Inscribed Quadrilaterals

A quadrilateral is inscribed in a circle (or cyclic) when all four vertices lie on the circle. Each of its angles is an inscribed angle, so Inscribed Angles tells us about all four.

Por qué funciona

In inscribed quadrilateral $ABCD$, $\angle A$ intercepts the arc $\widehat{BCD}$ and the opposite angle $\angle C$ intercepts the arc $\widehat{DAB}$. Those two arcs together make the whole circle, $360^\circ$. Each angle is half its arc, so $m\angle A + m\angle C = \frac{1}{2} \cdot 360^\circ = 180^\circ$. The same holds for $\angle B$ and $\angle D$.

Drag the corners around the circle. Opposite angles always add to $180^\circ$.

Idea clave

The opposite angles of a quadrilateral inscribed in a circle are supplementary. (The converse is true too: if opposite angles are supplementary, the quadrilateral can be inscribed in a circle.)

Ejemplo resuelto: One angle

$ABCD$ is inscribed in a circle and $m\angle B = 78^\circ$. Find $m\angle D$.

$\angle B$ and $\angle D$ are opposite: $m\angle D = 180 - 78 = 102^\circ$.

Respuesta: $102^\circ$

Ejemplo resuelto: With algebra

$PQRS$ is inscribed in a circle, $m\angle P = (2x + 16)^\circ$ and $m\angle R = (3x - 1)^\circ$. Find $m\angle P$.

$(2x + 16) + (3x - 1) = 180$, so $5x + 15 = 180$ and $x = 33$.
$m\angle P = 2(33) + 16 = 82^\circ$.

Respuesta: $82^\circ$

Ejemplo resuelto: Which shapes fit?

Can a rhombus that is not a square be inscribed in a circle? Can an isosceles trapezoid?

In a non-square rhombus, opposite angles are equal but not $90^\circ$, so they cannot add to $180^\circ$: no.
In an isosceles trapezoid, base angles are equal and consecutive angles between the bases are supplementary, so each pair of opposite angles adds to $180^\circ$: yes.

Respuesta: The rhombus cannot; the isosceles trapezoid can.

Not every quadrilateral is cyclic. Every rectangle is (its opposite angles add to $90^\circ + 90^\circ$), and so is every isosceles trapezoid. A parallelogram that is not a rectangle is not: its opposite angles are equal, so they add to $180^\circ$ only when both are right angles. A kite is cyclic exactly when its two equal angles are right angles.

Error común

Adding consecutive angles (like $\angle A$ and $\angle B$) to $180^\circ$. In an inscribed quadrilateral it is the opposite angles that are supplementary. Consecutive angles are supplementary in a parallelogram, which is a different fact.

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