55. THEOREM. The angles opposite the equal sides of an isosceles triangle are equal.

Given: ▲ ABC, AB = AC. To Prove: LB=LC. Proof: Suppose AX is drawn dividing BAC into two equal angles, and meeting BC at X. In A BAX and CAX, AX= AX (Identical); AB AC (Given); ▲ BAX

X

=≤CAX. (Because ▲ made them=.) ..▲ ABX=▲ ACX. (Why?) (52.) ..ZB=ZC. (Why?) (See 27.)

56. THEOREM. An equilateral triangle is equiangular. (See 55.)

57. THEOREM. The line bisecting the vertex-angle of an isosceles triangle is perpendicular to the base, and bisects the base.

Prove ▲ ABX and ACX equal as in 55. Then, AXB =AXC. (Why?) (27.) .. AXB and AXC are rt. 4 (16).

.. AX is to BC. (Why?) (17.) And, also, BX = CX. (Why?) (27.)

58. THEOREM. Two triangles are equal, if the three sides of one are equal respectively to the three sides of the other.

Proof: Place ▲ ABC in the position of ▲ AST so that the longest equal sides (BC and ST) coincide and 4 is opposite ST from R. Draw RA. RS = AS (Given). .. ASR is an isosceles A. (Def. 24.)'

.. ≤ SRA= ≤ SAR. (Why?) (55.) Likewise TR AT (?) and

▲ TRA= / TAR. (Why?)

▲ SRT = 2 SAT (Ax.2).

That is, A RST=▲ ABC.

Adding these equals we obtain
..▲ RST=▲ AST (52).
(Substitution, Ax. 6.)

59. Elements of a theorem. Every theorem contains two parts, the one is assumed to be true and the other results from this assumption. The one part contains the given conditions, the other part states the resulting truth.

The assumed part of a theorem is called the hypothesis. The part whose truth is to be proved is the conclusion. Usually the hypothesis is a clause introduced by the word "if." When this conjunction is omitted, the subject of the sentence is known and its qualities, described in the qualifying words, constitute the "given conditions." Thus, in the theorem of 58, the assumed part follows the word "if," and the truth to be proved is: "Two triangles are equal."

60. Elements of a demonstration. All correct demonstrations should consist of certain distinct parts, namely:

1. Full statement of the given conditions as applied to a particular figure.

2. Full statement of the truth which it is required to prove. 3. The Proof. This consists in a series of successive statements, for each of which a valid reason should be quoted. (The drawing of auxiliary lines is sometimes essential, but this part is accomplished by imperatives for which no reasons are necessary.)

4. The conclusion declared to be true.

The letters "Q.E.D." are often annexed at the end of a demonstration and stand for "quod erat demonstrandum," which means, "which was to be proved."

MODEL DEMONSTRATIONS

The angles opposite the equal sides of an isosceles triangle are equal.

:.^ ABX = ▲ ACX. (Two A are if two sides and the included of one are = respectively to two sides and the included of the other.) Hence, ≤ B = ≤ C. (Homologous parts of equal figures are equal.) Q.E.D.

Two triangles are equal if the three sides of one are equal respectively to the three sides of the other.

sides (BC and ST) coincide, and A is opposite ST from R. Draw RA. RS AS (Hypothesis).

▲ASR is isosceles. (An isosceles ▲ is a ▲ two sides of which are equal.) :. 2 SRA = 2 SAR... (1) .. (The opp. the

sides of an isos. ▲ are =.) AT (Hypothesis).

▲ TRA is isosceles. (Same reason as before.) < TRA = < TAR... (2).. (Same reason as for (1).) Adding equations (1) and (2).

2 SRT = 2 SAT. (If ='s are added to ='s the results are =.) Consequently, the ^ RST = ▲ AST. (Two ▲ are = if two sides and the included of one are = respectively to two sides and the included of the other.)

That is, ▲ RST = ▲ ABC. (Substitution; ▲ ABC is the same as ▲ AST.)

Q.E.D.

The preceding form of demonstration will serve to illustrate an excellent scheme of writing the proofs. It will be observed that the statements are at the left of the page and their reasons at the right. This arrangement will be found of great value in the saving of time, both for the pupil who writes the proofs and for the teacher who reads them.

61. The converse of a theorem is the theorem obtained by interchanging the hypothesis and conclusion of the original theorem. Consult 44 and 45; 79, 80, and others.

Every theorem which has a simple hypothesis and a simple conclusion has a converse, but only a few of these converses are actually true theorems.

For example: Direct theorem: "Vertical angles are equal."

Converse theorem: "If angles arc equal, they are vertiThis statement cannot be universally true.

cal."

The theorem of 120 is the converse of that of 55.

62. Auxiliary lines. Often it is impossible to give a simple demonstration without drawing a line (or lines) not described in the hypothesis. Such lines are usually dotted for no other reason than to aid the learner in distinguishing the lines mentioned in the hypothesis and conclusion from lines whose use is confined to the proof. Hence, lines mentioned in the hypothesis and conclusion should never be dotted. (The figure used in 57 should contain no dotted line.)

63. Converse of definitions. The converse of a definition is true. It is often advantageous to quote the converse of a definition, as a reason, instead of the definition itself.

64. Homologous parts. Triangles are proved equal in order that their homologous sides, or homologous angles, may be proved equal. This is a very common method of proving lines equal and angles equal.

65. The distance from one point to another is the length of the straight line joining the two points.

66. THEOREM. If lines be drawn from any point in a perpendicular erected at the midpoint of a straight line to the ends of the line,

I. They will be equal.

II. They will make equal angles with the perpendicular.

III. They will make equal angles with the line.

Given AB to CD at its

midpoint, B; P any point in AB; PC and PD.

To Prove: I. PC=PD;

II. ≤ CPB = ≤ DPB; and

III. < C = L D.

Proof: In rt. A PBC and

PBD, BC = BD (Hyp.); BP = BP (Iden.). ..▲ PBC = ▲ PBD. (Why?) (53.)

.. I. PC=PD (Why?) (27;) II. ≤ CPB = ≤ DPB (Why?); III. C=≤D (Why?).

Q.E.D.

67. THEOREM. Any point in the perpendicular bisector of a line is equally distant from the extremities of the line. (See 66, I.)

68. THEOREM. Any point not in the perpendicular bisector of a line is not equally distant from the extremities of the line.

DO+OP > PD. (Why?) (Ax. 12.) But co = OD (67). .. CO+OP > PD. (Substitution; Ax. 6.)

That is, PC > PD, or PC is not = PD.

Q.E.D.