To solve for the angles $x$, $y$, and $z$ in the given diagram, let's break it down step by step.

Step 1: Identify the Known Angles

• The angle at the point $O$ (central angle) is $55^\circ$.
• The angle $20^\circ$ is given between the tangent and the chord.

Step 2: Use the Properties of the Circle

1. Angle $x$ (Angle between tangent and chord):

   • The angle between the tangent and the chord $x$ is equal to the angle subtended by the chord on the opposite side of the circle. This is known as the "Alternate Segment Theorem."
   • Therefore, $x = 55^\circ$.

2. Angle $y$ (Central angle):

   • The central angle $y$ is twice the inscribed angle that subtends the same arc.
   • In this case, the angle subtended by the arc at the center is $\gamma$.
   • Since $x = 55^\circ$, and $\gamma$ and $x$ subtend the same arc, the angle $y$ at the center should be: $y = 2 \times \gamma = 2 \times 55^\circ = 110^\circ$

3. Angle $z$ (Remaining angle in the triangle):

   • The sum of angles in a triangle is always $180^\circ$.
   • Therefore, in the triangle with angles $55^\circ$, $20^\circ$, and $z$: $z = 180^\circ - 55^\circ - 20^\circ = 105^\circ$

Final Answers:

• $x = 55^\circ$
• $y = 110^\circ$
• $z = 105^\circ$

Would you like more details on any of the steps, or do you have any other questions?

Relative Questions:

1. What is the Alternate Segment Theorem and how is it used in circle geometry?
2. Can the angle sum property be applied to other geometric shapes?
3. What are the properties of angles subtended by the same chord in a circle?
4. How does the Central Angle Theorem work in this scenario?
5. What other circle theorems might be useful for similar problems?

Tip:

Remember that the sum of angles in any triangle is always $180^\circ$, which is a useful fact in solving many geometric problems.