This is the converse of the preceding theorem, and is half of Euclid I, 28, his theorem being divided for the reason above stated. There are several typical pairs of equal or supplemental angles that would lead to parallel lines, of which Euclid uses only part, leaving the other cases to be inferred. This accounts for the number of corollaries in this connection in later textbooks. Surveyors make use of this proposition when they wish, without using a transit instrument, to run one line parallel to another. [Illustration] For example, suppose two boys are laying out a tennis court and they wish to run a line through _P_ parallel to _AB_. Take a 60-foot tape and swing it around _P_ until the other end rests on _AB_, as at _M_. Put a stake at _O_, 30 feet from _P_ and _M_. Then take any convenient point _N_ on _AB_, and measure _ON_. Suppose it equals 20 feet. Then sight from _N_ through _O_, and put a stake at _Q_ just 20 feet from _O_. Then _P_ and _Q_ determine the parallel, according to the proposition just mentioned. THEOREM. _If two parallel lines are cut by a transversal, the exterior-interior angles are equal._ This is also a part of Euclid I, 29. It is usually followed by several corollaries, covering the minor and obvious cases omitted by the older writers. While it would be possible to dispense with these corollaries, they are helpful for definite reference in later propositions. THEOREM. _The sum of the three angles of a triangle is equal to two right angles._ Euclid stated this as follows: "In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles." This states more than is necessary for the basal fact of the proposition, which is the constancy of the sum of the angles. The theorem is one of the three most important propositions in plane geometry, the other two being the so-called Pythagorean Theorem, and a proposition relating to the proportionality of the sides of two triangles. These three form the foundation of trigonometry and of the mensuration of plane figures. The history of the proposition is extensive. Eutocius (_ca._ 510 A.D.), in his commentary on Apollonius, says that Geminus (first century B.C.) testified that "the ancients investigated the theorem of the two right angles in each individual species of triangle, first in the equilateral, again in the isosceles, and afterwards in the scalene triangle." This, indeed, was the ancient plan, to proceed from the particular to the general. It is the natural order, it is the world's order, and it is well to follow it in all cases of difficulty in the classroom.