The angles of a triangle

In every triangle, the three angles add up to $\displaystyle 180^{\circ}$.

A sheet of paper shows it:

• Cut out a triangle, and mark its angles $\displaystyle \alpha$, $\displaystyle \beta$ and $\displaystyle \gamma$.
• Tear off the triangle’s three corners.
• Lay the corners side by side, so that the three vertices meet at one point.

Together the three corners fill a straight angle, that is $\displaystyle 180^{\circ}$:

• This holds in every triangle: big or small, narrow or wide.
• With a protractor too: measure the three angles of a triangle and add them up. You get $\displaystyle 180^{\circ}$, or a number very close to it – a measurement is never quite exact.

In triangle $\displaystyle ABC$: $\displaystyle {\angle A}={50^{\circ}}$, $\displaystyle {\angle B}={70^{\circ}}$.
We find $\displaystyle \angle C$.

• The two known angles together: $\displaystyle {50^{\circ}+70^{\circ}}={120^{\circ}}$.
• The angles of a triangle add up to $\displaystyle 180^{\circ}$, so $\displaystyle {\angle C}={180^{\circ}-120^{\circ}}={60^{\circ}}$.

In one calculation: $\displaystyle {\angle C}={180^{\circ}-(50^{\circ}+70^{\circ})}={60^{\circ}}$.

Two angles of triangle $\displaystyle ABC$ are given.
Calculate the third angle, $\displaystyle \angle C$.

$\displaystyle {\angle A}={35^{\circ}}$, $\displaystyle {\angle B}={85^{\circ}}$.

$\displaystyle {\angle A}={90^{\circ}}$, $\displaystyle {\angle B}={27^{\circ}}$.

$\displaystyle {\angle A}={112^{\circ}}$, $\displaystyle {\angle B}={41^{\circ}}$.

By its angles, a triangle is one of three kinds:

• An acute-angled triangle: all three of its angles are acute, like triangle $\displaystyle ABC$.
• A right-angled triangle: one of its angles is a right angle, like triangle $\displaystyle DEF$ (the angle $\displaystyle D$).
• An obtuse-angled triangle: one of its angles is obtuse, like triangle $\displaystyle KLM$ (the angle $\displaystyle M$).

Every triangle has at least two acute angles. It has at most one right or obtuse angle.

Try to build a triangle with two right angles, at $\displaystyle B$ and at $\displaystyle C$:

• The two sides leaving $\displaystyle B$ and $\displaystyle C$ go straight up, side by side, and never meet – there is no third vertex.
• In numbers: two right angles already make $\displaystyle {90^{\circ}+90^{\circ}}={180^{\circ}}$, and nothing is left for the third angle.
• Two obtuse angles make more than $\displaystyle 180^{\circ}$, so they cannot be in one triangle either.

By their sides, there are three kinds of triangles:

• A scalene triangle: its three sides have different lengths, like $\displaystyle ABC$.
• An isosceles triangle: two of its sides are equal, like $\displaystyle DEF$.
• An equilateral triangle: all three of its sides are equal, like $\displaystyle KLM$.

In a drawing, a small dash on a side marks its length: sides with the same number of dashes are equal.

Two angles of a triangle are given.
Find the third angle.
Write which kind of triangle it is by its angles: acute-angled, right-angled or obtuse-angled.

$\displaystyle 25^{\circ}$ and $\displaystyle 65^{\circ}$.

$\displaystyle 30^{\circ}$ and $\displaystyle 45^{\circ}$.

$\displaystyle 55^{\circ}$ and $\displaystyle 70^{\circ}$.

Is there such a triangle?
Write yes or no, and explain.

A triangle with two obtuse angles.

A triangle whose angles are $\displaystyle 30^{\circ}$, $\displaystyle 60^{\circ}$ and $\displaystyle 100^{\circ}$.

A triangle each of whose angles is $\displaystyle 60^{\circ}$.

The drawing shows three triangles: $\displaystyle KLM$, $\displaystyle DEF$ and $\displaystyle PQR$.
The sides of triangle $\displaystyle PQR$: $\displaystyle {PQ}={4}$ cm, $\displaystyle {PR}={3.3}$ cm, $\displaystyle {QR}={3.7}$ cm.
For each of them, write which kind of triangle it is by its sides, and which by its angles.