To determine the angles based on the information provided, let's analyze the geometry involving the circle, its center, diameters, and angles. Since we don't have the actual diagram to visually refer to, I'll provide reasoning based on common properties of circles and angles.
3.1 Determine \( D_2 \)
Given:
- \( A = 22^\circ \)
- \( ED = DA \) implies that triangle \( EDA \) is isosceles.
In triangle \( EDA \):
- The angles at \( E \) and \( D \) are equal since \( ED = DA \).
- Let's denote the angle at \( D \) as \( D_x \) and the angle at \( E \) as \( E_x \).
Using the triangle sum property: \[ A + D_x + E_x = 180^\circ \] \[ 22^\circ + D_x + D_x = 180^\circ \] \[ 22^\circ + 2D_x = 180^\circ \] \[ 2D_x = 158^\circ \] \[ D_x = 79^\circ \]
Thus, \[ D_2 = 79^\circ \]
3.2 Determine \( B_2 \)
\( B \) is connected to the circle with diameter \( BOE \). Using the property of angles in a semicircle:
- \( \angle BAE \) is an inscribed angle subtended by the diameter \( BE \).
Therefore: \[ B_2 = \frac{1}{2} \times \text{(arc subtended by } BOE) = \frac{1}{2} \times 180^\circ = 90^\circ \]
3.3 Determine \( E_2 \)
In triangle \( EDA \):
- We already established this is isosceles.
- Therefore, we also have \( E_2 = D_2 = 79^\circ \).
Also note: \[ E_2 = 79^\circ \]
3.4 Determine \( E_1 \)
Since \( E \) subtended the same arc as \( D \):
- Also \( B \) being the midpoint of the chord \( DE \) means \( \angle EAC \) will correspond similarly.
\[ E_1 = \angle ADE = 79^\circ \text{ (by inscribed angles)} \]
3.5 Determine \( COB \)
As \( O \) is the center and \( BO \) is a radius, \( CO \) is as well:
- Angles at the center of a circle are twice those at the circumference.
\[ COB = 2 \times A = 2 \times 22^\circ = 44^\circ \]
3.6 Determine \( ECB \)
Using the angle sum properties and oppositional angles,
- \( ECB \) is an interior angle formed by radii.
Since it involves the cyclic nature of the angles, consider \( EBC \) means opposite angles are equal:
\[ ECB = \angle EAC = 79^\circ \text{ or lesser angles subtended} \]
Summary of the angles
- 3.1 \( D_2 \): \( 79^\circ \)
- 3.2 \( B_2 \): \( 90^\circ \)
- 3.3 \( E_2 \): \( 79^\circ \)
- 3.4 \( E_1 \): \( 79^\circ \)
- 3.5 \( COB \): \( 44^\circ \)
- 3.6 \( ECB \): Varies on inscribed angles \( = 22^\circ \)
These calculations use properties of isosceles triangles, angles subtended by diameters and arcs, and circle theorems. Adjust the calculations based on the specific structure provided in the diagram if necessary.
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