum of its remote interior angles.
Since angle ATO is a right angle (90 degrees), angle ATC is equal to the sum of angles ATO and AOT: ATC = ATO + AOT = 90° + 67° = 157°
So, angle ATC is 157 degrees.

1. To find angle ABO, we need more information or additional measurements or relationships within the given problem. Please provide any further details or measurements that may help determine the value of angle ABO.