If any side (AB) of a triangle (ABC) be produced (to D), the external angle (CBD) is equal to the sum of the two internal non-adjacent angles (A, C), and the sum of the three internal angles is equal to two right angles.
Dem.—Draw BE parallel to AC [xxxi.]. Now since BC intersects the parallels BE, AC, the alternate angles EBC, ACB are equal [xxix.]. Again, since AB intersects the parallels BE, AC, the angle EBD is equal to BAC [xxix.]; hence the whole angle CBD is equal to the sum of the two angles ACB, BAC: to each of these add the angle ABC and we have the sum of CBD, ABC equal to the sum of the three angles ACB, BAC, ABC: but the sum of CBD, ABC is two right angles [xiii.]; hence the sum of the three angles ACB, BAC, ABC is two right angles.
Cor. 1.—If a right-angled triangle be isosceles, each base angle is half a right angle.
Cor. 2.—If two triangles have two angles in one respectively equal to two angles in the other, their remaining angles are equal.
Cor. 3.—Since a quadrilateral can be divided into two triangles, the sum of its angles is equal to four right angles.
Cor. 4.—If a figure of n sides be divided into triangles by drawing diagonals from any one of its angles there will be (n − 2) triangles; hence the sum of its angles is equal 2(n − 2) right angles.
Cor. 5.—If all the sides of any convex polygon be produced, the sum of the external angles is equal to four right angles.
Cor. 6.—Each angle of an equilateral triangle is two-thirds of a right angle.
Cor. 7.—If one angle of a triangle be equal to the sum of the other two, it is a right angle.
Cor. 8.—Every right-angled triangle can be divided into two isosceles triangles by a line drawn from the right angle to the hypotenuse.