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. Exercises. 1. Trisect a right angle. 2. Any angle of a triangle is obtuse, right, or acute, according as the opposite side is greater than, equal to, or less than, twice the median drawn from that angle. 3. If the sides of a polygon of n sides be produced, the sum of the angles between each alternate pair is equal to 2(n − 4) right angles. 4. If the line which bisects the external vertical angle be parallel to the base, the triangle is isosceles. 5. If two right-angled △s ABC, ABD be on the same hypotenuse AB, and the vertices C and D be joined, the pair of angles subtended by any side of the quadrilateral thus formed are equal. 6. The three perpendiculars of a triangle are concurrent. 7. The bisectors of two adjacent angles of a parallelogram are at right angles. 8. The bisectors of the external angles of a quadrilateral form a circumscribed quadrilateral, the sum of whose opposite angles is equal to two right angles. 9. If the three sides of one triangle be respectively perpendicular to those of another triangle, the triangles are equiangular. 10. Construct a right-angled triangle, being given the hypotenuse and the sum or difference of the sides. 11. The angles made with the base of an isosceles triangle by perpendiculars from its extremities on the equal sides are each equal to half the vertical angle. 12. The angle included between the internal bisector of one base angle of a triangle and the external bisector of the other base angle is equal to half the vertical angle. 13. In the construction of Prop. xviii. prove that the angle DBC is equal to half the difference of the base angles. 14. If A, B, C denote the angles of a △, prove that (A + B), (B + C), (C + A) will be the angles of a △ formed by any side and the bisectors of the external angles between that side and the other sides produced. PROP. XXXIII.—Theorem. The right lines (AC, BD) which join the adjacent extremities of two equal and parallel right lines (AB, CD) are equal and parallel.