For if BD is not in a straight line with BC, let BE be in a straight line with BC. Then because AB stands on the straight line CBE, therefore the angles ABC, ABE are together equal to two right angles; I. 2. but the angles ABC, ABD are also together equal to two right angles; Hyp. therefore the angles ABC, ABE are together equal to the angles ABC, ABD. Ax. c. Take away the common angle ABC c; then the angle ABE is equal to the angle ABD, Ax. e. the part to the whole, Ax. a. which is impossible; therefore BE is not in a straight line with BC. In the same way it can be shown that no other straight line than BD is in a straight line with BC, therefore BC and BD are in one straight line. Q.E.D. THEOR. 4. If two straight lines cut one another, the vertically opposite angles are equal to one another. Let the two straight lines AB, CD cut one another at : then shall the angle AOC be equal to the angle BOD, and the angle BOC to the angle AOD. Because AO stands upon CD, therefore the angles AOC, AOD are together equal to two right angles; again, because DO stands upon AB, I. 2. therefore the angles AOD, DOB are together equal to two right angles; therefore the angles AOC, AOD are together equal to the angles AOD, DOB; I. 2. Ax. c. then the angle AOC is equal to the angle BOD. In the same way it may be proved that the angle BOC is Ax. e. equal to the angle AOD. Q.E.D. Ex. 3. The bisectors of two vertically opposite angles are in one straight line. SECTION II. TRIANGLES. DEF. 18. A plane figure is a portion of a plane surface inclosed by a line or lines. DEF. 19. Figures that may be made by superposition to coincide with one another are said to be identically equal; or they are said to be equal in all respects. DEF. 20. The area of a plane figure is the quantity of the plane surface inclosed by its boundary. DEF. 21. A plane rectilineal figure is a portion of a plane surface inclosed by straight lines. When there are more than three inclosing straight lines the figure is called a polygon. DEF. 22. A polygon is said to be convex when no one of its angles is reflex. DEF. 23. A polygon is said to be regular when it is equilateral and equiangular; that is, when its sides and angles are equal. DEF. 24. A diagonal is the straight line joining the vertices of any angles of a polygon which have not a common arm. DEF. 25. The perimeter of a rectilineal figure is the sum of its sides. DEF. 26. A quadrilateral is a polygon of four sides, a pentagon one of five sides, a hexagon one of six sides, and so on. DEF. 27. A triangle is a figure contained by three straight lines. DEF. 28. Any side of a triangle may be called the base, and the opposite angular point is then called the vertex. DEF. 29. An isosceles triangle is that which has two sides equal ; the angle contained by those sides is called the vertical angle, the third side the base. THEOR. 5. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles included by these sides equal, then the triangles are identically equal, and of the angles those are equal which are opposite to the equal sides. Let ABC, DEF be two triangles having the side AB equal to the side DE, the side AC to the side DF, and the angle BAC to the angle EDF: AAA B then shall the triangles be identically equal, having the side BC equal to the side EF, the angle ACB to the angle DFE, and the angle ABC to the angle DEF. Let the triangle ABC be applied to the triangle DEF, so that the point A may fall on the point D, the side AB along the side DE, and the point C on the same side of DE as the point F; then B will fall on E, since AB is equal to DE, Hyp. AC will fall along DF, since the angle BAC is equal to the angle EDF, Hyp. and, AC falling along DF, C will fall on F, since AC is equal to DF. Hyp. Hence, B falling on E, and C on F, BC will coincide with Ax. 2. EF, and the triangle ABC will coincide with the triangle DEF, and is therefore identically equal to it, Ax. I. the side BC equal to the side EF, the angle ACB to the angle DFF, and the angle ABC to the angle DEF. Q.E.D. Ex. 4. The straight line which bisects the vertical angle of an isosceles triangle bisects the base. Ex. 5. Any point on the bisector of the vertical angle of an isosceles triangle is equidistant from the extremities of the base. Ex. 6. The straight line which bisects the vertical angle of an isosceles triangle is perpendicular to the base. Ex. 7. Any point D is taken on the bisector of an angle BAC; prove that, if AB is equal to AC, then the angle ADB is equal to the angle ADC. Ex. 8. The straight lines drawn from the extremities of the base of an isosceles triangle to the middle points of the opposite sides are equal to one another. Ex. 9. On one arm of an angle whose vertex is A points B and D are taken, and on the other arm points C and E, such that AB is equal to AC, and AD to AE: shew that BE is equal to CD. |