Key Points

A triangle is a closed shape with three vertices, three sides, and three interior angles. A triangle with vertices A, B, and C is denoted as $\triangle ABC$.

For a triangle to be possible with side lengths $a, b,$ and $c$, the sum of the lengths of any two sides must be greater than the length of the third side. This means $a + b > c$, $a + c > b$, and $b + c > a$.

To quickly check if three side lengths can form a triangle, verify that the sum of the two shorter sides is greater than the longest side. If this holds true, a triangle can be formed.

The sum of the measures of the three interior angles of any triangle is always $180^\circ$. For $\triangle ABC$, this means $\angle A + \angle B + \angle C = 180^\circ$.

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two opposite interior angles. For example, an exterior angle at vertex C is equal to $\angle A + \angle B$.

To construct a triangle given three sides, draw one side as the base. Use a compass to draw arcs from each end of the base with radii equal to the other two side lengths. The intersection of the arcs is the third vertex.

To construct a triangle given two sides and the angle between them, draw one side as the base. At one vertex, construct the given angle. Mark the length of the second side on the new arm of the angle to find the third vertex.

To construct a triangle given two angles and the side between them, draw the side as a base. At each end of the base, construct the given angles. The point where the arms of the angles intersect is the third vertex.

A triangle can be formed with two given angles only if their sum is less than $180^\circ$. If the sum is $180^\circ$ or more, the other two sides will be parallel or diverge, and will not form a triangle.

Triangles can be classified based on their side lengths: Equilateral (all three sides are equal), Isosceles (two sides are equal), and Scalene (all three sides have different lengths).

Triangles can be classified based on their angles: Acute-angled (all angles are less than $90^\circ$), Right-angled (one angle is exactly $90^\circ$), and Obtuse-angled (one angle is greater than $90^\circ$).

An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side. The length of the altitude is considered the height of the triangle corresponding to that base.

An equilateral triangle has all three sides of equal length and all three interior angles equal to $60^\circ$.

An isosceles triangle has two sides of equal length. The angles opposite the equal sides are also equal in measure.