Lines can be divided proportional to each other in the manner described above and 340:3 gives the details where AC is to be divided in the same proportion as AB. The method is also useful in dividing a given line into any number of equal parts (as in setting out dovetails) where AB (340:4), which is assumed to be 51/2 units (inches or millimetres) long, has to be divided into four equal parts. AB is first drawn the correct length and AC any convenient length easily divisible by four. Parallel lines from AC will then equally divide AB.

Where a line has to be divided into two equal parts, i. e. bisected, then 340:7 shows the method in which arcs with equal radii are struck from A and B to intersect at C and D, and a vertical line drawn through CD will then bisect AB. This method can also be used for striking a perpendicular to a given line, and 340:8 shows the method in which a semicircle is described from point A on BC, and using the same radius throughout describing arcs D and E from B and C and intersecting arcs from D and E to F; a line drawn from F to A will then be truly perpendicular. If compasses are not available then perpendiculars can be drawn using the theorem attributed to Pythagoras (scale of equal parts) in which the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides. Thus in 340:9 if AB is drawn 3 units long, then measurements of 4 units from B and 5 units from A (or any multiples of all three measurements) will only coincide at point C on a line which is at true right angles to A.


Angles are formed by the convergence of two straight lines, or more precisely by the rotation of a straight line about a fixed point, and the method of bisecting them is shown in 341:1 where an arc is struck from centre A and further equal arcs from B and C. From the point at which these two arcs intersect a line drawn to A will bisect the angle. This method can be used to strike all the commonly used angles, e. g. a right angle of 90° can be bisected to produce a true mitre of 45° and again bisected to produce a half-mitre of 221/2°. Angles of 60° can be struck by describing an arc BC from A (341:2) and a similar arc AC from B. If lines are drawn from C to A and B then the figure thus formed will be an equilateral triangle with all sides and all angles equal. As the sum of the angles contained in a triangle must equal 180°, then each angle will be 60°, and bisecting them further gives angles of 30°.


Triangles are shown in 341:3. Right-angled triangles (A) have two sides at right angles to










341 Angles, triangles and parallelograms

each other, the third side is known as the 'hypotenuse' and the angles so formed by this line are equal in sum to one right angle, i. e. 45° each. Equilateral triangles (B) have three equal sides and therefore three equal angles of 60°. Isosceles triangles (C) have two equal sides and two equal angles, and scalene triangles—acute scalene (D) and obtuse scalene (E)—have three unequal sides and therefore three unequal angles.

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