Circle Level - 2
Q1–10 of 30In a cyclic quadrilateral $ABCD$, $\angle A = (3x + 15)^\circ$ and $\angle C = (x + 25)^\circ$. Find the measure of $\angle A$.
Verification:
$(3x + 15) + (x + 25) = 180 \implies 4x + 40 = 180 \implies 4x = 140 \implies x = 35$.
$\angle A = 3(35) + 15 = 105 + 15 = 120^\circ$.
In cyclic quadrilateral $ABCD$, the ratio of angles $\angle A : \angle B : \angle C = 2 : 3 : 7$. What is the measure of $\angle D$?
Verification:
$\angle A + \angle C = 180^\circ \implies 2k + 7k = 180^\circ \implies 9k = 180^\circ \implies k = 20^\circ$.
$\angle B = 3(20^\circ) = 60^\circ$.
$\angle D = 180^\circ - \angle B = 180^\circ - 60^\circ = 120^\circ$.
In a cyclic quadrilateral $ABCD$, side $AB$ is a diameter of the circumcircle. If $\angle ADC = 130^\circ$, find the measure of $\angle BAC$.
Verification:
$\angle ABC = 180^\circ - 130^\circ = 50^\circ$.
In right $\triangle ABC$, $\angle BAC = 90^\circ - 50^\circ = 40^\circ$
Quadrilateral $ABCD$ is inscribed in a circle. If $\angle A = 2\angle C$ and $\angle B = 3\angle D$, what is the measure of $\angle B$?
Verification:
$\angle A + \angle C = 180^\circ \implies 2\angle C + \angle C = 180^\circ \implies \angle C = 60^\circ, \angle A = 120^\circ$.
$\angle B + \angle D = 180^\circ \implies 3\angle D + \angle D = 180^\circ \implies \angle D = 45^\circ$.
$\angle B = 3(45^\circ) = 135^\circ$.
In cyclic quadrilateral $ABCD$, $AB \parallel CD$. If $\angle A = 70^\circ$, what is the measure of $\angle B$?
Verification:
$AB \parallel CD \implies \angle A + \angle D = 180^\circ \implies \angle D = 180^\circ - 70^\circ = 110^\circ$.
Cyclic quad $\implies \angle B + \angle D = 180^\circ \implies \angle B = 180^\circ - 110^\circ = 70^\circ$
In a cyclic quadrilateral $ABCD$, side $AB$ is extended past $B$ to point $E$. If exterior angle $\angle CBE = 115^\circ$, what is the measure of interior angle $\angle ADC$?
Verification:
$\angle ABC + \angle CBE = 180^\circ \implies \angle ABC = 180^\circ - 115^\circ = 65^\circ$.
$\angle ADC = 180^\circ - \angle ABC = 180^\circ - 65^\circ = 115^\circ$.
In cyclic quadrilateral $ABCD$, side $BC$ is extended to $F$. If $\angle DCF = (2x + 10)^\circ$ and $\angle BAD = (3x - 20)^\circ$, find the value of $x$.
Verification:
$2x + 10 = 3x - 20 \implies 3x - 2x = 10 + 20 \implies x = 30$.
Side $AB$ of cyclic quad $ABCD$ is extended to $E$. If $\angle CBE = 85^\circ$ and $\angle BAC = 35^\circ$, what is the measure of $\angle BDC$?
Verification:
Chord $BC$ subtends both $\angle BAC$ and $\angle BDC$ at the circumference.
Therefore, $\angle BDC = \angle BAC = 35^\circ$.
In cyclic quadrilateral $ABCD$, side $AD$ is extended to $G$. If exterior angle $\angle CDG = 100^\circ$ and $\angle DBC = 40^\circ$, find $\angle ABD$.
Verification:
Exterior angle $\angle CDG = \angle ABC = 100^\circ$.
$\angle ABC = \angle ABD + \angle DBC \implies 100^\circ = \angle ABD + 40^\circ \implies \angle ABD = 60^\circ$.
Sides $AB$ and $DC$ of a cyclic quadrilateral $ABCD$ are extended to meet at point $P$. If exterior angle $\angle PBC = 75^\circ$ and $\angle PCB = 50^\circ$, find $\angle ADC$.
Verification:
Exterior angle at vertex $B$ is $\angle PBC = 75^\circ$.
Interior opposite angle to $\angle ABC$ is $\angle ADC \implies \angle ADC = \angle PBC = 75^\circ$.