PROPOSITION XXX. THEOREM. Straight lines which are parallel to the same straight line are parallel to one another. = P B D F Let the st. lines AB, CD be each || to EF. Then must AB be || to CD. Draw the st. line GH, cutting AB, CD, EF in the pts. O, P, Q. Then GH cuts the || lines AB, EF, And GH cuts the lines CD, EF, and these are alternate angles; ... AB is to CD. I. 29. I. 29. I. 27. Q. E. D. The following Theorems are important. They admit of easy proof, and are therefore left as Exercises for the student. 1. If two straight lines be parallel to two other straight lines, each to each, the first pair make the same angles with one another as the second. 2. If two straight lines be perpendicular to two other straight lines, each to each, the first pair make the same angles with one another as the second. PROPOSITION XXXI. PROBLEM. To draw a straight line through a given point parallel to a given straight line. Let A be the given pt. and BC the given st. line. In BC take any pt. D, and join AD. Make DAE= L ADC. Produce EA to F. Then EF shall be ] to BC. I. 23. For AD, meeting EF and BC, makes the alternate angles equal, that is, ▲ EAD= ▲ ADC, .. EF is to BC. .. a st. line has been drawn through A || to BC. I. 27. Q. E. F. Ex. 1. From a given point draw a straight line, to make an angle with a given straight line that shall be equal to a given angle. Ex. 2. Through a given point A draw a straight line ABC, meeting two parallel straight lines in B and C, so that BC may be equal to a given straight line. PROPOSITION XXXII. THEOREM. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of every triangle are together equal to two right angles. Then will I II. LACD=LS ABC, BAC together. ▲ s ABC, BAC, ACB together=two rt. 4 s. I. 31. Then I. . BD meets the Is EC, AB, .. extr. 4 ECD=intr. ▲ ABC. And AC meets the Is EC, AB, .. ACE alternate BAC. .. 48 ECD, ACE together= 4s ABC, BAC together; then 4s ABC, BAC, ACB together= ▲ s ACD, ACB together, .. 48 ABC, BAC, ACB together two rt. 4 s. I. 13. Q. E. D. I. 29. I. 29. Ex. 1. In an acute-angled triangle, any two angles are greater than the third. Ex. 2. The straight line, which bisects the external vertical angle of an isosceles triangle is parallel to the base. Ex. 3. If the side BC of the triangle ABC be produced to D, and AE be drawn bisecting the angle BAC and meeting BC in E; shew that the angles ABD, ACD are together double of the angle AED. Ex. 4. If the straight lines bisecting the angles at the base of an isosceles triangle be produced to meet; shew that they will contain an angle equal to an exterior angle at the base of the triangle. Ex. 5. If the straight line bisecting the external angle of a triangle be parallel to the base; prove that the triangle is isosceles. The following Corollaries to Prop. 32 were first given in Simson's Edition of Euclid. COR. 1. The sum of the interior angles of any rectilinear figure together with four right angles is equal to twice as many right angles as the figure has sides. B E Let ABCDE be any rectilinear figure. Take any pt. F within the figure, and from F draw the st. lines FA, FB, FC, FD, FE to the angular pts. of the figure Then there are formed as many s as the figure has sides. The threes in each of these As together two rt. 4 s. ..all the s in these As together twice as many right 4s as there are ▲s, that is, twice as many right ▲ s as the figure has sides. Now angles of all the Ass at A, B, C, D, E and s at F, that is, = 48 of the figure and ▲ s at F, and ... = 48 of the figure and four rt. 4 s. I. 15. Cor. 2. ..s of the figure and four rt. 4 s=twice as many rt. 4 s as the figure has sides. COR. 2. The exterior angles of any convex rectilinear figure, made by producing each of its sides in succession, are together equal to four right angles. Every interior angle, as ABC, and its adjacent exterior angle, as ABD, together are two rt. 4 s. D B .. all the intr. 4s together with all the extr. 4 s all the intr. 4s together with all the extr. 48 .. all the extr. 4 s=four rt. 4 s. NOTE. The latter of these corollaries refers only to convex figures, that is, figures in which every interior angle is less than two right angles. When a figure contains an angle greater than two right angles, as the angle marked by the dotted line in the diagram, this is called a reflex angle. See p. 149. Ex. 1. The exterior angles of a quadrilateral made by producing the sides successively are together equal to the interior angles. |