Therefore the angle formed by the base and the altitude upon an arm of an isosceles triangle equals one-half the vertical angle.

161. REMARK.

Geometrical propositions may be demon

strated by algebraical computation.

Although a direct geometrical proof is always preferable, the above method is very useful, especially for establishing relations between angles.

The propositions to be used are mainly the proposition of the sum of the angles of a polygon (and triangle), and the proposition of the exterior angles of a polygon (and triangle).

If the algebraical solution should be too difficult for the beginner, numerical examples ought to be solved at first.

Ex. 243. If the bisectors of two adjacent angles are perpendicular to each other, the angles are supplementary.

Ex. 244. The bisectors of vertical angles are in a straight line.

Ex. 245. Perpendiculars drawn from a point within an angle, upon the sides, include an angle which is the supplement of the given angle.

Ex. 246. If the vertical angles of two isosceles triangles are supplementary, the base angles are complementary.

Ex. 247. If the base angles of two isosceles triangles are complementary, the vertical angles are supplementary.

Ex. 248. The exterior angle at the base of an isosceles triangle is equal to a right angle increased by one-half the vertical angle.

Ex. 249. If each side of an equilateral triangle be trisected, and the points of division nearest to each vertex be joined respectively, a hexagon is formed which is equiangular and equilateral.

Ex. 250. Homologous medians of equal triangles are equal.

Ex. 251. Homologous altitudes of equal triangles are equal.

Ex. 252. Two isosceles triangles are equal if the vertical angle and the altitude upon an arm of the one are respectively equal to the vertical angle and the homologous altitude of the other.

Ex. 253. If a median of a triangle is perpendicular to the base, the triangle is isosceles. Ex. 254. If in the pentagon ABCDE AB= BC, AE = CD, and ▲ A = ≤ C, then BE = BD, and ELD.

Ex. 255. If the opposite sides of a hexagon are parallel, and one pair of opposite are equal, all opposite sides are

sides equal.

A

A

Ex. 259. Two equilateral triangles are equal if the altitude of one equals the altitude of the other.

Ex. 260. Any straight line that passes through the midpoint of one of the diagonals of a parallelogram, bisects the parallelogram.

Ex. 261. The number of all diagonals of a polygon of n sides is n(n−3).

2

Ex. 262. If a perpendicular be dropped from the vertex to the base of a triangle, each segment of the base will be smaller than the adjacent side of the triangle.

Ex. 263. How many sides has a polygon, the sum of whose interior angles is equal to three times the sum of the angles of a hexagon ?

Ex. 264. How many sides has an equiangular polygon, whose exterior angle equals the interior angle of an equilateral triangle ?

Ex. 265. Prove the proposition of the sum of the interior angles of a polygon by joining any point within to the vertices of the polygon.

Ex. 266. If the vertices of a triangle lie in the sides of another triangle, the perimeter of the first is less than the perimeter of the second.

Ex. 267. The perpendiculars from two vertices of a triangle upon the median drawn from the third vertex are equal.

Ex. 268. The altitude upon the hypotenuse of a right triangle divides the figure into two triangles which are mutually equiangular.

Ex. 269. If the upper base of an isosceles trapezoid is equal to the arms, the diagonals bisect the angles at the lower base.

Ex. 270. The bisectors of the four angles of a parallelogram enclose a rectangle.

Ex. 271. The lines joining the midpoints of the sides of a rectangle, taken in order, enclose a rhombus.

Ex. 272. The lines joining the midpoints of the sides of a rhombus, taken in order, enclose a rectangle.

Ex. 273. The lines joining the midpoints of the sides of any quadrilateral, taken in order, enclose a parallelogram. (Ex. 212.)

Ex. 274. The lines joining the midpoints of opposite sides of any quadrilateral, bisect each other. (Ex. 273.)

Ex. 275. If the vertical angle of an isosceles triangle is one-half of a base angle, a bisector of a base angle divides the figure into two isosceles triangles.

* Ex. 276. If a line from one end of the base of an isosceles triangle to the opposite side divides the figure into two isosceles triangles, then the line is a bisector of the base angle, and each base angle equals the double of the vertical angle,

* Ex. 277. The midpoints of two opposite sides of a quadrilateral and the midpoints of the diagonals determine the vertices of a parallelogram.

* Ex. 278. The lines joining the midpoints of the opposite sides of a quadrilateral and the line joining the midpoints of the diagonals meet in a point. (Exs. 277, 274.)

Ex. 279. The bisectors of the exterior angles of a quadrilateral form a quadrilateral, the sum of whose opposite angles is equal to one straight angle.

Ex. 280. A line from the vertex of an isosceles triangle to any point in the base is smaller than the arms.

In order to prove that the sum of two lines,

a and b, equals a third line, c, either

(a) Construct the sum of a and b, and prove the line so obtained is equal to c, or