PROPOSITION V. THEOREM.

51. From a point without a line, there can be but one perpendicular to that line.

B

Given: The point A, the line BC, AB to BC, and AC any other line from A to BC;

To Prove:

AC is not perpendicular to BC.

Turn the figure ABC about BC till A takes the position a' in the original plane. Mark the point 4', restore ABC to its first position, and join A'B, A'C.

Then since AB and AC can be made to coincide with A'B

and 'C, while BC retains its position,

(Const.)

(6)

(29)

.. ACA' is not a straight line,

.. ZACA' is not a straight 4,

ZACB, the half of ▲ ACA', is not a right .

Q.E.D.

EXERCISE 9. If in the diagram for Prop. III., angle CAD is of a right angle, what angle must BA make with AC so that BA shall be in the same straight line with AD?

10. If, in a line MN, a point P be taken, and two lines PQ, PR, be drawn so that angle QPM = angle RPN, then QR is a straight line. 11. In the diagram for Prop. V., the angles A and A' are the supplements of what angles? Show that these angles are equal.

12. In this diagram, show that the exterior angles at C are equal.

TRIANGLES.

52. A triangle is a portion of a plane bounded by three straight lines.

53. The lines that bound a triangle are called its sides; the angles formed by the sides, its angles; and the vertices of the angles, the vertices or angular points of the triangle.

If all the sides of a triangle be produced both ways (see triangle C below), nine new angles will be formed in addition to those properly called the angles of the triangle, or by way of distinction, its interior angles. Of the nine outer angles, the six angles that are supplementary to the interior angles, are called exterior angles. Interior angles are always meant whenever we refer to the angles of a triangle without any distinguishing epithet.

54. The base of a triangle is the side on which it is supposed to stand; the angle opposite the base is sometimes referred to as the vertical angle.

AAA

55. An equilateral triangle has three equal sides; as 4.

56. An isosceles triangle has two equal sides; as B.

57. A scalene triangle has no two sides equal; as C.

E

D

F

58. A right triangle has a right angle; as D.

59. An obtuse triangle has an obtuse angle; as E.

60. An acute triangle has all its angles acute; as F.

While lines and angles can be equal only in one way, it is different in regard to triangles and other inclosed figures. For as will afterwards be seen, two triangles that are equal as regards surface may differ greatly as to their sides and angles. Hence it is necessary to define the term equal in regard to figures in general.

61. Equal figures are such as can be made to coincide exactly; that is, are equal in every respect.

62. THEOREM. Two triangles are equal if their angular points can be made to coincide.

For if the angular points coincide, the sides terminated by those points must coincide (14); hence, also, the angles formed by the sides, and the surfaces bounded by them, must coincide. It is evident that the theorem may be extended so as to apply to figures bounded by any number of straight lines, the reasoning being exactly the same.

EXERCISE 13. Show, by Prop. V., that no triangle can have two of its angles right angles.

14. If triangle A has all its angles equal, show that its six exterior angles are all equal.

15. If a triangle has two equal angles, those angles must be acute. What about the third angle ?

16. If a triangle has no two angles equal, it has three different pairs of equal exterior angles.

17. If, in triangle D, the sides containing the right angle be produced through its vertex, the sum of the exterior angles thus formed is equal to a straight angle.

18. If, in triangle E, the sides containing the obtuse angle be produced through its vertex, the sum of the exterior angles thus formed is less than a straight angle.

B

PROPOSITION VI. THEOREM.

63. Two triangles are equal if a side and the including angles of the one are respectively equal to a side and the including angles of the other.

ДДД

C B

C B

Given: In triangles ABC, A'B'C', BC equal to B'C', angle B equal to angle B', and angle C equal to angle c';

To Prove Triangle ABC is equal to triangle A'B'C'.

If ▲ ABC be placed upon ▲ A'B'C',

so that BC B'C',

then since B = LB',

(Post. 5)

(Hyp.)

BA will take the direction of B'A',

(25)

Also since C = ≤ c',

(Hyp.)

(25)

and 4 will lie on B'A' or B'A' produced.

CA will take the direction of c'A',

and 4 will lie on c'A' or C'A' produced.

Then since 4 lies on both B'A' and c'A',
A must coincide with 4',

the only point common to B'A' and c'A',

.. AABC=AA'B'C'.

(6)

Q.E.D. (62)

64. SCHOLIUM. If the including angles are all equal, i.e., if CB = ≤ C' =▲ B', as in the right hand pair of triangles, it is manifest that AABC can be made to coincide with ▲ A'B'C' in the reverse position also, so that AB will coincide with A'c', and AC with A'B'.

PROPOSITION VII. THEOREM.

65. If two angles of a triangle are equal, the sides opposite those angles are also equal.

Given: In triangle ABC, angle C equal to angle B;
AB is equal to AC.

To Prove:

Turn ▲ ABC about its base BC till A takes the position A' (Post. 5). Mark the point A', restore ABC to its first position, and join A'B, A'c (Post. 1), so as to form the ▲ A'BC.

Since their angular points can be made to coincide, (Const.)

and A'BAB, A'C= AC, A'BC = ZB,

(62)

A'CB = C. (14)

EXERCISE 19. In the diagram for Prop. VII., if AA' be drawn, cut

ting BC in O, show that when triangle ABC is folded over on triangle A'BC, AO must coincide with A'O.