Angles Inside and Outside Circles: Practice

Points $A,B,C,D$ occur in order around a circle. Chords $AC,BD$ meet at $X$. Side lines $AB,DC$ meet at external $P$ with orders $P,A,B$ and $P,D,C$. Far arc $BC$ excludes $A,D$; near arc $DA$ excludes $B,C$. Angle $AXB$ is $86^\circ$, and angle $APD$ is $41^\circ$. Find the far arc $BC$.

Points $A,B,C,D$ occur in order around a circle. Chords $AC,BD$ meet at $X$. Side lines $AB,DC$ meet at external $P$ with orders $P,A,B$ and $P,D,C$. Far arc $BC$ excludes $A,D$; near arc $DA$ excludes $B,C$. Angle $AXB$ is $98^\circ$, angle $APD$ is $27^\circ$, and consecutive arc $AB$ is $67^\circ$. Find the inscribed angle $BAD$.

Ray $PT$ is tangent at $T$, and a secant has order $P,A,B$. Near arc $TA$ excludes $B$; far arc $BT$ excludes $A$. The degree measures of near and far arcs are $3x+2$ and $7x+18$. Angle $TPA$ is $48^\circ$. Find arc $AB$ excluding $T$.

Rays $PT,PU$ are tangent to one circle at distinct $T,U$, with $P$ strictly outside. In triangle $PTU$, angle $PTU$ measures $(2x+3)^\circ$, and angle $TPU$ measures $(3x+6)^\circ$. Point $R$ lies on the major arc $TU$. Find major arc $TRU$.

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$. Angles $APB,BPC$ measure $(4x-2)^\circ,(2x+8)^\circ$. Consecutive arc $AB$ is $88^\circ$. Find consecutive arc $CD$.

Two secants from external $P$ have orders $P,A,B$ and $P,D,C$. The circle order is $A,B,C,D$. Far arc $BC$ excludes $A,D$; near arc $DA$ excludes $B,C$. The consecutive arcs $AB,BC,CD,DA$ measure $30^\circ,270^\circ,30^\circ,30^\circ$. Find the angle between rays $PA$ and $PD$.

Ray $PT$ is tangent at $T$, and a secant has order $P,A,B$. Near arc $TA$ excludes $B$; far arc $BT$ excludes $A$. The near arc is $42^\circ$ and arc $AB$ excluding $T$ is $96^\circ$. Find $m\angle TPA$.

Two secants from external $P$ have orders $P,A,B$ and $P,D,C$. The circle order is $A,B,C,D$. Far arc $BC$ excludes $A,D$; near arc $DA$ excludes $B,C$. Angle $APD$ is $54^\circ$. The far and near arc measures, in degrees, are $5x+41$ and $2x+17$. Find the far arc.

Rays $PT,PU$ are tangent to one circle at distinct $T,U$, with $P$ strictly outside. Angle $TPU$ is $116^\circ$. Point $R$ lies on the major arc $TU$. Find the inscribed angle $TRU$.

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$. Angle $APB$ is $96^\circ$. The degree measures of consecutive arcs $AB,CD$ are $2x+14$ and $4x+22$. Find the inscribed angle $ACB$.

Two secants from external $P$ have orders $P,A,B$ and $P,D,C$. The circle order is $A,B,C,D$. Far arc $BC$ excludes $A,D$; near arc $DA$ excludes $B,C$. Consecutive arcs $AB,CD$ are $73^\circ,105^\circ$. The degree measures of near arc $DA$ and far arc $BC$ are $2x+6$ and $4x+14$. Find $m\angle APD$.

Ray $PT$ is tangent at $T$, and a secant has order $P,A,B$. Near arc $TA$ excludes $B$; far arc $BT$ excludes $A$. The near arc is $58^\circ$, and angle $TPA$ is $55^\circ$. Point $V$ lies on the arc $AB$ excluding $T$. Find the inscribed angle $TVB$.