Cyclic quadrilateral questions with answers

Questions

1. Question 1Opposite angles

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle. Find $x$.

   Show the worked solution

   1. Step 1: Find x$\mathstrut x = 180^\circ - 79^\circ$ (∠s in opp. segments)$x = 101$
2. Question 2Exterior angle

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle and $BC$ is produced to $E$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠BCD$\mathstrut \angle BCD = 180^\circ - 94^\circ$ (∠s in opp. segments)$\angle BCD = 86^\circ$
   2. Step 2: BCE is a straight line$\mathstrut x = 180^\circ - 86^\circ$ (adj. ∠s on a str. line)$x = 94$So the exterior angle equals the interior opposite angle (ext. ∠ of cyclic quad.)
3. Question 3With the angle at the centre

   In the diagram, $A$, $B$, $C$ and $D$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ADC$\mathstrut \angle ADC = \dfrac{152^\circ}{2}$ (∠ at centre = 2 ∠ at circumference)$\angle ADC = 76^\circ$
   2. Step 2: Find x$\mathstrut x = 180^\circ - 76^\circ$ (∠s in opp. segments)$x = 104$
4. Question 4Do the four points lie on a circle?

   The diagram shows a quadrilateral $ABCD$. Do $A$, $B$, $C$ and $D$ lie on a circle? Choose the answer with its reason.

   Show the worked solution

   1. Step 1: Find angle BCD (angle sum of quadrilateral)$\angle BCD = 360^\circ - 104^\circ - 72^\circ - 108^\circ$$\phantom{\angle BCD} = 76^\circ$ (∠ sum of quad.)
   2. Step 2: Add the opposite angles DAB and BCD$\angle DAB + \angle BCD = 104^\circ + 76^\circ$$\phantom{\angle DAB + \angle BCD} = 180^\circ$
   3. Step 3: The points lie on a circle exactly when the opposite angles add up to 180°Yes: angle DAB + angle BCD = 180°.