Angles Inside and Outside Circles: Practice Level 1 | Geometry

Level 1 · Foundational 0 / 12 answered

Question 1 of 12

1

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$ . Consecutive arcs $AB$ and $CD$ measure $62^\circ$ and $104^\circ$ . Find $m\angle APB$ .

Answer choices for question 1 A $83^\circ$ B $42^\circ$ C $166^\circ$ D $97^\circ$
2

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$ . Which two consecutive arcs determine $\angle BPC$ by a half-sum?

Answer choices for question 2 A Arcs $AB$ and $CD$ B Arcs $AB$ and $BC$ C Arcs $BC$ and $DA$ D Arcs $AC$ and $BD$
3

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 $188^\circ$ and the near arc is $52^\circ$ . Find $m\angle APD$ .

Answer choices for question 3 A $120^\circ$ B $68^\circ$ C $136^\circ$ D $94^\circ$
4

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 $44^\circ$ and the far arc is $156^\circ$ . Find $m\angle TPA$ .

Answer choices for question 4 A $100^\circ$ B $112^\circ$ C $78^\circ$ D $56^\circ$
5

Rays $PT,PU$ are tangent to one circle at distinct $T,U$ , with $P$ strictly outside. The minor arc $TU$ measures $122^\circ$ . Find $m\angle TPU$ .

Answer choices for question 5 A $61^\circ$ B $58^\circ$ C $122^\circ$ D $119^\circ$
6

Two chords cross strictly inside a circle, away from its center. Which rule gives an angle at their intersection?

Answer choices for question 6 A Half the sum of the arcs facing its vertical pair B Half the difference of those two arcs C Half just one of those arcs D The full sum of those two arcs
7

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$ . Angle $APB$ is $87^\circ$ , and consecutive arc $CD$ is $98^\circ$ . Find consecutive arc $AB$ .

Answer choices for question 7 A $38^\circ$ B $185^\circ$ C $76^\circ$ D $174^\circ$
8

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 exterior angle is $42^\circ$ , and the near arc is $56^\circ$ . Find the far arc.

Answer choices for question 8 A $98^\circ$ B $84^\circ$ C $28^\circ$ D $140^\circ$
9

Points $A,V,B$ lie on a circle. The arc from $A$ to $B$ excluding $V$ measures $146^\circ$ . Find $m\angle AVB$ .

Answer choices for question 9 A $73^\circ$ B $146^\circ$ C $107^\circ$ D $292^\circ$
10

Points $A,B,C,D$ occur in that order around a circle. Chords $AC,BD$ cross at interior $P$ . Angle $APB$ measures $113^\circ$ . Find the adjacent angle $BPC$ .

Answer choices for question 10 A $113^\circ$ B $56.5^\circ$ C $226^\circ$ D $67^\circ$
11

Rays $PT,PU$ are tangent to one circle at distinct $T,U$ , with $P$ strictly outside. Angle $TPU$ measures $109^\circ$ . Find the minor arc $TU$ .

Answer choices for question 11 A $109^\circ$ B $142^\circ$ C $71^\circ$ D $251^\circ$
12

Two secants from external $P$ have orders $P,A,B$ and $P,D,C$ . The circle order is $A,B,C,D$ . Which route is the far arc in the half-difference rule for $\angle APD$ ?

Answer choices for question 12 A From $A$ to $D$ without passing through $B$ or $C$ B From $B$ to $C$ without passing through $A$ or $D$ C From $A$ to $B$ without passing through $C$ or $D$ D From $B$ to $C$ through $A$ and $D$