Circle Level - 1
Q1–10 of 50An arc $AB$ subtends an angle of $110^\circ$ at the center $O$ of a circle. What is the measure of the angle subtended by the same arc at any point $C$ on the major arc of the circle?
Hint: Inscribed Angle Theorem: The angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining circumference ($\angle ACB = \frac{1}{2} \angle AOB = \frac{110^\circ}{2} = 55^\circ$)
In a circle with center $O$, points $A, B, C,$ and $D$ lie on the circumference such that $ABCD$ is a cyclic quadrilateral. If $\angle ABC = 108^\circ$, find $\angle ADC$.
Hint: Opposite angles of a cyclic quadrilateral are supplementary ($\angle ABC + \angle ADC = 180^\circ \implies \angle ADC = 180^\circ - 108^\circ = 72^\circ$).
$AB$ is a diameter of a circle with center $O$. $C$ is a point on the circumference such that $\angle BAC = 35^\circ$. What is the measure of $\angle ABC$?
Hint: Angle in a semicircle is a right angle ($\angle ACB = 90^\circ$). In $\triangle ABC$, $\angle ABC = 180^\circ - (90^\circ + 35^\circ) = 55^\circ$.
In a cyclic quadrilateral $ABCD$, side $AB$ is extended to a point $E$. If external angle $\angle CBE = 82^\circ$, what is the measure of $\angle ADC$?
Hint: The exterior angle of a cyclic quadrilateral equals the interior opposite angle ($\angle CBE = \angle ADC = 82^\circ$).
In a circle with center $O$, chords $AB$ and $CD$ intersect at an internal point $P$. If $\angle AOC = 50^\circ$ and $\angle BOD = 70^\circ$ (where $A, C, B, D$ lie in cyclic order), find $\angle APC$.
Hint: The angle formed by two intersecting chords inside a circle is half the sum of their intercepted central arcs: $\angle APC = \frac{\angle AOC + \angle BOD}{2} = \frac{50^\circ + 70^\circ}{2} = 60^\circ$.
Points $A, B, C, D$ lie on a circle in order. If $\angle ADB = 42^\circ$, what is the measure of $\angle ACB$?
Hint: Angles subtended by the same arc ($AB$) in the same segment of a circle are equal ($\angle ACB = \angle ADB = 42^\circ$).
$AB$ is a chord of a circle subtending a central angle of $80^\circ$. $P$ is a point on the minor arc $AB$. What is $\angle APB$?
Hint: Angle on major arc $= \frac{80^\circ}{2} = 40^\circ$. Minor arc point $P$ and major arc point $Q$ form a cyclic quadrilateral, so $\angle APB = 180^\circ - 40^\circ = 140^\circ$.
In a circle, arc $AB$ is one-sixth of the total circumference. What is the measure of the angle subtended by arc $AB$ at any point on the remaining part of the circle?
Hint: Central angle $= \frac{360^\circ}{6} = 60^\circ$. Inscribed angle $= \frac{60^\circ}{2} = 30^\circ$.
$A, B, C$ are three points on a circle with center $O$. If $\angle AOB = 90^\circ$ and $\angle BOC = 120^\circ$, find $\angle ABC$ assuming $O$ lies inside $\triangle ABC$.
Hint: Central angle $\angle AOC = 360^\circ - (90^\circ + 120^\circ) = 150^\circ$. Inscribed angle $\angle ABC = \frac{150^\circ}{2} = 75^\circ$.
In a circle, $PQ$ is a chord equal to the radius $R$ of the circle. What is the angle subtended by $PQ$ at the major circumference?
Hint: Triangle formed by radius $OP$, $OQ$, and chord $PQ$ is equilateral, so central angle $\angle POQ = 60^\circ$. Angle at major circumference $= \frac{60^\circ}{2} = 30^\circ$.