The problem asks to find the value of the angle $x$ in each diagram based on circle geometry principles. Let’s analyze each case:

Diagram (1):

• Known: Central angle $\angle 62^\circ$.
• By the inscribed angle theorem, the inscribed angle $x$ is half of the central angle subtended by the same arc. $x = \frac{62^\circ}{2} = 31^\circ$

Diagram (2):

• Known: Angle $35^\circ$ is an inscribed angle, subtended by the arc across the circle.
• $x$ is the exterior angle at the vertex of the triangle, equal to the sum of the opposite interior angles: $x = 35^\circ + 35^\circ = 70^\circ$

Diagram (3):

• Known: The inscribed angle $\angle 56^\circ$.
• Using the inscribed angle theorem again, $x$ is half of the central angle subtended by the same arc: $x = \frac{56^\circ}{2} = 28^\circ$

Final Answers:

1. $x = 31^\circ$
2. $x = 70^\circ$
3. $x = 28^\circ$

Let me know if you need detailed explanations or have questions!

Related Questions:

1. How does the inscribed angle theorem work for semi-circles?
2. What are the properties of angles subtended by the same arc?
3. How to determine the angle $x$ in a cyclic quadrilateral?
4. How to solve problems where chords and tangents meet at a point?
5. What is the relationship between central angles and arcs?

Tip:

Always check if the given angle is a central angle, an inscribed angle, or an exterior angle to apply the correct geometric property.