Triangles and their angles

Classifying triangles

By sides:

• equilateral: all three sides congruent;
• isosceles: at least two sides congruent (the legs); the third side is the base;
• scalene: no sides congruent.

By angles:

• acute: three acute angles;
• right: one right angle; the sides that form it are the legs and the side opposite it is the hypotenuse;
• obtuse: one obtuse angle;
• equiangular: three congruent angles.

The triangle sum theorem

Here's why. Through one vertex, draw the line parallel to the opposite side. Three angles line up along that line to form a straight angle. The middle one is the triangle's angle at that vertex; the outer two are congruent to the other two angles of the triangle (they're alternate interior angles, as you'll see in the lesson on parallel lines).

Two consequences:

• a triangle has at most one right or obtuse angle;
• the acute angles of a right triangle are complementary.

The exterior angle theorem

Extend one side of a triangle past a vertex. The angle formed outside the triangle is an exterior angle. The two triangle angles that aren't next to it are its remote interior angles.

The triangle inequality

Three segments don't always make a triangle. Try sides of \(5\), \(6\) and \(11\): the two shorter sides laid end to end reach exactly \(11\), so they lie flat along the third side and enclose nothing.

In practice, check that the two shorter sides add up to more than the longest side. So if two sides are \(a\) and \(b\), the third side \(c\) satisfies

Isosceles triangles

In an isosceles triangle with a \(72°\) base angle, the other base angle is also \(72°\) and the vertex angle is \(180° - 144° = 36°\).

Practice