Triangles
Classifying Triangles
A triangle is a three-sided polygon, and triangles are usually classified two ways: by their side lengths and by their angle measures.
By Side Length
- Scalene triangle — all three sides have different lengths (and all three angles are different too).
- Isosceles triangle — at least two sides are equal in length. The angles opposite those equal sides are also equal to each other.
- Equilateral triangle — all three sides are equal in length, which forces all three angles to equal 60^{\circ}.
By Angle Measure
- Acute triangle — all three angles are acute (less than 90^{\circ}).
- Right triangle — one angle is exactly 90^{\circ}. The side opposite the right angle is called the hypotenuse, and it is always the longest side.
- Obtuse triangle — one angle is greater than 90^{\circ}.
The Triangle Angle Sum Theorem
One of the most useful facts in all of geometry: the three interior angles of any triangle always add up to 180^{\circ}. This holds no matter how the triangle is shaped — tall and thin, short and wide, it never changes.
Symbolically, for a triangle with angles A, B, and C:
A + B + C = 180^{\circ}The Exterior Angle Theorem
If you extend one side of a triangle past a vertex, you create an exterior angle at that vertex. This exterior angle forms a linear pair with the triangle's interior angle at that same vertex (so together they sum to 180^{\circ}).
The Exterior Angle Theorem states that an exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles. If a triangle has interior angles A, B, C, and you extend the side to create an exterior angle at vertex C, then that exterior angle equals A + B.
Example
In triangle ABC, angle A measures 52^{\circ} and angle B measures 65^{\circ}. Find angle C, and then find the measure of the exterior angle at vertex C.
- By the Angle Sum Theorem: 52^{\circ} + 65^{\circ} + C = 180^{\circ}
- Solve: C = 180^{\circ} - 117^{\circ} = 63^{\circ}
- The exterior angle at C is supplementary to the 63^{\circ} interior angle: 180^{\circ} - 63^{\circ} = 117^{\circ}
- Check with the Exterior Angle Theorem directly: the exterior angle should equal the sum of the two remote interior angles, 52^{\circ} + 65^{\circ} = 117^{\circ}. It matches!
Key Takeaways
- By sides: scalene (no equal sides), isosceles (two equal sides), equilateral (three equal sides).
- By angles: acute (all angles under 90^{\circ}), right (one 90^{\circ} angle), obtuse (one angle over 90^{\circ}).
- The three interior angles of a triangle always sum to 180^{\circ}.
- An exterior angle equals the sum of the two remote (non-adjacent) interior angles.