Arcs, chords, tangents, inscribed angles Knowledge-Based Practice

Questions

1. 1.

   Two tangent segments, $PA$ and $PB$, touch a circle at points $A$ and $B$ from an external point $P$. If $\angle APB = 46^\circ$ and point $C$ lies on the major arc $\overparen{AB}$, what is the measure of the inscribed angle $\angle ACB$?

   • A) $46^\circ$
   • B) $54^\circ$
   • C) $67^\circ$
   • D) $73^\circ$
2. 2.

   A quadrilateral $ABCD$ is inscribed in a circle such that diagonal $BD$ bisects $\angle ABC$. If the measure of $\angle BCD$ is $104^\circ$ and the measure of arc $\overparen{AB}$ is $110^\circ$, what is the measure of arc $\overparen{CD}$?

   • A) $98^\circ$
   • B) $76^\circ$
   • C) $88^\circ$
   • D) $54^\circ$
3. 3.

   From an external point $P$, a tangent line touches a circle at point $T$, and a secant line intersects the circle at points $A$ and $B$ such that $A$ lies between $P$ and $B$. If the measure of arc $\overparen{BT}$ is $140^\circ$ and $\angle BPT = 36^\circ$, what is the measure of arc $\overparen{AT}$?

   • A) $68^\circ$
   • B) $72^\circ$
   • C) $104^\circ$
   • D) $52^\circ$
4. 4.

   In a circle, chords $AB$ and $CD$ intersect at point $P$ inside the circle. If the measure of arc $\overparen{AC}$ is $42^\circ$ and the measure of arc $\overparen{BD}$ is $78^\circ$, what is the measure of $\angle APD$?

   • A) $60^\circ$
   • B) $110^\circ$
   • C) $120^\circ$
   • D) $138^\circ$
5. 5.

   Line $AT$ is tangent to a circle at point $A$, and chord $AC$ forms an angle of $54^\circ$ with line $AT$. If point $B$ lies on the major arc $\overparen{AC}$ such that $AB = BC$, what is the measure of $\angle BAC$?

   • A) $54^\circ$
   • B) $63^\circ$
   • C) $72^\circ$
   • D) $126^\circ$

Answers and step-by-step solutions appear after you finish the test.