A Tale of Three Intersecting Lines

A triangle is the simplest closed shape possible, built from exactly three vertices and three sides. Suppose you need to construct a triangle with sides 4 cm, 5 cm and 6 cm, but all you have is a compass and a ruler — no protractor. Trying this freehand by trial and error works eventually, but a compass makes it exact on the very first try.

Start by drawing one side as the base — say, the 4 cm side, with endpoints A and B. The third vertex, C, must be exactly 5 cm from A and exactly 6 cm from B at the same time. To find every point that is 5 cm from A, draw an arc of radius 5 cm centred at A — every point on that arc is a valid candidate. Separately, every point that is 6 cm from B lies on an arc of radius 6 cm centred at B. The vertex C has to satisfy both conditions simultaneously, so it must be a point where the two arcs cross.

Draw both arcs, and mark their point of intersection as C. Joining AC and BC completes the triangle — and by construction, AC is guaranteed to be exactly 5 cm (since C sits on the arc of radius 5 cm from A) and BC is guaranteed to be exactly 6 cm (since C also sits on the arc of radius 6 cm from B). An equilateral triangle, where all three sides are equal — say, 4 cm each — is simply a special case of this same method, using the same length for both arcs.

Triangles are also classified by their side lengths: an equilateral triangle has all three sides equal, an isosceles triangle has exactly two sides equal, and a scalene triangle has three sides that are all different lengths.

Try to construct a triangle with sides 3 cm, 4 cm and 8 cm, and the two arcs you need to draw (of radius 3 cm and 4 cm, from the ends of the 8 cm base) never meet at all — they are simply too short to reach each other across such a long base. This isn't a drawing mistake; it reveals a genuine mathematical limit on which three lengths can actually form a triangle.

The reasoning behind this limit comes from a simple real-world fact: the shortest distance between any two points is always the straight line directly between them. Any route that detours through a third point is at least as long, and usually longer. Applied to a triangle with vertices A, B and C, this means the direct path between any two vertices (one side of the triangle) must be shorter than the roundabout path through the third vertex (the sum of the other two sides) — and this must hold for all three possible comparisons at once.

This is called the triangle inequality: a triangle with side lengths a, b and c can exist only if each length is smaller than the sum of the other two. Checking the lengths 5 cm, 7 cm and 10 cm: is 5 < 7+10=17? Yes. Is 7 < 5+10=15? Yes. Is 10 < 5+7=12? Yes. All three checks pass, so a triangle with these sides does exist. Now check 4 cm, 6 cm and 11 cm: the largest length, 11, needs to be less than 4+6=10 — but 11 is not less than 10, so this set of lengths fails the triangle inequality, and no triangle can be built from them.

In practice, you only ever need to check one comparison: whether the longest side is shorter than the sum of the other two. If that single check passes, the other two comparisons are automatically satisfied as well, since the longest side is always the hardest one to beat.

A triangle can also be constructed when you're given a mix of sides and angles instead of all three sides. One common case is two sides and the angle included between them. To construct a triangle with AB = 5 cm, AC = 4 cm and the angle between them, angle A = 45°: draw AB as the base, construct a 45° angle at A using a second arm, mark point C exactly 4 cm along that arm, then join BC. Because every measurement is fixed directly by the construction, exactly one triangle results — there's no ambiguity.

The other common case is two angles and the side included between them. To construct a triangle with AB = 5 cm, angle A = 45° and angle B = 80°: draw AB as the base, then draw rays from A and from B at the given angles. Since the two rays are drawn in fixed directions from fixed points, they meet at exactly one point, giving exactly one triangle — again, with no ambiguity.

Not every combination of angle measurements is possible, though. If an "included" angle is 180° or more, the two sides meeting there would lie flat on a straight line (or fold back on themselves) instead of forming a proper corner, so no triangle exists. Similarly, for two angles and their included side, the two base angles must add to less than 180° — if their sum is 180° or more, the two rays drawn from the base either never meet or point away from each other entirely, so again, no triangle forms. For example, base angles of 90° and 95° add to 185°, which is too large — no triangle is possible with those two angles, regardless of the side length between them. But base angles of 80° and 99° add to only 179°, which is less than 180°, so a triangle with those two base angles is possible.

One of the most useful facts about any triangle is that its three interior angles always add up to exactly 180°, whatever shape or size the triangle is. This is called the angle sum property. It can be proved neatly by drawing a line through one vertex that is parallel to the opposite side. In triangle ABC, draw a line XY through vertex A, parallel to side BC. Because XY is parallel to BC, angle XAB equals angle B (alternate angles with transversal AB), and angle YAC equals angle C (alternate angles with transversal AC). But angle XAB, angle BAC and angle YAC together form a straight line through A, so they must add up to 180°. Substituting the alternate-angle equalities: angle B + angle BAC + angle C = 180°, which is exactly the angle sum property.

Using this property, if you know two angles of a triangle, you can always find the third without measuring anything: just subtract both known angles from 180°. For a triangle with angles 55° and 65°, the third angle is 180° − 55° − 65° = 60°. For a triangle with angles 100° and 45°, the third angle is 180° − 100° − 45° = 35°.

Extending one side of a triangle beyond a vertex creates an exterior angle at that vertex. Since the exterior angle and the triangle's interior angle at that same vertex together form a straight line (180°), and the angle sum property already says the interior angle equals 180° minus the other two interior angles, it follows that the exterior angle equals the sum of the two "remote" interior angles — the two angles not at that vertex. For a triangle with interior angles 70° and 50° at two vertices, the third interior angle is 180° − 70° − 50° = 60°, and the exterior angle at that third vertex is 180° − 60° = 120° — which indeed equals 70° + 50° = 120°, confirming the exterior angle property.

The altitude from a vertex is the perpendicular line segment dropped from that vertex to the line containing the opposite side; its length is what we usually mean by the "height" of a triangle. Every triangle has three altitudes, one from each vertex. To draw an altitude accurately, a ruler alone isn't precise enough for a true 90° angle — instead, place a set square against a ruler aligned with the base, then slide the set square along the ruler until its perpendicular edge touches the opposite vertex, and draw the altitude along that edge.

In some triangles, the foot of the altitude doesn't land between the two base vertices at all — it lands on the extension of the base, outside the triangle itself. This happens in triangles with an obtuse angle (an angle greater than 90°): the altitude from one of the vertices adjacent to the obtuse angle has to be drawn to the extended base line, not the base segment itself.

Besides classifying triangles by their sides (equilateral, isosceles, scalene), triangles are also classified by their angles. An acute-angled triangle has all three angles smaller than 90°. A right-angled triangle has exactly one angle equal to 90°; in a right triangle, two of the three sides are also two of the triangle's own altitudes, since they already meet at a right angle. An obtuse-angled triangle has exactly one angle greater than 90°.

Because the three angles of any triangle must add to 180°, a triangle can never have two right angles or two obtuse angles at once — there simply wouldn't be enough "angle budget" left for the third angle to be a positive number. This is a direct, useful consequence of the angle sum property from the previous section.