PQRSTU is a regular hexagon.
Question:

PQRSTU is a regular hexagon. Determine each angle of ΔPQT.

Solution:

A regular hexagon is made up of 6 equilateral triangles.

So, $\angle \mathrm{PQT}=60^{\circ}$ and $\angle \mathrm{QTP}=30^{\circ}$

Since the sum of the angles of $\triangle \mathrm{PQT}$ is $180^{\circ}$, we have:

$\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{T}=180^{\circ}$

$\Rightarrow \angle \mathrm{P}+60^{\circ}+30^{\circ}=180^{\circ}$

$\Rightarrow \angle \mathrm{P}=180^{\circ}-90^{\circ}$

$\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{T}=180^{\circ}$

$\Rightarrow \angle \mathrm{P}+60^{\circ}+30^{\circ}=180^{\circ}$

$\Rightarrow \angle \mathrm{P}=180^{\circ}-90^{\circ}$

$\Rightarrow \angle \mathrm{QPT}=90^{\circ}$

$\therefore$ The angles of the triangle are $90^{\circ}, 60^{\circ}$ and $30^{\circ}$.

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