Circles

2010018002

Level: 
A
In the figure the cut of a regular polygon with unspecified number of vertices is shown. The red angle is the central angle of the polygon, the blue angle is the interior angle of the polygon. Suppose we consider a regular polygon with the central angle of \(30^{\circ}\), then find the measure of the interior angle of this polygon.
\(150^{\circ}\)
\(180^{\circ}\)
\(90^{\circ}\)
\(210^{\circ}\)

2010012808

Level: 
A
A regular nonagon \( ABCDEFGHI \) is inscribed in a circle. Calculate the measures of all interior angles of the quadrilateral \( BDGI \). (See the picture.)
\( \alpha=100^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=100^{\circ} \)
\( \alpha=110^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=90^{\circ} \)
\( \alpha=110^{\circ};\ \beta=70^{\circ};\ \gamma=70^{\circ};\ \delta=110^{\circ} \)
\( \alpha=120^{\circ};\ \beta=80^{\circ};\ \gamma=80^{\circ};\ \delta=120^{\circ} \)

2010012807

Level: 
A
A regular dodecagon \( ABCDEFGHIJKL \) is inscribed in a circle. Find the measures of all interior angles of the quadrilateral \( BFIL \). (See the picture.)
\( \alpha=90^{\circ};\ \beta=75^{\circ};\ \gamma=90^{\circ};\ \delta=105^{\circ} \)
\( \alpha=90^{\circ};\ \beta=60^{\circ};\ \gamma=80^{\circ};\ \delta=130^{\circ} \)
\( \alpha=80^{\circ};\ \beta=75^{\circ};\ \gamma=90^{\circ};\ \delta=115^{\circ} \)
\( \alpha=90^{\circ};\ \beta=105^{\circ};\ \gamma=90^{\circ};\ \delta=105^{\circ} \)

2010012806

Level: 
A
Points \( A \) and \( B \) divide the circle \( k \) into two arcs whose lengths are in the ratio \( 3:12 \). Point \( C \) is an interior point of the longer arc. What is the degree measure of the angle \( ACB \)?
\( 36^{\circ}\)
\( 72^{\circ}\)
\( 24^{\circ}\)
\( 45^{\circ}\)