Through the other points of division on AG draw parallels to GB. These lines divide AB into n equal parts. [The proof is left to the student.] Ex. 369. A line bisecting one non-parallel side of a trapezoid, and parallel to the base, bisects the other non-parallel side. PROPOSITION XL. THEOREM 155. A line which joins the mid-points of two sides of a triangle is parallel to the third side and equal to half of it. F and Proof. Produce ED by its own length to F. Draw FB. But Ex. 370. If the mid-points of the three sides of a triangle are joined, four equal triangles are formed. Ex. 371. A line joining the mid-points of two adjacent sides of a quadrilateral is equal and parallel to the line joining the mid-points of the other two. Ex. 372. The lines joining the mid-points of the sides of any quadrilateral, taken in order, inclose a parallelogram. Ex. 373. If the mid-points of the arms of an isosceles triangle are joined to the mid-points of the base, an equilateral parallelogram is formed. Ex. 374. If in ▲ ABC, the medians AD and BE meet in O, the line joining the mid-points of AO and BO is equal to DE. Ex. 375. In quadrilateral ABCD, the line connecting the mid-points of AD and AC is equal to the line joining the mid-points of BD and BC. Ex. 376. If the points A' and B' AC and BC of AABC, CA' = A'B' = { (AB). are lying respectively in the sides (CA) and CB' = } (CB), then * Ex. 377. The median of a trapezoid is parallel to the base, and equal to half the sum of the bases. Ex. 378. Given three fixed points A, B, and C, not in a straight line, construct a triangle the mid-points of whose sides are A, B, and C. [See practical problems, p. 290, No. 26.] 156. DEF. An equiangular polygon is a polygon all of whose angles are equal. An equilateral polygon is a polygon all of whose sides are equal. A polygon of five sides is called a pentagon; one of six sides, a hexagon; seven sides, a heptagon; eight sides, an octagon; ten sides, a decagon; twelve sides, a dodecagon. All the polygons discussed are understood to be convex, i.e. such that no side produced will enter the polygon. 157. Polygons that are mutually equiangular and equilateral are congruent, for they can be made to coincide. Ex. 379. Draw two mutually equiangular quadrilaterals that are not mutually equilateral. Ex. 380. Draw two mutually equilateral quadrilaterals that are not mutually equiangular. PROPOSITION XLI. THEOREM 158. The sum of all the angles of a polygon of n sides is equal to (n-2) straight angles. B Given ABCDE a polygon of n sides. To prove Connect any point within, as O, with all the vertices. There will be n▲ thus formed. Since the sum of the angles of each ▲ But the sum of all about 0 .. 2A+2B+ < c + 1 st. 4, (110) (52) ... = ··· = ( n − 2) st. s. 2 st. s. 159. COR. Each angle of an equiangular polygon of n sides Ex. 381. How many straight angles are in the sum of the angles of a polygon of 9 sides? of 12 sides? of 100 sides?. Ex. 382 How many right angles are in a polygon of 20 sides? of 41 sides? of 200 sides? of 1000 sides? Ex. 383. How many degrees are in the sum of the angles of a polygon of 4 sides? of 5 sides? of 6 sides? Ex. 384. How many sides has a polygon the sum of whose angles is 1000 st. ? 200 rt. ? 24 rt. ? 720° ? Ex. 385. How many degrees are in each angle of an equiangular quadrilateral? pentagon? hexagon? decagon? W PROPOSITION XLII. THEOREM 160. If the sides of any polygon be successively produced, forming one exterior angle at each vertex, the sum of these angles is equal to two straight angles. B. Given ABCDE, a polygon of n sides, with the ext. a, b, (Ax. 2) :: (ZA + ZB+···)+(Za+Zb+.....) =n st. 4. But (4+B+ ...) .. <a+2b+c=2 st. 4. Ex. 386. How many degrees are in each exterior angle of an equiangular polygon of 10 sides? of 9 sides? of 36 sides? of 72 sides? Ex. 387. How many sides has a polygon each of whose exterior angles equals 30°? one right angle? 60^? 45°? Ex. 388. How many sides has a polygon each of whose interior angles equals 160°? 179°? 135°? rt. ¿? Ex. 389. How many sides has an equiangular polygon, three of whose exterior angles are together equal to 90° ? Ex. 390. How many sides has an equiangular polygon, four of whose angles are together equal to seven right angles? Ex. 391. How many sides has a polygon, the sum of whose interi angles equals twice the sum of its exterior angles? Ex. 392., How many sides has a polygon each of whose interior angles equals eight times the adjacent exterior angle? 161. DEF. Three or more lines are concurrent if they meet in a common point. PROPOSITION XLIII. THEOREM 162. The bisectors of the angles of a triangle are concurrent in a point which is equidistant from the sides of the triangle. B Given ▲ ABC, and AD, BE, and CF, the bisectors of A, B, and C respectively. To prove (a) AD, BE, and CF meet in a point, as 0. (b) o is equidistant from AB, BC, and CA. Proof. Since AD and BE cannot be II, AD and BE meet in a point, as 0. (107) From o draw Ox, or, and oz respectively ▲ AB, BC, and CA. .'. AD, BE, and CF meet in point 0, and 0 is equidistant from AB, BC, and CA. Q. E. D. Ex. 393. Find by a construction a point equidistant from the three sides of an obtuse triangle. |