Ex. 34. If a diagonal of a quadrilateral bisects those angles whose vertices it joins, the diagonal divides the figure into two equal triangles.

PROPOSITION III. THEOREM

B

C

72. Two triangles are equal if two sides and the included angle of the one are equal respectively to two sides and the included angle of the other (s.a.s.s.a. s.).

Α'

B

Hyp. In ABC and A'B'C',

To prove

Proof. with B'C'.

AB=

A'B', BC = B'C', and ≤ B= Z B'.

ΔΑΒΟ =AA'B'C'.

Apply ▲ ABC to ▲ A'B'C' so that BC shall coincide

74. REMARK.

If the lines and angles whose equality is to be proved are not parts of triangles, try to construct such triangles.

Ex. 42. If in triangle ABC, AB = BC, then A = C.

PROPOSITION IV. THEOREM

75. An exterior angle of a triangle is greater than either remote interior angle.

Hyp.

To prove

4

LBCD is an ext. Z of ▲ ABC.

< BCD><A or B.

Proof. Let E be the midpoint of BC. Draw AE and produce it its own length to F. Draw FC.

In A ABE and FCE, AE= EF and BE = EC.

(Con.)

(Ax. 9)

By joining the midpoint of AC to B, it follows in the same

Ex. 43. If four points, A, B, C, D, in a straight

line be joined to a point, E, without, then

(1) ZABE><ACE,

(2) ZABE>< CED,

(3) LABE>LADE.

A B C D

Ex. 44. If in quadrilateral ABCD the diagonals AC and BD meet in E, find four angles smaller than angle BEA.

Ex. 45. If from any point D in ▲ ABC DA and DB are drawn, then LD> <C.

A

E

a\b

c\d

76. DEF. When two straight lines, AB and CD, are cut by a third straight line, EF, called a transversal, then the angles a, b, g, and h are called exterior angles, the angles c, d, e, and ƒ are called interior angles, the angles a and e, b and f, c and g, and d and h are called corresponding angles, the angles c and f, and d and e are called alternate interior angles, and the angles b and g, a and h, are called alternate exterior angles.

e

gh

PARALLEL LINES

77. DEF. Parallel lines are lines which lie in the same plane and do not meet if produced indefinitely.

parallel.

78. AXIOM 11. Two intersecting lines cannot be both parallel

to a third straight line.

AB and CD are

79. THEOREM. Two straight lines which are parallel to a third straight line are parallel to each other.

For if the two lines should meet, we would have two intersecting straight lines parallel to a third straight line, which contradicts Axiom 11.

C

PROPOSITION V. THEOREM

80. Two lines are parallel if a transversal to these lines makes the alternate interior angles equal.

D

B

с

G

F

Hyp. AC and DF are intersected by BE so that

Proof. AC and DF either meet or are parallel.

Suppose they meet in G.

Then BEG is a triangle whose exterior Z ABE is equal to a remote interior ▲ BEG, which is impossible. Hence AC and DF are parallel.

Q.E.D.

81. SCHOLIUM. That the lines AC and DF cannot meet on the side of A and D can be proved by the same method.

82. REMARK. In order to demonstrate that two lines are parallel, prove the equality of a pair of alternate interior angles

Ex. 48. In the same diagram, if / AHC=ZEIH, CD is parallel to EF.

Ex. 49. Lines are parallel if a transversal makes the alternate exterior angles equal.