COR. Not more than two equal straight lines can be drawn from a given point to a given straight line.

THEOR. 16. If two triangles have two sides of the one equal to two sides of the other, each to each, but the included angles unequal, then the bases are unequal, the base of that which has the greater angle being greater than the base of the other.

THEOR. 17. If two triangles have two sides of the one equal to two sides of the other, each to each, but the bases unequal, then the included angles are unequal, the angle of that which has the greater base being

greater than the angle of the other. [By Rule of Conversion.]

THEOR. 18. If two triangles have the three sides of the one equal to the three sides of the other, each to each, then the triangles are identically equal, and of the angles those are equal which are opposite to equal sides. [Alternative proofs, (i) by Theors. 16 and 5. (ii) By Theors. 7 and 5.]

THEOR. 19. If two triangles have two angles of the one equal to two angles of the other, each to each, and have likewise the sides opposite to one pair of equal

angles equal, then the triangles are identically equal, and of the sides those are equal which are opposite to equal angles. [By Superposition and Theor. 9.]

THEOR. 20. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise the angles opposite to one of the equal sides in each equal, then the angles opposite to the other two equal sides are either equal or supplementary, and in

1

the former case the triangles are identically equal,

[By Superposition.]

COR. Two such triangles are identically equal

(1) If the two angles given equal are right angles o obtuse angles.

(2) If the angles opposite to the other two equal sides are both acute, or both obtuse, or if one of them is a right angle.

(3) If the side opposite the given angle in each triangle is not less than the other given side.

SECTION 3.

PARALLELS AND PARALLELOGRAMS.

DEF. 35. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet.

DEF. 36. A trapezium is a quadrilateral that has only one pair of opposite sides parallel.

This figure is sometimes called a trapezoid.

DEF. 37. A parallelogram is a quadrilateral whose opposite sides are parallel.

DEF. 38. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to one another.

Thus in the figure 1, 2, 7, 8

are called exterior angles,

and 3, 4, 5, 6, interior

angles; again, 4 and 6, 3

and 5 are called alternate

2/1

3/4

angles; lastly, I and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles.

DEF. 39. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted

between perpendiculars let fall on it from the extremities of the former.

THEOR. 21. If a straight line intersects two other straight lines and makes the alternate angles equal, the straight

lines are parallel. [Contrapositive of Theor. 9.]

THEOR. 22. If a straight line intersects two other straight lines and makes either a pair of alternate angles equal, or a pair of corresponding angles equal, or a pair of interior angles on the same side supplementary; then, in each case, the two pairs of alternate angles are equal, and the four pairs of corresponding angles are equal, and the two pairs of interior angles on the same side are supplementary. COR. If a straight line intersects two other straight lines

and makes a pair of corresponding angles equal, or

a pair of interior angles on the same side supplementary, the straight lines are parallel.

AXIOM 3. Through the same point there cannot be more than one straight line parallel to a given straight line.

THEOR. 23. If two straight lines are parallel, and are intersected by a third straight line, the alternate angles are equal. [By Rule of Identity, using Ax. 3.]

COR. I. If a straight line intersects two parallel straight lines, and is perpendicular to one of them, it is also perpendicular to the other.

COR. 2. If a straight line intersects two parallel straight lines, it makes the corresponding angles equal, and the interior angles on the same side supplementary.

THEOR. 24. Straight lines that are parallel to the same straight line are parallel to one another. [Contrapositive of Ax. 3.]

THEOR. 25. If a side of a triangle is produced, the exterior angle is equal to the two interior opposite angles; and the

three interior angles of a triangle are together equal to two right angles.

COR. In a right-angled triangle the two acute angles are complementary.

THEOR. 26. All the interior angles of any convex polygon together with four right angles are equal to twice as many right angles as the polygon has sides.

THEOR. 27. The exterior angles of any convex polygon made by producing the sides in order are together equal to four right angles.

THEOR. 28. The adjoining angles of a parallelogram are supplementary, and the opposite angles are equal.

COR. If ore of the angles of a parallelogram is a right angle, all its angles are right angles.

DEF. 40. The figure is then called a rectangle.

THEOR. 29. The opposite sides of a parallelogram are equal to one another, and each diagonal divides it into two

identically equal triangles.

COR. If the adjoining sides of a parallelogram are equal,

all its sides are equal.

DEF. 41. The figure is then called a rhombus.

DEF. 42. A square is a rectangle that has all its sides equal. THEOR. 30. If two parallelograms have two adjoining sides of the one respectively equal to two adjoining sides of the other, and likewise an angle of the one equal to an angle of the other; the parallelograms are identically equal. [By Superposition.]

COR. Two rectangles are equal, if two adjoining sides of the one are respectively equal to two adjoining sides

of the other; and two squares are equal, if a side of

the one is equal to a side of the other.

THEOR. 31. If a quadrilateral has two opposite sides equal and parallel, it is a parallelogram.

THEOR. 32. Straight lines that are equal and parallel have equal projections on any other straight line; conversely,

parallel straight lines that have equal projections on another straight line are equal.

THEOR. 33. Equal straight lines that have equal projections on another straight line are parallel to that line, or make equal angles with it.

THEOR. 34. If there are two pairs of straight lines all of which are parallel, and the intercepts made by each pair on

a straight line that cuts them are equal, then the intercepts on any other straight line that cuts them are also equal.

COR. I. If there are three parallel straight lines, and the intercepts made by them on any straight line that

cuts them are equal, then the intercepts on any other straight line that cuts them are also equal.

COR. 2. The straight line drawn through the middle point of one of the sides of a triangle parallel to the base passes through the middle point of the other side. COR. 3. The straight line joining the middle points of two sides of a triangle is parallel to the base. [Cor. 2. and Rule of Identity.]

SECTION 4.

PROBLEMS.

PROB. I. To bisect a given angle.

PROB. 2. To draw a perpendicular to a given straight line from

a given point in it.