To determine the sizes of the angles or segments listed (3.1 to 3.6) and provide reasons, I will work through each one based on the geometric properties of the circle, given angles, and the layout from the image.

3.1 $\angle D_2$

• Analysis: $D_2$ typically denotes an angle at a specific point in the diagram. However, based on the image's geometry, it appears to be associated with point $D$. Unfortunately, due to the small size of the image, more precise details about $D_2$ cannot be provided without further information.

• Reasoning: If we assume $D_2$ refers to an angle formed by two intersecting lines at $D$, then it would depend on the specific lines or circle arcs involved. More context is needed for exact reasoning.

3.2 $\angle B_2$

• Analysis: $B_2$ might represent another angle at point $B$. The label might suggest it is related to the angle $\angle AOB$ in the circle.

• Reasoning: If $B_2$ refers to an angle formed by $OB$ with some other line or arc, and given $OB$ is the radius, this angle would typically relate to properties like central angle or inscribed angle, depending on the given geometry.

3.3 $\angle E_2$

• Analysis: Similar to $D_2$ and $B_2$, $E_2$ likely refers to an angle associated with point $E$. Further context from the diagram is needed to determine its exact size.

• Reasoning: $E_2$ could relate to an angle involving line segments or arcs passing through $E$. The angle's size depends on the relationship of these segments or arcs with the circle.

3.4 $\angle E_1$

• Analysis: Similar to $E_2$, but this refers to a different angle at $E$.

• Reasoning: Without more detail, the exact size can't be determined, but it would follow the same geometric principles as $E_2$.

3.5 $\angle COE$

• Analysis: $\angle COE$ refers to the angle subtended by points $C$ and $E$ at the center $O$ of the circle.

• Reasoning: Since $O$ is the center and $OB$ is given as 6 cm, $\angle COE$ would be a central angle. The size of this angle depends on the length of the arc $CE$ on the circle. The central angle is directly proportional to the length of the arc it subtends. Without specific arc measurements or more details, an exact angle can't be determined.

3.6 $\angle EC$

• Analysis: $\angle EC$ likely refers to an angle involving points $E$ and $C$, possibly at point $C$.

• Reasoning: If $EC$ is the arc or segment, then the angle's size depends on where the vertex of the angle is located relative to the circle (center or circumference). If it's an inscribed angle, it would be half the size of the corresponding central angle.

Summary:

Due to the small size of the image and lack of precise details, a detailed determination of these angles and their exact sizes isn't possible without more context. However, each angle's size would depend on the geometric relationships in the diagram, such as central angles, inscribed angles, or angles between lines and tangents.

Would you like to provide more context or ask another question related to this geometry?

Related Questions:

1. What is the inscribed angle theorem, and how does it relate to angles in a circle?
2. How do you calculate the measure of a central angle given the arc length?
3. What is the relationship between a tangent and a radius in a circle?
4. How can you use trigonometry to find unknown angles in a circle?
5. What are the properties of cyclic quadrilaterals?

Tip: When working with circle geometry, always remember the key relationships like the central angle being twice the inscribed angle on the same arc.