The angles of a quadrilateral

A diagonal of a quadrilateral is a segment joining two vertices that are not neighbours.

• The quadrilateral $\displaystyle ABCD$ has two diagonals: $\displaystyle AC$ and $\displaystyle BD$.
• The sides join neighbouring vertices: $\displaystyle AB$, $\displaystyle BC$, $\displaystyle CD$ and $\displaystyle DA$.

In every quadrilateral, the four angles add up to $\displaystyle 360^{\circ}$.

The diagonal $\displaystyle AC$ cuts the quadrilateral into two triangles: $\displaystyle ABC$ and $\displaystyle ACD$.

• The angles of triangle $\displaystyle ABC$ are $\displaystyle \alpha$, $\displaystyle \beta$ and $\displaystyle \gamma$, and they add up to $\displaystyle 180^{\circ}$.
• The angles of triangle $\displaystyle ACD$ are $\displaystyle \delta$, $\displaystyle \varepsilon$ and $\displaystyle \varphi$, and they add up to $\displaystyle 180^{\circ}$ too.
• Together these six angles make up exactly the four angles of the quadrilateral: $\displaystyle {\angle A}={\alpha+\delta}$, $\displaystyle {\angle B}={\beta}$, $\displaystyle {\angle C}={\gamma+\varepsilon}$, $\displaystyle {\angle D}={\varphi}$.

So the angles of a quadrilateral add up to $\displaystyle {180^{\circ}+180^{\circ}}={360^{\circ}}$:

A rectangle fits: four right angles, $\displaystyle {4\cdot90^{\circ}}={360^{\circ}}$.

In quadrilateral $\displaystyle ABCD$: $\displaystyle {\angle A}={80^{\circ}}$, $\displaystyle {\angle B}={95^{\circ}}$, $\displaystyle {\angle C}={110^{\circ}}$.
We find $\displaystyle \angle D$.

• The three known angles together: $\displaystyle {80^{\circ}+95^{\circ}+110^{\circ}}={285^{\circ}}$.
• The angles of a quadrilateral add up to $\displaystyle 360^{\circ}$, so $\displaystyle {\angle D}={360^{\circ}-285^{\circ}}={75^{\circ}}$.

Three angles of a quadrilateral are given.
Calculate the fourth angle.

$\displaystyle 70^{\circ}$, $\displaystyle 100^{\circ}$ and $\displaystyle 120^{\circ}$.

$\displaystyle 90^{\circ}$, $\displaystyle 90^{\circ}$ and $\displaystyle 65^{\circ}$.

$\displaystyle 135^{\circ}$, $\displaystyle 45^{\circ}$ and $\displaystyle 135^{\circ}$.

In the kite-shaped quadrilateral $\displaystyle ABCD$: $\displaystyle {\angle B}={\angle D}$.
$\displaystyle {\angle A}={50^{\circ}}$, $\displaystyle {\angle C}={110^{\circ}}$.
We find $\displaystyle \angle B$ and $\displaystyle \angle D$.

• $\displaystyle \angle A$ and $\displaystyle \angle C$ together: $\displaystyle {50^{\circ}+110^{\circ}}={160^{\circ}}$.
• The other two angles get $\displaystyle {360^{\circ}-160^{\circ}}={200^{\circ}}$.
• They are equal, so each of them is $\displaystyle {200^{\circ}:2}={100^{\circ}}$.

In the quadrilateral $\displaystyle ABCD$ in the drawing: $\displaystyle {\angle B}={\angle D}$.
$\displaystyle {\angle A}={70^{\circ}}$, $\displaystyle {\angle C}={130^{\circ}}$.
Find $\displaystyle \angle B$ and $\displaystyle \angle D$, and explain.

Answer, and explain.

Is there a quadrilateral all four of whose angles are acute?

Three angles of a quadrilateral are right angles. What is the fourth angle?

All four angles of a quadrilateral are equal. How many degrees is each of them?

Three angles of a quadrilateral are equal, and the fourth angle is $\displaystyle 60^{\circ}$.
Find the three equal angles.