Angles Inside and Outside Circles: Practice

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$. The degree measures of arcs $AB,CD$ are $3x+8$ and $5x+12$. Angle $APB$ is $74^\circ$. Find $x$.

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 far and near arc measures, in degrees, are $8x+30$ and $2x+6$. Angle $APD$ is $66^\circ$. Find $x$.

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 $46^\circ$, and angle $TPA$ is $67^\circ$. Find the far arc.

Rays $PT,PU$ are tangent to one circle at distinct $T,U$, with $P$ strictly outside. The minor arc $TU$ measures $(4x+15)^\circ$, and angle $TPU$ measures $(3x+18)^\circ$. Find $x$.

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$. Angle $APB$ is $114^\circ$, and consecutive arcs $AB,BC$ are $95^\circ,70^\circ$. Find $m\angle BPC$.

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 far arc is $231^\circ$, and the near arc is $49^\circ$. 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 $38^\circ$, and angle $TPA$ is $92^\circ$. Find the arc $AB$ excluding $T$.

Rays $PT,PU$ are tangent to one circle at distinct $T,U$, with $P$ strictly outside. The minor arc $TU$ measures $(5x-10)^\circ$, and angle $TPU$ measures $(2x+15)^\circ$. Find the major arc from $T$ to $U$.

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$. Consecutive arcs $AB,CD$ measure $86^\circ,124^\circ$. Angle $BPC$ measures $(3x+9)^\circ$. Find $x$.

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 far arc is $186^\circ$ and the near arc is $54^\circ$. Find the exterior angle $BDC$ of triangle $PBD$.

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 and far arcs measure $72^\circ$ and $188^\circ$. Point $Y$ lies on the secant line with order $Y,P,A,B$. Find $m\angle TPY$.

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$. Angle $APB$ is $98^\circ$, and consecutive arc $BC$ is $84^\circ$. Find consecutive arc $DA$.