216. The sum of all the angles of any polygon is twice as many right angles as the polygon has sides, less four right angles.

Given polygon ABCDE..., any polygon having n sides.
To prove the sum of its = 2 n rt. — 4 rt. s.

217. Cor. Each angle of an equiangular polygon of

2(n-2)

n sides is equal to

right angles.

n

218. If the sides of any polygon are prolonged in succession one way, no two adjacent sides being prolonged through the same vertex, the sum of the exterior angles thus formed is four right angles.

Given polygon P with ≤1, Z2, Z3, Z4, ..... its successive exterior angles.

To prove <1 + Z2 + Z3 + Z4 +

... = 4 rt. s.

REASONS

1. If one str. line meets another str. line, the sum of

the two adj. is 2 rt. .

$ 65.

2. If equals are multiplied by equals, the products are equal. $ 54, 7 a.

3. Arg. 2.

4. The sum of all the of any polygon = 2 n rt.

-4rt. . § 216.

5. If equals are subtracted from equals, the remainders are equal. § 54, 3.

219. Note. The formula 2 n rt. 4 – 4 rt.. (§ 216) is sometimes more useful in the form (n − 2) 2 rt. 4.

Ex. 247. Find the sum of the angles of a polygon of 7 sides; sides; of 10 sides.

of 8

Ex. 248. Prove Prop. XXIX by drawing as many diagonals as possible from one vertex.

Ex. 249. How many diagonals can be drawn from one vertex in a polygon of 8 sides? of 50 sides? of n sides? Show that the greatest number of diagonals possible in a polygon of n sides (using all vertices) is n(n-3)

2

Ex. 250. How many degrees are there in each angle of an equiangular quadrilateral? in each angle of an equiangular pentagon?

Ex. 251. How many sides has a polygon the sum of whose angles is 14 right angles ? 20 right angles? 540° ?

Ex. 252. How many sides has a polygon the sum of whose interior angles is double the sum of its exterior angles ?

Ex. 253. Is it possible for an exterior angle of an equiangular polygon to be 70° ? 72°? 140° ? 144° ?

Ex. 254. How many sides has a polygon each of whose exterior angles equals 12° ?

Ex. 255. How many sides has a polygon each of whose exterior angles is one eleventh of its adjacent interior angle?

Ex. 256. How many sides has a polygon the sum of whose interior angles is six times the sum of its exterior angles?

Ex. 257. How many sides has an equiangular polygon if the sum of three of its exterior angles is 180° ?

Ex. 258. Tell what equiangular polygons can be put together to make a pavement. How many equiangular triangles must be placed with a common vertex to fill the angular magnitude around a point?

QUADRILATERALS. PARALLELOGRAMS

QUADRILATERALS CLASSIFIED WITH RESPECT TO PARALLELISM

220. Def. A parallelogram is a quadrilateral whose opposite sides are parallel.

221. Def. A trapezoid is a quadrilateral having two of its opposite sides parallel and the other two not parallel.

222. Def. A trapezium is a quadrilateral having no two of its sides parallel.

PARALLELOGRAMS CLASSIFIED WITH RESPECT TO ANGLES

223. Def. A rectangle is a parallelogram having one right angle.

It is shown later that all the angles of a rectangle are right angles.

224. Def. A rhomboid is a parallelogram having an oblique angle.

It is shown later that all the angles of a rhomboid are oblique.

225. Def. A rectangle having two adjacent sides equal is

a square.

It is shown later that all the sides of a square are equal.

226. Def. A rhomboid having two adjacent sides equal is

a rhombus.

It is shown later that all the sides of a rhombus are equal. 227. Def. A trapezoid having its two non-parallel sides equal is an isosceles trapezoid.

228. Def. Any side of a parallelogram may be regarded as its base, and the line drawn perpendicular to the base from any point in the opposite side is then the altitude.

229. Def. The bases of a trapezoid are its parallel sides, and its altitude is a line drawn from any point in one base perpendicular to the other.

230. Any two opposite angles of a parallelogram are equal, and any two consecutive angles are supplementary.

B

C

Given

A

ABCD.

To prove: (a) ZA = ZC, and ≤B=ZD;

(b) any two consecutive, as A and B, sup.

231. Cor. All the angles of a rectangle are right angles, and all the angles of a rhomboid are oblique angles.

Ex. 259. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.

Ex. 260. If an angle of one parallelogram is equal to an angle of another, the remaining angles are equal each to each.

Ex. 261. The bisectors of the angles of a parallelogram (not a rhombus or a square) inclose a rectangle.