COROLLARIES. 1. If two corresponding angles are equal,

the same three conclusions follow.

=

For if a <a', then,

Zac, it follows that <c = <a',

i.e. two alternate angles are equal.

2. If two interior or two exterior angles on the same side of the transversal are supplemental, the same three conclusions follow.

For if d < a' = st. <, then, ·. Zd + c = st. Z, it follows that ca', i.e. two alternate angles are equal.

EXERCISES. 89. Prove the theorem, given that b = d.

90. Prove cor. 1, given that < c =

c.

91. Prove cor. 2, given that b + c = st. Z.

92. In the figure of th. 15, suppose a = c′ = 120° 30′, how large is each of the other angles ?

93. In the same figure, suppose a + d' = st. <, and a = 2 d, how large is each of the other angles?

94. If a transversal cuts two lines making the sum of the two interior angles on the same side of the transversal a straight angle, one of them being 30° 27', how large is each of the other angles?

95. If a transversal of two lines makes two corresponding angles equal, then every angle made by the transversal is equal to or supplementary to every other angle made by it.

Theorem 16. If a transversal of two lines makes a pair of alternate angles equal, the two lines are parallel.

a' would

2. P, P' cannot meet towards P, for then

be greater than

c.

Why?

3. ... P, P' cannot meet at all, and P || P'. Def. parallel. 4. Similarly any other two alt. may be taken equal.

COROLLARIES.

lines are parallel.

1. If two corresponding angles are equal, the

For then two alt. s are equal. Th. 15, cor. 1, which says — (?)

2. If two interior or two exterior angles on the same side of the transversal are supplemental, the lines are parallel.

For then two alt. s are equal. Th. 15, cor. 2, which says - (?)

3. Two lines perpendicular to the same line are parallel. (Why?)

POSTULATE 5. It now becomes necessary to assume a fifth postulate, and upon it rests much of the elementary theory of parallels. It is: Two intersecting straight lines cannot both be parallel to the same straight line.

COROLLARY. A line cutting one of two parallel lines, cuts the other also, the lines being unlimited.

(Show that the corollary is necessarily true if the postulate is.)

EXERCISES. 96. In th. 16 would lines bisecting a and c be parallel? Prove it. 97. Show that if a draughtsman's square slides along a ruler, as in the annexed figure, B1C1 | B2C2, and A1C1 || A2C2.

A

B, A2/

A

B2

Theorem 17. The alternate angles formed by a transversal with two parallels are equal.

Given P and P' two parallels,

and T, a transversal.

P

figure, making an equal to a'.

2. Then Q would be parallel to P'.

3. But this is impossible, ... P II P'.

Why?

Post. 5

(Two intersecting straight lines cannot both be parallel to the same straight line.)

4. Similarly it is absurd to suppose that < a' >< c. ../c=

a'.

COROLLARIES. 1. A line perpendicular to one of two paral lels is perpendicular to the other also.

For it cuts the other (post. 5, cor.) and the alternate angles are equal right angles.

2. A line cutting two parallels makes corresponding angles equal, and the interior, or the exterior, angles on the same side of the transversal supplemental. (Th. 15.)

3. If the alternate or the corresponding angles are unequal, or if the interior angles on the same side of the transversal are not supplemental, then the lines are not parallel, but meet on that side of the transversal on which the sum of the interior angles is less than a straight angle.

For the lines cannot be parallel, by th. 17 and cor. 2.

.. P and P' cannot meet towards P', for then c would be greater thana', th. 5.

Let the student give the proof in full form, in steps.

4. Two lines respectively perpendicular to two intersecting

lines cannot be parallel.

For, in the annexed figure, let AB LX, CD LY;

join A and C. Then

BAC <rt., and ACD

<rt. ; .. their sum is <st. ; .. cor. 3 applies.

Give proof in full form in steps.

5. If the arms of one angle are parallel to the arms of another, the angles are equal or supplemental.

The proof is left to the student.

Theorem 18. Lines parallel to the same line are parallel to each other.

EXERCISES. 98. In th. 18, if T cuts A, must it necessarily cut M? Why? If it cuts M, must it necessarily cut B? Why?

99. A line parallel to the base of an isosceles triangle makes equal angles with the sides or the sides produced. (The line may pass above, through, or below the triangle, or through the vertex.)

100. If through any point equidistant from two parallels, two transversals are drawn, they will cut off equal segments of the parallels.

101. ABC is a triangle, and through P, the point of intersection of the bisectors of B and ≤ C, a line is drawn parallel to BC meeting AB at M, and CA at N. Prove that MN MB + CN.

=

Theorem 19. In any triangle, (1) any exterior angle equals the sum of the two interior non-adjacent angles; (2) the sum of the three interior angles is a straight angle.

5. But

6.

NOTES.

Why?

Why?

XBC, Za+c, which

2x+2y+b= st. Z.

..Za+Zb+c=st. Z,

by substituting 4 in 5, which proves (2).

1. Th. 19, (2), is attributed to Pythagoras. 2. The theorem is one of the most important of geometry. to its corollaries (p. 44) frequent reference is hereafter made.

Ax. 2

Def. st.

To it and

EXERCISES. 102. PQR is a triangle having PQ = PR; RP is produced to S so that PS= RP; QS is drawn. Prove that QS 1 RQ.

103. Show that th. 19, (2), is true by laying a ruler along AB, revolv

ing it about A through A so that it lies along AC, then

about C through C so that it lies along CB, then about

B through B so that it lies along BA; thus show that

the ruler has turned through a straight angle and A through A+ZB+ZC. (An illustration, not a proof.)

B

104. As in ex. 103, show that the sum of the exterior angles of any polygon equals a perigon.

105. Prove th. 19, (2), by drawing through C, in the figure given, a line | AB.

106. Also by assuming any point P on AB, drawing PC, and showing that BPC+ CPA = st. 2, and also equals the sum of the interior angles.

107. State the reciprocal of th. 8, and prove or disprove it.