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

B

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 I), 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.

AC will fall along DF, since the angle BAC is equal to the

angle EDF,

Hyp.

and, AC falling along DF,

C will fall on F, since AC is equal to DF.

Hyp.

Hence, B falling on E, and C on F, BC will coincide with

Ax. 2.

EF,

and the triangle ABC will coincide with the triangle DEF, and is therefore identically equal to it, the side BC equal to the side EF, the angle ACB to the angle DFE, and the angle ABC to the angle DEF.

Ax. 1.

Q.E.D.

Ex. 4. The straight line which bisects the vertical angle of an isosceles triangle bisects the base.

Ex. 5. Any point on the bisector of the vertical angle of an isosceles triangle is equidistant from the extremities of the base.

Ex. 6. The straight line which bisects the vertical angle of an isosceles triangle is perpendicular to the base.

Ex. 7. Any point D is taken on the bisector of an angle BAC; prove that, if AB is equal to AC, then the angle ADB is equal to the angle ADC.

Ex. 8. The straight lines drawn from the extremities of the base of an isosceles triangle to the middle points of the opposite sides are equal to one another.

Ex. 9. On one arm of an angle whose vertex is A points B and D are taken, and on the other arm points C and E, such that AB is equal to AC, and AD to AE: shew that BE is equal to CD.

THEOR. 6. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides between the vertices of those angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to the equal angles.

Let ABC, DEF be two triangles having the angle ABC equal to the angle DEF, the angle ACB to the angle DFE, and the side BC to the side EF:

B

then shall the triangles be identically equal, having the angle BAC equal to the angle EDF, the side AC to the side DF, and the side AB to the side DE.

Let the triangle ABC be applied to the triangle DEF, so that the point B may fall on the point E, the side BC along the side EF, and the point A on the same side of EF as the point

D;

then C will fall on F, since BC is equal to EF,

Нур.

BA will fall along ED, since the angle CBA is equal to the angle FED,

Нур.

and CA will fall along FD, since the angle BCA is equal to the angle EFD;

Нур.

hence A, which is the point of intersection of BA and CA, will fall on D, which is the point of intersection of ED and FD,

and the triangle ABC will coincide with the triangle DEF, and is therefore identically equal to it,

Ax. I.

the angle BAC equal to the angle EDF, the side AC to the side DF, and the side AB to the side DE.

Q.E.D.

Ex. 10. If the bisector of an angle of a triangle is also perpendicular to the opposite side, the triangle is isosceles.

THEOR. 7. If two sides of a triangle are equal, the angles opposite to those sides are equal.

Let ABC be a triangle having the side AB equal to the side AC:

да

then shall the angle ACB be equal to the angle ABC.

Let A'B'C' be a triangle identically equal to the triangle ABC, the points A',B',C' corresponding respectively to the points A, B, C.

Then in the triangles ABC, A'C'B',

AB is equal to A'C', since it is equal to AC,
and AC is equal to A'B', since it is equal to AB,

Hyp.

Hyp.

and the angle BAC is equal to the angle C'A'B',