337. If the sum of two angles is equal to two right angles (180°), each angle is called the supplement of the other. Thus the angles BCF and ACF (Fig. 7) are supplementary angles.

338. Acute and obtuse angles in distinction from right angles are called oblique angles, and intersecting lines not perpendicular to each other are called oblique lines.

339. When a cord with a weight attached at one end is freely suspended from the other end and is at rest, the line of direction of the cord is a vertical line (generally called. a plumb line).

340. A line perpendicular to a vertical line is called a horizontal line.

341. If two angles have the same vertex, and if the sides of the one prolonged coincide with the sides of the other, they are called vertical angles. Thus, the angles a and b (Fig. 8) are called vertical angles.

342. THEOREM. Two vertical angles are equal. It is required to show that a equals b.

(The sum of the angles about a point in a straight line on the same side of the lines equals 180°.)

Therefore a +c=b+c.

(When two magnitudes are each equal to a third magnitude, they are equal to each other.)

α

C

FIG. 8.

Taking away the angle c from each side of the equality,

343. THEOREM. A perpendicular is the shortest distance from a point to a straight line.

A

It is required to show that the perpendicular CD (Fig. 9) is less than any oblique line CE drawn from C to AB.

C

F

D E

FIG. 9.

B

On AB as an axis turn the figure CDE until DC takes the position of DF, and EC the position of EF; then turn the figure CDE back to its original posi, tion. CF is less than CE+ EF (§ 323).

That is, 2 CD is less than 2CE, or CD is less than CE.

By regarding BA as perpendicular to CF, since EC is equal to EF, and CD equal to DF, it follows that,

344. Two oblique lines drawn from a point in a perpendicular, and cutting off equal distances from the foot of the perpendicular, are equal.

345. If two equal oblique lines are drawn from a point in a perpendicular, they cut off equal distances from the foot of the perpendicular.

346. If a perpendicular is drawn through the middle point of a straight line, every point in the perpendicular is equally distant from the extremities of the straight line.

C

A

О

347. Two points each equidistant from the extremities of a straight line determine the perpendicular to the middle of that line.

FIG. 10.

348. A set-square (Fig. 10) is a piece of wood with three straight edges, two of which contain a right angle A. AB is the base edge, AC the perpendicular edge, and BC the hypotenuse.

349. PROBLEM. At a given point in a straight line to erect a perpendicular to this line, with the aid of a set-square.

Place a ruler so that its edge will coincide with the straight line, and then place the base edge of the square against the ruler, and slide the square along until the point is reached; then draw a line,

keeping the pencil in contact with the perpendicular edge of the square. The line thus drawn is the line required.

350. PROBLEM. To let fall a perpendicular to a given straight line from a point without the line, with the aid of a set-square.

FIG. 11.

Place the ruler and square in the positions described in the last problem, and slide the square along until its perpendicular edge touches the point. Draw a line from the point to the given straight line, keeping the pencil in contact with the perpendicular edge of the square. The line thus drawn is the line required.

351. PROBLEM. At a given point in a given straight line to erect a perpendicular to this

line, with the aid of compasses.
Let AB be the given line,
and P the given point.
It is required to erect a
perpendicular at the point P.

Take the equal distances
PE and PF, and from E and

A

E

Fas centres with the same opening of the two arcs intersecting at some point, as C.

兴

P

FIG. 12.

B

compasses describe

The line drawn

through C, Pis the perpendicular required (§ 347).

352. At a given point in a given line only one perpendicular to the line can be erected.

353. PROBLEM. From a given point to let fall a perpen

dicular to a given straight line, with the aid of compasses.

Let AB (Fig. 13) be the given line, and C the given point.

It is required to let fall a perpendicular from C to the straight line AB.

From C as a centre, with an opening of the compasses sufficiently great, describe an

arc, cutting AB at E and F. From E and Fas centres, with the same opening of the compasses, describe two arcs intersecting at D. A line drawn through the points Cand D to the given straight line is the line required. For C and D, being two points equally distant from E and F, determine the position of the perpendicular to the middle of EF, and consequently perpendicular to the line AB, § 347.

354. From a given point without a straight line only one perpendicular can be let fall to that line.

355. PROBLEM. To divide a straight line into two equal parts, that is, to bisect a straight line.

It is required to bisect AB (Fig. 14). From A and B as centres, with the same opening of the compasses, describe arcs intersecting at C and D. Draw CD. Now C and D, being two points equally distant from A and B, determine the perpendicular to the middle of AB, § 347.

CIRCUMFERENCES.

356. A circumference is a curved line all points of which are equally distant from a point within called the

centre.

357. To trace a circumference, fix upon paper one of the points of the compasses open, and

turn the instrument about the fixed point in such a way that the tracing point does not leave the paper. The line thus formed is called a circumference, since all points are at the same distance from the immovable point which is the centre.

358. The radius is the distance OA from the centre O to any point of the circumference (Fig. 16).

A

FIG. 15.

359. The diameter is a straight line passing through the centre and terminated by the circumference, as BOC. The diameter is double the radius, and divides the circumference into two equal parts.

360. An arc is a portion of the circumference, as EFG.

361. A chord is a straight line which joins the extremities of an

B

F

FIG. 16.

G

arc, as EG. A chord is said to subtend its arc, and an

arc is said to be subtended by its chord.

362. A circle is a portion of the plane which is bounded

by the circumference.

363. A sector is a part of a circle contained by two radii and the intercepted arc, as AOB.