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side MI, opposite to the equal angle d, in the triangle MPI. In precisely the same manner I can prove that RS is equal to MI, and consequently equal to OP; and so I might go on, and shew that every perpendicular, dropped from the line AB to the parallel line CD, is equal to RS, MI, OP, &c. The two parallel lines AB and CD will therefore, throughout, be at an equal distance from each other; and the same can be proved of other parallel lines.
If two lines are parallel to a third line, what relation must they bear to each other ?
A. They must be parallel to each other.
A. From the line CD being parallel to AB, it follows that every point in the line CD must be at an equal distance from the line AB; and because EF is also parallel to AB, every point in the line EF must also be at an equal distance from the line AB; and therefore (in Fig. I.) the whole distance between the lines CD and
EF, or (in Fig. II.) the difference between the equal distances, must be equal ; that is, the lines CD, EF, must likewise be equidistant; and consequently, parallel to each other.
What is the sum of all the angles in every triangle equal to ?
A. To two right angles.
Q. How do you prove this?
A. By drawing, through the vertex of the angle B/d b, a straight line parallel to the basis BC, the angle a will be equal to the angle d, and the angle c to the angle e (query 11 ); and as the sum of the three angles a, b, c, is equal to two right angles (query 4), the sum of the three angles d, b, e, in the triangle, must also be equal to two right angles.
Q. Can you now find out the relation, which the exterior angle e, bears to the two interior angles a and b?
* The teacher may give his pupils an ocular demonstratian of this truth, by cutting the three angles b, d, e, from a triangle, and then placing them along side of each other they will be in a straight line.
angle e, will be equal to the sum of the two interior angles, a and b.
Q. How do you prove this?
A. Because, by adding the angle c to the two angles a and b, it will make with them two right angles; and by adding it to the angle e alone, the sum of the two angles, e and c, will also be equal to two right angles, which could not be, if the angle e were not equal to the two angles a and b together. (Truth 2.)
Q. What other truths can you derive from the two which you have just now advanced ?
A. 1. That the exterior angle e is greater than either of the interior opposite ones, a orb.
2. If two angles of a triangle are known, the third angle is also determined.
3. When two angles of a triangle are equal to two angles of another triangle, the third angle in the one, will be equal to the third angle in the other.
4. No triangle can contain more than one right angle.
5. No triangle can contain more than one obtuse angle.
6. In a right-angular triangle, the right angle is equal to the sum of the two other angles.
Q. How can you convince me of the truth of each of these assertions?
RECAPITULATION OF THE TRUTHS CONTAINED IN
THE FIRST SECTION.
Can you now repeat the different principles of straight lines and angles which you have learned in this section.
Ans. 1. Two straight lines can cut each other only in one point.
2. Two straight lines, which have two points common, must coincide with each other throughout.
3. The sum of the two adjacent angles, which one straight line makes with another, is equal to two right angles.
4. The sum of all the angles, made by any number of straight lines meeting in the same point, and on the same side of a straight line, is equal to two right angles.
5. Opposite angles at the vertex are equal.
6. The sum of all the angles, made by the meeting of ever so many straight lines around the same point, is equal to four right angles.
7. When a triangle has one side and the two adjacent angles, equal to one side and the two adjacent angles in another triangle, the two triangles must be equal.
8. In equal triangles the equal angles are opposite to the equal sides.
9. If two straight lines are perpendicular to a third line, they must be parallel.
10. If one of two parallel lines is perpendicular to a third line, the other line must also be perpendicular to it.
11. If two lines are cut by a third line at equal angles, or so as to make the alternate angles equal, or so that the sum of the two interior angles, formed by the intersection of a third line, is equal to two' right angles, the two lines must be parallel.
12. If two parallel lines are cut by a third line, the alternate angles are equal.
13. Parallel lines are throughout equidistant.
14. If two lines are parallel to a third line, they must be parallel to one another.
15. The sum of the three angles in a triangle is equal to two right angles.
16. If one of the sides of a triangle is extended, the exterior angle is equal to the sum of the two interior opposite angles.
17. The exterior angle is greater than any one of the interior opposite angles.
18. If two angles of a triangle are given, the third is determined.
19. There can be but one right angle, or one obtuse angle, but never a right angle and obtuse angle, in the same triangle.
20. In a right-angular triangle, the right angle is equal to the sum of the two other angles.*
* The teacher may now ask his pupils to repeat the demonstrations of these principles.