Proposition XL

155. 1. Draw two parallelograms such that two sides of one, and the angle between them, shall be equal to the corresponding parts of the other. How do the parallelograms compare?

2. How do two rectangles compare, if the base and altitude of one are equal to the corresponding parts of the other?

Theorem. Two parallelograms are equal, if two sides and the included angle of one are equal to two sides and the included angle of the other, each to each.

Data: Any two parallelograms, as ABCD and EFGH, in which AB EF, AD = = EH, and angle ▲

=

angle E.

To prove parallelograms ABCD and EFGH equal.

Proof. Place EFGH upon ABCD so that EF coincides with its equal AB and E with its equal ▲ 4.

Then, § 70, and

Also,

and

EH coincides with its equal AD,

HG takes the direction of DC,
G falls upon DC, or upon DC produced.

FG takes the direction of BC,

G falls upon BC, or upon BC produced.

Since G falls upon both DC and BC, it must fall upon their point of intersection C, which is the only point common to DC

and BC.

Hence, § 36,
Therefore, etc.

ABCDEFGH.

Q.E.D.

156. Cor. Two rectangles are equal, if the base and altitude of one are equal to the base and altitude of the other, each to each.

Ex. 102. From any point in the base of an isosceles triangle lines are drawn parallel to the equal sides and produced until they meet the sides of the triangle. How does the sum of these two lines compare with one of the equal sides of the triangle ?

Proposition XLI

157. Draw three or more parallel lines intercepting equal parts on a transversal; draw any other transversal. How do the parts which the parallels intercept on the second transversal compare in length?

Theorem. If three or more parallel lines intercept equal parts on any transversal, they intercept equal parts on every transversal.

Proof. Draw AB, CD, and EF each parallel to HP.

Then, § 140, ABMH, CDKM, and EFPK are parallelograms, and, § 153, HM = AB, MK = CD, and KP = EF.

Now, in A ABC, CDE, and EFG,

K

P

Ex. 103. In the triangle ABC angle A is double angle B and the exterior angle at C is 105°. How many degrees are there in angles A and B

respectively?

Ex. 104. If one angle of a parallelogram is a right angle, what is the value of each of the other angles?

Ex. 105. One angle of a parallelogram is three times its supplement. What is the value of each angle of the parallelogram?

MILNE'S GEOM.-5

Proposition XLII

158. 1. Draw a triangle and a line parallel to the base, bisecting one of the sides. How does it divide the other side? How does the part of this line intercepted by the sides of the triangle compare in length with the base of the triangle?

2. Draw a triangle and a line connecting the middle points of two of its sides. What is the direction of this line with reference to the third side of the triangle?

Theorem. If a straight line drawn parallel to the base of a triangle bisects one of its sides, it bisects the other side, and is equal to one half of the base.

Data: Any triangle, as ABC, and a straight line DE drawn parallel to AB bisecting AC at D.

D

E

C

159. Cor. The line joining the middle points of two sides of a

triangle is parallel to the third side.

For if the line is not parallel to the third side, suppose a line drawn through D, the middle point of 4C, parallel to AB. By § 158, it will pass through E, the middle point of BC, and we shall have two straight lines drawn between the same two points, which by Ax. 12 is impossible. Consequently, the line joining the middle points of two sides of a triangle is parallel to the third side.

Proposition XLIII

160. Draw a trapezoid and a line connecting the middle points of the non-parallel sides. What is the direction of this line with reference to the bases of the trapezoid? How does it compare in length with the sum of the bases?

Theorem. The line which joins the middle points of the non-parallel sides of a trapezoid is parallel to the bases and is equal to one half their sum.

Data: Any trapezoid, as ABCD, and the line EF joining the middle points of the non-parallel sides AD and BC.

To prove EF parallel to AB and DC and equal to one half AB + DC.

A

D

E

K

H

B

Proof. Draw AC intersecting EF at K, and from H, the middle point of AC, draw HE and HF.

:. § 70,

But, data,

then,

and

Now,

and

HF DC;

HE DC;

EHF is a straight line parallel to AB and DC.

hence, Ax. 2,

Therefore, etc.

DC,

AB;
(AB+ DC).

Q.E.D.

POLYGONS

161. A portion of a plane bounded by any number of straight lines is called a Polygon.

The sum of the straight lines which bound a polygon is called its perimeter.

The term polygon is usually applied to figures of more than four sides.

162. A polygon of three sides is called a trigon or triangle; one of four sides, a tetragon or quadrilateral; one of five sides, a pentagon; one of six sides, a hexagon; one of seven sides, a heptagon; one of eight sides, an octagon; one of ten sides, a decagon; one of twelve sides, a dodecagon; one of fifteen sides, a pentadecagon.

163. A polygon such that none of its sides, if produced, extend within it is called a Convex Polygon.

164. A polygon such that two or more of its sides, if produced, extend within it is called a A Concave Polygon.

The reflex angle ABC is called a re-entrant angle. Unless otherwise stated, polygons considered hereafter will be understood to be convex.

Diagonal

B

165. A straight line joining the vertices of two non-adjacent angles of a polygon is called a Diagonal of the Polygon.

Proposition XLI

166. 1. Draw convex polygons, each having a different number of sides, and from any vertex of each draw its diagonals. How does the number of triangles into which each polygon is divided compare with the number of sides of the polygon?

To how many right angles is the sum of the angles of a triangle equal? To how many times two right angles is the sum of the interior angles of a polygon equal?