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and b, it makes with them two right angles; and by adding it to the angle e alone, the sum of the two angles, e and c, is also equal to two right angles (Query 3), which could not be, if the angle e alone were not equal to the two angles a and b together. (Truth III.)

Q. What other truths can you derive from the two which you have just now advanced?

A. 1. The exterior angle e is greater than either of the interior opposite ones, a or b.

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 is 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. No triangle can contain a right and an obtuse angle together.

7. In a right-angled 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, and form but one and the same straight line.

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, each to each, the two triangles are 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 are parallel to each other.

10. If two lines are cut by a third line at equal angles, or so as to make the alternate angles equal, or so as to make the sum of the two interior angles formed by the intersection of a third line, equal to two right angles, the two lines are parallel.

11. If two lines are cut by a third line at unequal angles; or so as to have the alternate angles unequal; or in such a way as to make the sum of the two interior angles less than two right angles, these two lines will, when sufficiently extended, cut each other.

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 are parallel to each other.

15. The sum of the three angles in any 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 either of the interior opposite ones.

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, and never a right angle and obtuse angle together, in the same triangle.

20. In a right-angled 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.

SECTION II.

OF EQUALITY AND SIMILARITY OF TRIANGLES.

PART I.

OF THE EQUALITY OF TRIANGLES.

Preliminary Remark. There are three kinds of equality to be considered in triangles, viz: equality of area, without reference to the shape; equality of shape, without reference to the area-similarity; and equality of both shape and area-coincidence. All questions, asked in this section, will refer only to the last two kinds of equality; and those in the first part, only to the coincidence of triangles.

QUERY I.

If two sides and the angle which is included by them in one triangle, are equal to two sides and the angle which is included by them in another triangle, each to each, what relation do these two triangles bear to each other?

Ans. They are equal to each other in all their parts, that is, they coincide with each other throughout.

Show me that this must be the case with any two triangles, ABC, abc, in which we will suppose the side AB=ab, AC=ac, and the angle at A equal to the angle

at a

A

B a

b

A. By placing the line ac upon its equal AC, the angle at a will coincide with the angle at A, because these two angies are equal; and the line ab will fall upon the line AB; and as ab= AB, the point b will fall upon B; that is, the three points of the triangle abc will fall upon the three points of the triangle ABC, thus:

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consequently these two triangles must coincide.

Q. What remark can you here make with respect to the sides and angles of equal triangles?

A. The equal sides, cb, CB, are opposite to the equal angles at a and A.

QUERY II.

If one side and the two adjacent angles in one triangle, are equal to one side and the two adjacent angles in another triangle, each to each, what relation do the two triangles bear to each other?

A. They are equal, and the angles opposite to the equal sides are also equal, as has been proved in the 1st Section. (Query 6.)

QUERY III.

What remark can you make with respect to the two angles at the basis of an isosceles triangle?

A. They are equal to each other.

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