75. The complements of two equal angles are equal (Art. 72 and Ax. 3); the supplements

of two equal angles are equal (Art. 69, Ax. 3).

76. The sum of all the angles about a point equals four right angles.

Thus, Za+2b+2c+ 2 d + Le= 4 rt. 4.

77. The sum of all the angles about a point on the same side of a straight line passing through the point equals two right angles.

Thus, 4p+4q+2r=2 rt. 4.

Fig. 2

Fig. 3

EXERCISES. GROUP 3

Ex. 1. How many different straight lines are determined by three points not in the same straight line?

Ex. 2. How many straight lines are determined by four points in a plane, no three of them being in the same straight line?

40°, 50°, 60°, 70° respec

Ex. 3. If, in Fig. 2 above, & a, b, c, d = 40°, 50°, 60°, tively, find ≤ e.

Ex. 4. If, in Fig. 3 above, the lines forming the angle q are perpendicular to each other and ≤ p = 47°, find the other angles of the figure.

Ex. 5. Measure Za of Fig. 1 on preceding page. out measuring it. Now measure Zb and compare the two results.

Ex. 6. Given QBL AB, PB 1 BC, and LABC=130°; find the other angles of the figure.

Ex. 7. Arrange five points in a plane

Find b with

سلا

so that the fewest number of straight lines may pass through them, no line to pass through more than three points.

BOOK I

RECTILINEAR FIGURES

PROPOSITION I. THEOREM

78. If one straight line intersects another straight line, the opposite or vertical angles are equal.

Given the straight lines AB and CD intersecting at the point 0.

To prove AOC = Z DOB and AOD = ≤ COB.

Proof.

2AOC+2A0D-2 rt. 6,

Art. 73.

(the sum of two adjacent angles formed by one straight line meeting another straight line equals two right angles),

:. LAOC+ LAOD Z BOD + LAOD,

(things equal to the same thing are equal to each other).

Ax. 1.

(if equals be subtracted from equals, the remainders are equal).

In like manner it may be proved that

LAOD L COB.

Ax. 3.

Q. E. D.

Ex. If in the above figure ZDOB=70°, find the other angles

without measuring them.

(29)

PROPOSITION II. THEOREM

79. If, from a point in a perpendicular to a given line, two oblique lines be drawn cutting off on the given line equal segments from the foot of the perpendicular, the oblique lines are equal and make equal angles with the perpendicular.

Given a line AB with CD to it at the point C, and PR and PQ drawn from any point as P in CD, cutting off QC on AB.

CR

Proof. Fold over the figure DCB about DC as an axis till it comes in the plane DCA.

Then

Geom. Ax. 2. Z DCB ZDCA (all right & are =). Art. 72.

.. line CB will take the direction of CA.

(only one straight line can be drawn connecting two given points).

And

CPR will coincide with CPQ.
[

... PR PQ, and CPR = CPQ,
(geometric figures which coincide are equal).

Art. 47.

Q. E. D.

Ex. 1. Point out the hypothesis and the conclusion in the general enunciation of Prop. I. Also point them out in the particular enunciation. Do the same for Prop. II.

Ex. 2. If three straight lines intersect at a point, how many of the angles formed is it necessary to measure, in order to determine all the angles?

PROPOSITION III. THEOREM

80. From a given point without a straight line but one perpendicular can be drawn to the line.

Given the straight line AB, P any point without AB, PQ 1 AB, and PR any other line drawn from P to AB. To prove that PR is not 1 AB.

Proof. Produce PQ to P' making QPPQ. Draw RP'.

(if, from a point in a to a given line, two oblique lines be drawn cutting off on the giv.n line equal segments from the foot of the 1, the oblique lines are equal and make equal & with the 1).

But PRP is not a straight line,

Art. 66.

(only one straight line can be drawn connecting two given points).

:. PRP is not a straight 2.

:. ZPRQ, the half of Z PRP', is not a right Z. Ax. 10. .. PR is not 1 AB.

. only one perpendicular can be drawn from P to AB.

Q. E. D.

Ex. Three straight lines intersect at a point. Two of the adjacent angles formed at the point are 30° and 40°. Find all the other angles at the point.

TRIANGLES

81. A triangle is a portion of a plane bounded by three straight lines, as the triangle ABC.

82. The sides of a triangle are the lines which bound it; the perimeter of a triangle is the sum of the sides; the angles of a triangle are the angles formed by the sides, as the angles A, B

A

B

C

and C; the vertices of a triangle are the vertices of the angles of the triangle.

B

83. An exterior angle of a triangle is an angle formed by one side and by another side produced, as the angle BCD. With reference to the angle BCD, the angles A and B are termed the opposite interior angles.

D

C

84. Classification of triangles according to relative length of the sides. A scalene triangle is a triangle in which no two sides are equal. An isosceles triangle is one in which two sides are equal. An equilateral triangle is one in which

all three sides are equal.

Scalone

Isosceles

Equilateral