Special triangles

In the isosceles triangle $\displaystyle ABC$ two sides are equal: $\displaystyle {AB}={AC}$. Each part of the triangle has a name:

• The legs: the two equal sides, $\displaystyle AB$ and $\displaystyle AC$.
• The base: the third side, $\displaystyle BC$.
• The apex angle: the angle between the legs, $\displaystyle \angle A$.
• The base angles: the two angles at the base, $\displaystyle \angle B$ and $\displaystyle \angle C$.

In an isosceles triangle, the two base angles are equal. Folding shows it:

• Cut an isosceles triangle $\displaystyle ABC$ out of paper.
• Fold it so that the leg $\displaystyle AB$ lies exactly on the leg $\displaystyle AC$.
• The fold line starts at $\displaystyle A$ and reaches the middle of the base, the point $\displaystyle D$.

• $\displaystyle B$ falls on $\displaystyle C$, and the angle $\displaystyle B$ covers the angle $\displaystyle C$ exactly: $\displaystyle {\angle B}={\angle C}$.
• The fold line $\displaystyle AD$ is the triangle’s axis of symmetry: its two halves cover each other exactly.

The base angles are equal, so one angle of an isosceles triangle is enough to calculate the other two.

From the apex angle to the base angles. The apex angle is $\displaystyle 40^{\circ}$.

• The two base angles together: $\displaystyle {180^{\circ}-40^{\circ}}={140^{\circ}}$.
• They are equal, so each of them is $\displaystyle {140^{\circ}:2}={70^{\circ}}$.

From a base angle to the apex angle. One base angle is $\displaystyle 65^{\circ}$.

• The other base angle is $\displaystyle 65^{\circ}$ too.
• The apex angle: $\displaystyle {180^{\circ}-2\cdot65^{\circ}}={180^{\circ}-130^{\circ}}={50^{\circ}}$.

In triangle $\displaystyle ABC$: $\displaystyle {AB}={AC}$.
Calculate.

$\displaystyle {\angle A}={80^{\circ}}$. Calculate $\displaystyle \angle B$.

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

$\displaystyle {\angle C}={50^{\circ}}$. Find $\displaystyle \angle B$ and $\displaystyle \angle A$.

In an equilateral triangle all three angles are equal, and each of them is $\displaystyle 60^{\circ}$.

• An equilateral triangle is isosceles from every side: for any two of its sides, the angles opposite them are base angles, so they are equal.
• Three equal angles that add up to $\displaystyle 180^{\circ}$: each of them is $\displaystyle {180^{\circ}:3}={60^{\circ}}$.

In the right-angled triangle $\displaystyle ABC$ the angle $\displaystyle C$ is a right angle. Its sides have names too:

• The legs: the two sides that form the right angle, $\displaystyle AC$ and $\displaystyle BC$.
• The hypotenuse: the side opposite the right angle, $\displaystyle AB$. The hypotenuse is the longest side of the triangle.

The two acute angles of a right-angled triangle add up to $\displaystyle 90^{\circ}$: the right angle takes $\displaystyle 90^{\circ}$ of the $\displaystyle 180^{\circ}$, and $\displaystyle 90^{\circ}$ is left for the other two.

Calculate the missing angle of a right-angled triangle.

In triangle $\displaystyle ABC$: $\displaystyle {\angle C}={90^{\circ}}$, $\displaystyle {\angle A}={35^{\circ}}$.
Calculate $\displaystyle \angle B$.

In triangle $\displaystyle KLM$: $\displaystyle {\angle K}={90^{\circ}}$, $\displaystyle {\angle L}={61^{\circ}}$.
Write which sides are the legs and which is the hypotenuse.
Find $\displaystyle \angle M$.

In a right isosceles triangle, each of the base angles is $\displaystyle 45^{\circ}$.

• Its legs are the two equal sides $\displaystyle CA$ and $\displaystyle CB$, and its apex angle is the right angle.
• The two base angles together: $\displaystyle {180^{\circ}-90^{\circ}}={90^{\circ}}$, and each of them is $\displaystyle {90^{\circ}:2}={45^{\circ}}$.
• A square napkin folded along its diagonal makes such a triangle.

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

An isosceles triangle whose base angle is $\displaystyle 90^{\circ}$.

An isosceles triangle whose apex angle is $\displaystyle 100^{\circ}$.

A right-angled triangle that is also equilateral.

In triangle $\displaystyle ABC$: $\displaystyle {AB}={AC}$, $\displaystyle {\angle B}={68^{\circ}}$.
Find $\displaystyle \angle C$ and $\displaystyle \angle A$, and explain.