Chapter 7 — A Tale of Three Intersecting Lines

Chapter 7, 'A Tale of Three Intersecting Lines', is about constructing triangles using a compass and ruler, understanding when a triangle can exist (the triangle inequality), proving that the angles of any triangle sum to 180° (the angle sum property), constructing altitudes, and classifying triangles by side lengths and angle measures.

Draw the base AB of the required length. From A, draw an arc of the same length as a radius. From B, draw another arc of the same radius. The point C where the two arcs intersect is the third vertex; join AC and BC to complete the equilateral triangle.

The triangle inequality says that in any set of three lengths, each length must be smaller than the sum of the other two. For example, the set 3, 4, 5 satisfies the triangle inequality, whereas the set 10, 15, 30 does not (because 30 > 10 + 15). A triangle exists with given side lengths if and only if those lengths satisfy the triangle inequality.

Check whether the longest side is less than the sum of the other two sides. If it is, the triangle exists; if it equals or exceeds that sum, no triangle is possible. Only this one comparison is needed because the two smaller lengths will automatically each be less than the sum of the remaining pair.

When the longest side is taken as the base and circles (or arcs) of the two smaller radii are drawn from its endpoints, satisfying the triangle inequality means the sum of the two radii is greater than the base length. This causes the two circles to intersect internally at two points, and either intersection point serves as the required third vertex.

Take one of the given sides as the base. At one endpoint, construct the given included angle. Along the other arm of that angle, mark the second given side length to locate the third vertex. Join the third vertex to the other base endpoint to complete the triangle.

Draw the included side as the base. At each endpoint, construct the respective given angle. The point where the two new line segments meet is the third vertex. If the sum of the two given angles is 180° or more, the lines will be parallel or diverge and no triangle forms.

The angle sum property states that the sum of the three angles of any triangle is always 180°. The proof draws a line through the top vertex A parallel to the base BC. The two new angles formed equal angles B and C respectively (alternate angles with transversals AB and AC). Since the three angles at A together form a straight line (180°), we get ∠A + ∠B + ∠C = 180°. This method appears in Euclid's 'The Elements', attributed to the Greek mathematician Euclid who lived around 300 BCE.

An exterior angle is the angle formed between the extension of one side of a triangle and the adjacent side. From the angle sum property, the exterior angle equals the sum of the two non-adjacent (remote) interior angles. For example, if ∠A = 50° and ∠B = 60°, then ∠ACB = 70° and the exterior angle ∠ACD = 180° − 70° = 110°, which equals 50° + 60°.

An altitude is a perpendicular line segment drawn from a vertex to the opposite side (or its extension). Every triangle has three altitudes — one from each vertex. In a right-angled triangle, one of the sides is itself an altitude. Altitudes can be constructed accurately using a set square aligned with a ruler.

By sides: equilateral triangles have all three sides equal; isosceles triangles have two sides equal; scalene triangles have all three sides of different lengths. By angles: acute-angled triangles have all three angles acute; right-angled triangles have one 90° angle; obtuse-angled triangles have one obtuse angle.

A triangle exists when the sum of the two given angles is less than 180°. If the sum equals or exceeds 180°, no triangle is possible. The length of the included side has no effect on whether the triangle exists — it only affects the size of the triangle.

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