6. If two right-angled triangles have equal hypotenuses, and a side of one equal to a side of the other, they are congruent. 7. The bisectors of the three internal angles of a triangle are concurrent. 8. The bisectors of two external angles and the bisector of the third internal angle are concurrent. 9. Through a given point draw a right line, such that perpendiculars on it from two given points on opposite sides may be equal to each other. 10. Through a given point draw a right line intersecting two given lines, and forming an isosceles triangle with them. Parallel Lines. Def. i.—If two right lines in the same plane be such that, when produced indefinitely, they do not meet at any finite distance, they are said to be parallel. Def. ii.—A parallelogram is a quadrilateral, both pairs of whose opposite sides are parallel. Def. iii.—The right line joining either pair of opposite angles of a quadrilateral is called a diagonal. Def. iv.—If both pairs of opposite sides of a quadrilateral be produced to meet, the right line joining their points of intersection is called its third diagonal. Def. v.—A quadrilateral which has one pair of opposite sides parallel is called a trapezium. Def. vi.—If from the extremities of one right line perpendiculars be drawn to another, the intercept between their feet is called the projection of the first line on the second. Def. vii.—When a right line intersects two other right lines in two distinct points it makes with them eight angles, which have received special names in relation to one another. Thus, in the figure—1, 2; 7, 8 are called exterior angles; 3, 4; 5, 6, interior angles. Again, 4; 6; 3, 5 are called alternate angles; lastly, 1, 5; 2, 6; 3, 8; 4, 7 are called corresponding angles. PROP. XXVII.—Theorem. If a right line (EF) intersecting two right lines (AB, CD) makes the alternate angles (AEF, EFD) equal to each other, these lines are parallel. Dem.—If AB and CD are not parallel they must meet, if produced, at some finite distance: if possible let them meet in G; then the figure EGF is a triangle, and the angle AEF is an exterior angle, and EFD a non-adjacent interior angle. Hence [xvi.] AEF is greater than EFD; but it is also equal to it (hyp.), that is, both equal and greater, which is absurd. Hence AB and CD are parallel. Or thus: Bisect EF in O; turn the whole figure round O as a centre, so that EF shall fall on itself; then because OE = OF, the point E shall fall on F; and because the angle AEF is equal to the angle EFD, the line EA will occupy the place of FD, and the line FD the place of EA; therefore the lines AB, CD interchange places, and the figure is symmetrical with respect to the point O. Hence, if AB, CD meet on one side of O, they must also meet on the other side; but two right lines cannot enclose a space (Axiom x.); therefore they do not meet at either side. Hence they are parallel. PROP. XXVIII.—Theorem.