2. Those which lie on opposite sides of both lines are called Vertical Angles. Thus, a and b, lying on opposite sides of A B and on opposite sides of CD, are vertical; likewise, c and d.

Note.-Let the student see to it that he is able to give, in every instance, not only the number of the reference, but the reference itself, and that he understands clearly its use or application. It is all-important to his successful and pleasant study of Geometry that he should start well.

THEOREM II.

If two straight lines intersect, the vertical angles will be equal.

Therefore, if two straight lines intersect, the vertical angles will be equal.

Cor. I. If two straight lines intersect, the sum of the four angles formed about the point of intersection will be equal to four right angles.

Cor. II. The sum of all the angles that can be formed about a point is equal to four right angles.

D

h

リ

B

1. A Plane Surface, or simply a Plane, is a surface such that a straight line which joins any two of its points will lie wholly in the surface. Thus, it is evident that the straight line joining the points C and D will lie wholly in the surface of the paper- no part of it will pass through or pierce the paper.

C.

A

B

E

D

3. The bounding lines are called sides of the polygon. Thus, A B, BC, CD, DE, AE, are called sides of the polygon ABCDE.

4. The points in which the sides meet are called vertices of the polygon; as, A, B, C, etc.

5. A Plane Triangle is a portion of a plane surface bounded by three straight lines; that is, a plane triangle is a polygon of three sides.

6. Triangles are classified, with reference to their angles, as

2d. An Acute Triangle is one whose angles are all acute-each less than a right angle, as B.

B

A

3d. An Obtuse Triangle is one

which has one obtuse angle—an angle

greater than a right angle, as C.

C

7. Triangles are classified, with reference to their sides, as scalene, isosceles and equilateral.

3d. An Equilateral Triangle is one having all of its sides equal, as F.

8. An Equiangular Triangle is one hav

F

ing all of its angles equal.

9. Polygons, and hence triangles, are mutually equilateral when their sides, taken in the same order, are equal each to each.

They are mutually equiangular when their angles, taken in the same order, are equal each to each.

10. The Hypotenuse of a right triangle is the side opposite the right angle.

B

с

11. The Base of a triangle is the side on which it is made to stand as, A B. A

12. The angle opposite the base is called the Vertical Angle, as c.

A

B

13. A Broken Line is a line consisting of two or more straight lines, as A B C, consisting of the lines A B and B C.

Plane Geometry, therefore, may be defined as the science which treats of plane figures—figures whose points are all in the same plane.

THEOREM III.

Any side of a triangle is less than the sum of the other two, and greater than their difference.

A

B

Let A B C be any triangle; then:

First: Any side will be less

than the sum of the other two.

on the

For, the direct distance from A to C, measured straight line A C, is less than the distance between the same points measured on the broken line A B C

Ax. 8.

Second: Any side will be greater than the difference between

the other two.