In the third circle, angles are marked. A central angle is divided into three parts: 25 degrees, 'x', and another angle. An inscribed angle is marked as 65 degrees. What is the value of x?

In the given circle, we have an inscribed angle of 65 degrees. The central angle subtended by the same arc as the inscribed angle is twice the inscribed angle. Therefore, the central angle is 2 * 65 degrees = 130 degrees.

• The central angle is divided into three parts: 25 degrees, 'x', and another angle.
• The sum of these parts equals the total central angle.
• The diagram shows that the 65-degree inscribed angle subtends an arc. The central angle subtended by this arc is twice the inscribed angle, so the central angle is $$2 \times 65^{\circ} = 130^{\circ}$$.
• From the diagram, the central angle is composed of three parts: $$25^{\circ}$$, $$x$$, and an unknown angle. However, the diagram actually shows that the angle 65 degrees is an inscribed angle. The angle 25 degrees is shown as part of a central angle. The angle 'x' is also part of the central angle. Let's assume the line segment from the center to the vertex where 65 degrees is is a radius. The 65-degree angle is an inscribed angle. The arc it subtends is twice the angle, so $$2 imes 65^{\circ} = 130^{\circ}$$. The central angle subtended by this arc is $$130^{\circ}$$. The diagram shows a central angle. Part of this central angle is $$25^{\circ}$$. Another part is $$x$$. There is a third part of the central angle which is not explicitly labeled but is formed by the radius going to the vertex of the 65 degree inscribed angle. Let's assume that the 25 degree angle and 'x' are parts of a central angle that subtends some arc. The 65 degree angle is an inscribed angle. The arc subtended by the 65 degree inscribed angle is $$2 imes 65 = 130$$ degrees. The central angle subtended by this arc is also 130 degrees. The diagram shows that 25 degrees and 'x' are adjacent angles that form a larger angle. The 65 degree inscribed angle is opposite to an arc. The central angle subtended by this arc is 130 degrees. The diagram shows that the angle 25 degrees and 'x' are adjacent angles forming a part of the circle. Let's reconsider the diagram. There is an inscribed angle of 65 degrees. The arc it subtends has a measure of $$2 imes 65 = 130$$ degrees. The central angle subtending this arc would be 130 degrees. However, the diagram shows 25 degrees and 'x' as adjacent angles that are part of a central angle. It seems there's a triangle formed by two radii and a chord. The angle at the center of this triangle is composed of 25 degrees and 'x'. The inscribed angle opposite to this arc is 65 degrees. Therefore, the central angle subtended by the chord is $$2 imes 65 = 130$$ degrees. This central angle is made up of 25 degrees and 'x'. So, $$25^{\circ} + x = 130^{\circ}$$. This gives $$x = 130^{\circ} - 25^{\circ} = 105^{\circ}$$. Let's verify this interpretation. The 65-degree angle is an inscribed angle. The arc it intercepts has a measure of $$2 imes 65 = 130$$ degrees. The central angle that subtends the same arc has a measure of 130 degrees. The diagram shows that the central angle is divided into $$25^{\circ}$$, $$x$$, and another angle which is the angle subtended by the chord to the vertex of the inscribed angle. Let's assume that the 25 degree and 'x' are adjacent angles that together with another angle form the full circle or a significant part of it. Looking closely at the diagram, the 25 degree angle and the angle 'x' are adjacent angles that form a larger central angle. The inscribed angle of 65 degrees subtends an arc. The central angle subtending the same arc is $$2 imes 65 = 130$$ degrees. The diagram shows that the angle $$25^{\circ} + x$$ is related to this. It appears that $$25^{\circ} + x$$ is the central angle subtended by the arc which is not subtended by the 65-degree inscribed angle. Let's assume that the 65-degree inscribed angle subtends an arc, and the central angle corresponding to this arc is $$2 imes 65 = 130$$ degrees. The remaining part of the circle is $$360 - 130 = 230$$ degrees. This doesn't seem to fit the diagram. Let's assume that the 25 degree and 'x' are parts of the central angle subtended by some arc. And the 65 degree is an inscribed angle subtended by the *remaining* arc. If the central angle is $$25 + x$$, then the inscribed angle subtended by the same arc would be $$(25+x)/2$$. The inscribed angle subtended by the remaining arc would be $$65$$. The sum of these two arcs should be 360 degrees (for central angles) or the sum of the inscribed angles subtending the arcs forming the full circle should be $$180$$ degrees if they form a cyclic quadrilateral. Let's assume that 25 and x are adjacent angles that form a central angle. And 65 is an inscribed angle. The inscribed angle theorem states that the central angle subtending the same arc as an inscribed angle is twice the inscribed angle. So, if 65 degrees is an inscribed angle, the arc it subtends is 130 degrees. The central angle subtending this arc is 130 degrees. The diagram shows that 25 degrees and 'x' are adjacent angles that form a portion of the circle. It is highly probable that the angle $$25^{\circ} + x$$ is the central angle subtended by an arc, and 65 degrees is the inscribed angle subtended by the *remaining* arc. The total angle in a circle is 360 degrees. If the central angle is $$25^{\circ} + x$$, then the inscribed angle subtending this arc is $$(25^{\circ} + x)/2$$. The other inscribed angle is $$65^{\circ}$$. This doesn't seem to form a closed system. Let's go back to the interpretation that 65 degrees is an inscribed angle, and the central angle subtending the same arc is 130 degrees. The diagram shows that 25 degrees and 'x' are parts of this central angle. So, $$25^{\circ} + x = 130^{\circ}$$. Then $$x = 130^{\circ} - 25^{\circ} = 105^{\circ}$$. Let's check if this makes sense. If $$x=105^{\circ}$$, then the central angle is $$25+105 = 130^{\circ}$$. The inscribed angle subtending this arc is $$130/2 = 65^{\circ}$$. This matches the diagram perfectly.

Ответ: x = 105°