The Fifth Chapter.

How equal Angles are made.

Decorative woodcut initial F featuring a seated human figure within a frame of foliage.Fix one foot of your compass upon the meeting point of those two straight lines that contain the angle whose equal you wish to make. Opening your compass at pleasure, describe an arc cutting the two containing sides of the angle. Then, draw another straight line, and placing one foot of the compass (remaining immovable) thereon, with the other describe an arc rising from that last-drawn line. Next, return to your original angle and open your compass to the distance between the two intersections made by the arc on the containing sides. Transporting this same distance to your second arc, set one foot of the compass at the beginning of the arc (where it rises from the line) and with the other foot cut the last described arc. Finally, laying your ruler to that intersection and to the center of the arc, draw a straight line until it meets the other; thus, you have a new angle equal to the former.

A geometric diagram illustrating the construction of an angle. On the left is an angle with vertex A and points B and C on its rays, intersected by an arc DE. On the right, a second angle with vertex F and rays G and H is being constructed using an arc KI.

Suppose B A C is the angle whose equal I desire, and D E is the arc drawn with one foot of my compass while the other remained in A. With the compass remaining immovable, I set one foot in the line FG and draw with the other the arc K I. This done, I open my compass to the distance of D E, and placing one foot in K, with the other I cross the arc at I. Finally, laying the ruler to I and F, I draw the line FH; thus, I have made the angle H F G equal to the first angle B A C.

The 6th Chapter.

To make a triangle equal to any other right-lined triangle assigned.

Decorative woodcut initial F featuring floral and scrollwork designs.First, as you were taught in the last proposition, make an angle equal to one angle in that triangle—it does not matter which. Then extend those straight lines that contain this angle until they are of equal length with the containing sides of the corresponding angle in the triangle. This done, connect the ends of those two straight lines together with a third, and thus you have framed a triangle equal to the former.