Ax. c.

therefore the angles ABC, ABD are together equal to the angles EBC, EBD; therefore the angles ABC, ABD are together equal to two right angles.

Q.E.D.

COR. All the angles made by any number of straight lines drawn from a point, each with the next following in order, are together equal to four right angles.

Ex. 1. If two straight lines intersect, and one of the four angles so formed is a right angle, shew that the other three angles are also right angles.

*Ex. 2. The bisectors of the adjacent angles which one straight line makes with another include a right angle.

THEOR. 3. If the adjacent angles made by one straight line with two others are together equal to two right angles, these two straight lines are in one straight line.

Let the adjacent angles ABC, ABD which the straight line AB makes with the two other straight lines BC, BD be together equal to two right angles :

B

A

then shall BC and BD be in one straight line.

* Exercises marked with an asterisk are worth remembering as results, or with a view to the solution of subsequent exercises.

For if BD is not in a straight line with BC,

let BE be in a straight line with BC.

Then because AB stands on the straight line CBE,

therefore the angles ABC, ABE are together equal to two right angles;

I. 2.

but the angles ABC, ABD are also together equal to two right angles;

Hyp.

therefore the angles ABC, ABE are together equal to the angles ABC, ABD.

Ax. c.

Take away the common angle ABC;

then the angle ABE is equal to the angle ABD,

Ax. e.

the part to the whole,

Ax. a.

which is impossible ;

therefore BE is not in a straight line with BC.

In the same way it can be shown that no other straight line than

BD is in a straight line with BC,

therefore BC and BD are in one straight line.

Q.E.D.

THEOR. 4. If two straight lines cut one another, the vertically opposite angles are equal to one another.

Let the two straight lines AB, CD cut one another at O:

then shall the angle AOC be equal to the angle BOD,

and the angle BOC to the angle AOD.

Because AO stands upon CD,

therefore the angles AOC, AOD are together equal to two right angles;

again, because DO stands upon AB,

I. 2.

therefore the angles AOD, DOB are together equal to two right angles;

I. 2.

therefore the angles AOC, AOD are together equal to the angles AOD, DOB;

Ax. c.

Ax. e.

take away the common angle AOD;

then the angle AOC is equal to the angle BOD.

In the same way it may be proved that the angle BOC is

equal to the angle AOD.

Q.E.D.

Ex. 3. The bisectors of two vertically opposite angles are in one

straight line.

SECTION II.

TRIANGLES.

DEF. 18. A plane figure is a portion of a plane surface inclosed by

a line or lines.

DEF. 19. Figures that may be made by superposition to coincide with one another are said to be identically equal; or they

are said to be equal in all respects.

DEF. 20. The area of a plane figure is the quantity of the plane surface inclosed by its boundary.

DEF. 21. A plane rectilineal figure is a portion of a plane surface

inclosed by straight lines. When there are more than

three inclosing straight lines the figure is called a polygon. DEF. 22. A polygon is said to be convex when no one of its angles

is reflex.

DEF. 23. A polygon is said to be regular when it is equilateral and equiangular; that is, when its sides and angles are equal.

DEF. 24. A diagonal is the straight line joining the vertices of any angles of a polygon which have not a common arm. DEF. 25. The perimeter of a rectilineal figure is the sum of its

sides.

DEF. 26. A quadrilateral is a polygon of four sides, a pentagon one of five sides, a hexagon one of six sides, and so on.

DEF. 27. A triangle is a figure contained by three straight lines. DEF. 28. Any side of a triangle may be called the base, and the opposite angular point is then called the vertex.

DEF. 29. An isosceles triangle is that which has two sides equal; the angle contained by those sides is called the vertical angle, the third side the base.

THEOR. 5. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles included by these sides equal, then the triangles are identically equal, and of the angles those are equal which are opposite to the equal sides.

Let ABC, DEF be two triangles having the side AB equal to the side DE, the side AC to the side DF, and the angle BAC to the angle EDF:

AAA

E

E

then shall the triangles be identically equal, having the side BC equal to the side EF, the angle ACB to the angle DFE, and the angle ABC to the angle DEF.

Let the triangle ABC be applied to the triangle DEF, so that the point A may fall on the point D, the side AB along the side DE, and the point C on the same side of DE as the point F;

then B will fall on E, since AB is equal to DE,

Hyp.