Therefore, also, the sum of the three interior 8 of the AABC is equal to two rights.

In the same way it may be proved that the sum of the three interiors of any other ▲ is equal to two rights.

This important proposition involves an immense variety of consequences. Some of the most obvious are stated in the following corollaries :

B

:

1. If one of a ▲ is a right, the other two

Ls are together equal to a right .

2. If one

other two

3. An obtuse

of a ▲ is equal to the sum of the

8, that is a right .

is greater, and an acute is less

than the sum of the other two s.

4. If one of a ▲ is greater than the sum of the other two, it is obtuse; if it is less than the sum of the other two, it is acute.

5. If two isosceles As have their vertical sequal, thes at their, bases are also equal.

6. If the vertical

Z, each of the

由

of an isosceles is a right s at the base is half a right

7. The sum of all the internal 8 of any rectilinear figure, together with four rights, is equal to twice as many rights as the figure has sides. Let ABCDE be a rectilinear figure. Take a point F somewhere inside the figure, and join the points F and A F and B, F and C, F and D, F and E The whole figure is thus divided into as many ▲ as the figure has sides.

F

The sum of the three s of each of these As it two rights; consequently, the sum of all the 28 of all the As is twice as many rights as the figure has sides.

The

s at the point F together make up four

rights; the others of the As make up the 28 of the polygon ABCDE. Therefore the Ls of the polygon together with four right 8 are equal to twice as many rights as the figure has sides.

8. If the sides of any convex rectilinear figure be produced, the externals are together equal to four rights. For each external with the internal adjacent to it, makes up the two rights. Consequently, all the internals and all the external s taken together make up twice as many right s as the figure has sides. But all

the internals, together with four rights, make up twice as many rights as the figure has sides. It follows, therefore, that all the external s taken together, make up four rights.

9. Each of an equilateral ▲ is the third part of two rights, or two-thirds of one right . Therefore, if we bisect an of an equilateral A, we get the third part of a right. (See note on Prop. IX.)

PROPOSITION XXXIII.

If there are two straight lines which are equal and parallel to each other, and their corresponding* extremities be joined by straight lines, these straight lines will also be equal and parallel.

For the construction in this proposition we must be able to join two given points by a straight line.

By corresponding extremities are meant those which point in the same direction.

с

To prove the proposition by the help of the construction we must know,

1. That if two As have two sides of the one equal respectively to two sides of the other, and also the included between the said sides in the first equal to the included between the said sides in the other ▲, those As will also be equal to each other in every other respect. (Prop. IV.)

2. That if two straight lines are parallel and a third line meets them, the alternate s so formed are equal. (Prop. XXIX.)

3. That if the alternates, formed when two straight lines are met by a third, are equal, those two straight lines will be parallel. (Prop. XXVII.)

D

Suppose CD and A B to be two equal and parallel straight lines, and let the corresponding extremities C and A and D and B be joined by the straight lines CA and D B.

B

It has to be proved that the lines CA and D B are equal and parallel. Join by a straight line either pair of extremities of the equal and parallel straight lines, which do not correspond, as, for example, C and B.

We have thus formed two As, CDB and B AC. In these As the side CD is equal to the side BA. (This we know to start with.)

The side C B is common to the two As.

Also the s DCB and CBA are equal, because they are the alternate s formed by the parallel straight lines CD and A B when met by a third line, CB.

Thus the two As CDB and B A C have two sides of the one (namely DC and C B) equal respectively

to two sides of the other (namely A B and B C), and the included DCB of the one equal to the included ZABC of the other.

Consequently (according to the fourth proposition), these triangles are also equal in every other respect. Among these respects are:

1. That the side B D is equal to the side A C. (This is one part of the proposition to be proved.)

2. That the DBC is equal to the

ACB.

Now DBC and ACB are the alternate s formed by the two straight lines A C and BD when met by a third line, C B.

Hence, since these alternates are equal, the lines A C and B D are parallel. (Prop. XXVII.)

It has thus been shown that the lines A C and BD are both equal and parallel.

PROPOSITION XXXIV.

The opposite sides and angles of a parallelogram are equal to each other, and a parallelogram is bisected (that is, divided

into two equal parts) by each of its diagonals.

To prove this proposition we must know,—

1. That if equals be added to equals the sums are equal. (Ax. II.)

2. That if two As have two sides and the
between them in the one, equal respectively
to two sides and the between them in the
equal in every other

other, those As are also
respect. (Prop. IV.)
3. That if two As have
respectively to two
side of the first

two

s of the one equal 8 of the other, and a equal to a side of the

second, similarly placed with respect to the equals, those As are also equal in every other respect. (Prop. XXVI.)

4. That if two lines are parallel and a third line meets them, the alternate 8 so formed are equal. (Prop. XXIX.)

Let ABDC be a parallelogram, that is, suppose it to be a four-sided figure, whose B opposite sides are parallel, and let A D be one of its diagonals.

A

D

We have to prove that the opposite sides are equal:—namely, the side A B to the side CD, and the side A C to the side BD: and also that the opposites are equal, namely, the ZACD, and the CAB to the

ABD to the

BDC and

that the parallelogram is divided into two equal parts by its diagonal A D.

Proof. Because A B and CD are parallel, and the line A D meets them, it follows that the alternate s so formed, namely, BAD and AD C, are equal. (Prop. XXIX.)

Because A C and BD are parallel and the line A D meets them, it follows (Prop. XXIX.) that the alternates so formed, namely, CAD and ADB, are equal.

Hence, the two As ABD and CDA have the two s BAD and BDA of the one equal respectively to the two s ADC and CAD of the other, and the side A D common to the two As and similarly placed with respect to the equals.

It follows, therefore (Prop. XXVI.), that these ▲s are also equal in every other respect. That is to say, the ABD is equal to the ACD; the side A B to the side CD, the side B D to the side A C, and the area of the one to the area of the other.