Triangle basics

Every triangle has three sides and three angles.
No matter its shape, the three angles always add to $180^\circ$.
So if you know two of the angles, you can find the third by subtracting from $180^\circ$.

For any triangle, no matter its type, the sum of any two side lengths must be greater than the third side.
This is the triangle inequality.
If two sides are too short to reach across the third, they cannot meet to close the triangle.
For example, sides of $2$, $3$, and $6$ cannot form a triangle, because $2 + 3 = 5$, which is less than $6$.

The area of a triangle is $\frac{1}{2} \times \text{base} \times \text{height}$.
The height must be measured straight up from the base at a right angle, not along a slanted side.

Triangles are grouped by their sides.
An equilateral triangle has all three sides equal, an isosceles has two sides equal, and a scalene has no sides equal.
In an isosceles triangle, the two angles opposite the equal sides are also equal.

Three special lines can be drawn from a vertex.
A median goes from a vertex to the midpoint of the opposite side.
An altitude goes from a vertex straight to the opposite side at a right angle — it is the height.
An angle bisector cuts the vertex's angle into two equal halves.

In an equilateral triangle (all sides equal, all angles $60^\circ$), these lines all coincide.
From each vertex, the median, altitude, angle bisector, and perpendicular bisector are the very same line.
Its area has a special formula: $\text{area} = \frac{\sqrt{3}}{4}s^2$, where $s$ is the side length.

In an isosceles triangle, the line from the apex (the vertex between the two equal sides) down to the base is also all of these at once.
It is the median, the altitude, the angle bisector, and the perpendicular bisector of the base, all in one.
That single line splits the isosceles triangle into two identical halves.

The size of an angle matches the side across from it.
The largest angle sits opposite the longest side, and the smallest angle opposite the shortest side.

A right triangle has one $90^\circ$ angle.
Its longest side, called the hypotenuse, lies opposite that right angle.
The three sides obey the Pythagorean theorem, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Two angles of a triangle are $50^\circ$ and $70^\circ$. What is the third angle?
The three angles add to $180^\circ$.
So the third angle is $180^\circ - 50^\circ - 70^\circ = 60^\circ$.

Can three sticks of length $4$, $6$, and $11$ form a triangle?
Check the two shorter sides against the longest: $4 + 6 = 10$.
Since $10$ is less than $11$, the two shorter sides cannot reach across.
So no, they cannot form a triangle.

A triangle has a base of $10$ and a height of $6$. What is its area?
The area is $\frac{1}{2} \times \text{base} \times \text{height}$.
So it is $\frac{1}{2} \times 10 \times 6 = 30$.

An equilateral triangle has a side length of $6$. What is its area?
Use the equilateral area formula $\frac{\sqrt{3}}{4}s^2$ with $s = 6$.
So the area is $\frac{\sqrt{3}}{4}(6)^2 = \frac{\sqrt{3}}{4}(36) = 9\sqrt{3}$.

A right triangle has legs of length $9$ and $12$. How long is the hypotenuse?
Use the Pythagorean theorem $a^2 + b^2 = c^2$.
So $c^2 = 9^2 + 12^2 = 81 + 144 = 225$, giving $c = \sqrt{225} = 15$.

Two sides of a triangle are $5$ and $8$. What is the range of possible lengths for the third side?
By the triangle inequality, the third side must be less than the sum $5 + 8 = 13$.
It must also be more than the difference $8 - 5 = 3$.
So the third side is between $3$ and $13$.