PROPOSITION XVIII. The plane angles which contain any solid angle are together less than four right angles. First let the solid angle at A be contained by three plane angles BAF, FAC, CAB. Then shall these angles be together less than four right 4 s. In AB, AC, AF take any points B, C, F, Then since the solid 4 at B is contained by three plane angles; .. Ls ABC, ABF are together > ↳ CBF. (Prop. 17) Similarly s ACB, ACF are together > ▲ FCB, (Prop. 17) and s AFC, AFB > ¿aBFC; (Prop. 17) .. Ls ABC, ABF, ACB, ACF, AFC, AFB are together the 4s of a BCF, i. e. the 4s at the bases of the ▲s ABC, ACF, AFB are together > two rights. > (I. 24) Now all the 4s of the as ABC, ACF, AFB are = six rights; .. thes at the vertex A are < four right ≤ s. (I. 24) Next, let the solid angle at A be contained by any number of plane angles. Let AB, AC, &c. be cut by a plane in the points B, C, F, &c. and join BC, CF, FG, &c. Then S ACB, ACF are together > BCF, (Prop. 17) SO LS AFC, AFG are together > CFG: and so on. .. all the 4s at the bases of the As ABC, ACF, &c. are together the angles of the polygon BCF > Now all the 4s of the As ABC, ACF, &c. are = twice as many rights as there are AS, (I. 24) and all the s of the polygon BCF together with four rights are equal to twice as many rights as the figure has sides; (I. 30) .. all the 4s of the as ABC, ACF, &c. are the s of the polygon together with four rights; but the 4s at the bases of the As ABC, ACF, &c. are >thes of the polygon; .. thes at the vertex A are < four rights. GEOMETRICAL EXERCISES. DEFINITIONS. One angle is called the complement of another, when the two together make up a right angle. One angle is called the supplement of another, when the two together make up two right angles. When two straight lines are drawn from a point, they form an angle less than two right angles, and also an angle greater than two right angles. The latter is called a reflex angle, and together with the former makes up four right angles. When two sides of a polygon form a re-entrant angle, the reflex angle, not the smaller angle, is considered as one of the angles of the polygon. A scalene triangle is one contained by three unequal straight lines. An oblong has all its angles right angles, but has not all its sides equal. A rhombus has all its sides equal, but its angles are not right angles. A rhomboid has its opposite sides equal to each other, but all its sides are not equal nor its angles right angles. A trapezoid is a four-sided figure which has two of its sides parallel, and the remaining two not parallel. N.B. This figure is sometimes called a trapezium ; but Euclid calls all quadrilateral figures trapeziums which are not parallelograms. If a straight line AB be divided in P and Q so that AP : AB as PQ : QB, then AB is said to be divided harmonically. Figures whose areas are equal are said to be equivalent to one another. A tetrahedron is a solid contained by four planes. A polyhedron is a solid contained by more than four planes. A dihedral angle is formed by the intersection of two planes. I. PLANE GEOMETRY. BOOK I. IF on the same base and on the same side of it two isosceles triangles are drawn, the vertex of one triangle must fall within the other. 2. If on the same base and on opposite sides of it two isosceles triangles are drawn, the straight line joining their vertices shall bisect the base. 3. Let the equal sides AB, AC of an isosceles triangle be produced to F, G making AF = AG: join FC, BG intersecting in H, then AH will bisect the angle BAC. 4. Prove that in the previous figure the lines bisecting the s at F, G will meet AH in the same point. 5. If the angles of one triangle are not severally equal to those of another, neither are their sides equal. 6. Divide a given angle into four equal angles. 7. Describe a right-angled triangle whose hypothenuse shall be equal to one straight line, and one of its sides equal to another straight line. 8. Draw a straight line through a given point cutting off equal parts from the arms of a given angle. 9. Find a point in a straight line equidistant from two given points without it. 10. Trisect a given right angle. II. Construct a triangle having given the base, one of the angles at the base, and the sum of the sides. 12. Draw a straight line through a given point equally inclined to two given straight lines. 13. Find a point in the base of a triangle equidistant from the two sides. 14. Describe an isosceles ▲ having each of the sides double the base. 15. Describe an isosceles triangle having each of the sides three times the size of the base. |