177. COROLLARY 1. A straight line perpendicular to one of

two parallel lines is perpendicular to the other also.

178. COROLLARY 2. If two

179. COROLLARY 3. If two angles have their sides respec

tively parallel, they are equal or supplementary.

The following diagrams illustrate various relative positions of such angles.

EXERCISES

1. Give ten concrete illustrations of parallels. 2. In the figure, m || n, and one of the exterior angles is 30°. Find the values of the other angles.

3. Howmany sets of numerically different angles does the figure (Ex.2) contain?

n

4. If the exterior angles (Ex. 2) are in the ratio 1:2 (2:3;4:5), determine their values.

5. If the ratio of the angles in Ex. 4 is 1:1, what is the position

of the transversal with reference to the parallels?

6. Let the figure represent two intersecting streets. How many numerically different angles are there?

7. If one of the streets extends in an east-and-west direction, and the ratio of the angles is 1:3, what is the direction of the other street?

W

E

8. The T-square is an instrument constantly used in mechanical

drawing (see illustration). By its

means a series of parallel lines

may be drawn at any desired

intervals. Upon what theorem of

parallels does its use depend?

9. Copy the following Greek designs, called meanders. What

method of constructing parallels is used?

GGGG

10. Which laws of parallels are illustrated by the following

capital letters: Z, N, H, E, F, W ?

11. In Fig. 1, 11 ||12; prove that ∠c = 2a + ∠b.

12. In Fig. 2, 11 || 12; prove that Za + b + ∠ c = 2 st. &.

13. If 11, 12, etc. are different lines, construct the following figures:

In (1) what is the position of l2 with reference to lo? in (2)-(6)

what are the relative positions of the first line and the last line?

14. Construct three equilateral triangles in

the positions shown in the figure. Prove that

b || a. What other lines are parallel?

a

b

15. The following is a practical method of drawing through a given point a line parallel to a given line. Perform the construction as indicated, and prove that it leads to the desired result.

Given the line BC and the point A. Required to draw through A a line parallel to BC.

With A as a center and a radius sufficiently great describe a circle cutting the line BC in the point D. Draw AD. On BC lay off DE equal to AD, and in the circle draw a chord DF equal to the distance A E, so that DF and AE lie on the same side of BC, but on opposite sides of AD. Draw AF and produce it if necessary. AF is parallel to BC.

16. In the accompanying figure there are four lines which are drawn parallel to the diagonal of the square. Do they appear to be parallel to the diagonal and to one another?

ANGLE-SUM

PROPOSITION XVIII. THEOREM

180. The sum of the interior angles of a triangle equals a straight angle.

Given the triangle ABC, whose angles are m, n, and o.

181. COROLLARY 1. An exterior angle of a triangle is equal to the sum of the two remote interior angles, and is therefore greater than either of them.

182. COROLLARY 2. If two angles of one triangle are equal respectively to two angles of another triangle, the third angles are equal.

183. COROLLARY 3. Two triangles are congruent if a side and any two angles of one are equal respectively to a side and two angles, similarly situated, of the other.

184. COROLLARY 4. The sum of two angles of a triangle is less than a straight angle.

185. COROLLARY 5. If a triangle has one right angle or one obtuse angle, the other angles are acute.

186. COROLLARY 6. Every triangle has at least two acute

angles.

187. COROLLARY 7. The sum of the two acute angles of a right triangle equals a right angle.

188. COROLLARY 8. Each angle of an equilateral triangle is an angle of 60°.

189. COROLLARY 9. If one leg of a right triangle is half the hypotenuse, then the angle opposite that leg is an angle of 30°, and the other acute angle is an angle of 60°.

190. COROLLARY 10. If a right triangle contains acute angles of 30° and 60° respectively, then the leg opposite the angle of 30° is one half the hypotenuse.

191. COROLLARY 11. If the legs of a right triangle are equal, each acute angle is an angle of 45°.

192. COROLLARY 12. If one acute angle of a right triangle is an angle of 45°, then the other acute angle is also an angle of 45°, and the legs of the right triangle are equal.

193. COROLLARY 13. Each base angle of an isosceles triangle is half the supplement of the vertex angle.

EXERCISES

1. In mechanical drawing constant use is made of certain fixed right triangles which are made of wood or celluloid. One of these has acute angles of 30° and 60°, while the other has acute angles of 45° each. What angles can a draftsman draw with the aid of these two triangles, but without bisecting any angle?

2. Show how one of the above tri

angles, combined either with the other triangle or with the T-square (see Ex. 8, p. 100), may be used to draw a series of parallels.