This chapter is about triangles — their construction, existence and angle properties. Using a compass and ruler you construct equilateral triangles (all sides equal), isosceles triangles (two sides equal) and scalene triangles (all sides different) by drawing arcs from the endpoints of a base. A central idea is the triangle inequality: a triangle exists only when each side is shorter than the sum of the other two. So lengths like 10, 15, 30 cannot form a triangle because \(30 > 10 + 15\), whereas 4, 5, 8 can because \(8 < 4 + 5\). You also construct triangles from two sides and the included angle, or two angles and the included side, and discover that a triangle needs the two given angles to sum to less than \(180^\circ\). By drawing a line parallel to the base through the top vertex, you prove the angle sum property: the three angles of any triangle add to \(180^\circ\). You meet exterior angles, altitudes (perpendiculars from a vertex to the opposite side), and classify triangles by sides and by angles (acute, right, obtuse).
Constructing equilateral, isosceles and scalene trianglesTriangle inequality and existence of trianglesConstructing triangles from sides and anglesAngle sum property and exterior anglesAltitudes and classifying triangles
Key terms
Equilateral triangle
A triangle with all three sides equal (and all angles \(60^\circ\)).
Isosceles triangle
A triangle with exactly two sides equal.
Scalene triangle
A triangle with all three sides of different lengths.
Triangle inequality
Each side of a triangle is shorter than the sum of the other two sides.
Angle sum property
The three angles of any triangle add up to \(180^\circ\).
Exterior angle
The angle between a side of a triangle extended and the adjacent side.
Altitude
A perpendicular line segment from a vertex to the opposite side (or its extension).
Extra questions & answers
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The triangle inequality states that each side of a triangle is:
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