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

complementary.

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

angles as the figure has sides.

COR. All the exterior angles of any convex polygon are together equal to four right angles.

THEOR. 27. The adjoining angles of a parallelogram are supplementary, and the opposite angles are equal.

COR. If one 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. 28. The opposite sides of a parallelogram are equal to one another, and a 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. 29. 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. 30. If a quadrilateral has two opposite sides equal and parallel, it is a parallelogram.

THEOR. 31. 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, and equal straight lines that have equal projections on another straight line make equal angles with that line, or are parallel to it.

THEOR. 32. 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 equal.

COR. I. 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. 2. The straight line joining the middle points of two sides of a triangle is parallel to the base. [Cor. 1. 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.

PROB. 3. To draw a perpendicular to a given straight line from a given point outside it.

PROB. 4. To bisect a given straight line.

PROB. 5. At a given point in a given straight line to make an

angle equal to a given angle.

PROB. 6. To draw a straight line through a given point parallel to a given straight line.

PROB. 7. To construct a triangle having its sides equal to three given straight lines, any two of which are greater than

the third.

PROB. 8. To construct a triangle, having given two sides and the angle between them.

PROB. 9. To construct a triangle, having given two sides and an angle opposite to one of them.

PROB. 10. To construct a triangle, having given two angles and the side that forms their common arm.

PROB. II. To construct a triangle, having given two angles and a side opposite to one of them.

SECTION 5.
Loci.

I. If any and every point on a line or group of lines (straight or curved), and no other point, satisfies an assigned condition, that line or group of lines is called the locus of the point satisfying that condition.

2. In order that a line or group of lines A may be properly termed the locus of a point satisfying an assigned condition X, it is necessary and sufficient to demonstrate the two following associated Theorems :

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If a point is on A, it satisfies X.

If a point is not on A, it does not satisfy X.

It may sometimes be more convenient to demonstrate the contrapositive of either of these Theorems.

i. The locus of a point at a given distance from a given point is the circumference of a circle having a radius equal to the given distance and its centre at the given point.

20

A SYLLABUS OF PLANE GEOMETRY.

3.

ii. The locus of a point at a given distance from a given straight line is the pair of straight lines parallel to the

given line, at the given distance from it and on opposite sides of it.

iii. The locus of a point equidistant from two given points is the straight line that bisects, at right angles, the line joining the given points.

iv. The locus of a point equidistant from two intersecting straight lines is the pair of lines, at right angles to

one another, which bisect the angles made by the given lines.

Intersection of Loci. If A is the locus of a point satisfying the condition X, and B the locus of a point satisfying the condition Y; then the intersections of A and B, and these points only, satisfy both the conditions X and Y.

i. There is one and only one point in a plane which is equidistant from three given points not in the same straight line.

ii. There are four and only four points in a plane each of which is equidistant from three given straight lines that intersect one another but not in the same point.

BOOK II*.

EQUALITY OF Areas.

SECTION 1.
THEOREMS.

DEF. 1. The altitude of a parallelogram with reference to a given side as base. is the perpendicular distance between the base and the opposite side.

DEF. 2. The altitude of a triangle with reference to a given side as base is the perpendicular distance between the base and the opposite vertex.

OBS. It follows from the General Axioms (d) and (e) (page 3), as an extension of the Geometrical Axiom 1 (page 10), that magnitudes which are either the sum or the difference of identically equal magnitudes are equal, although they may not be identically equal.

THEOR. 1. Parallelograms on the same base and between the same parallels are equal.

*

COR. 1. The area of a parallelogram is equal to the area of a rectangle, whose base and altitude are equal to those of the parallelogram.

COR. 2. Parallelograms on equal bases and of equal altitude are equal; and of parallelograms of equal altitudes, that is the greater which has the greater base; and also of parallelograms on equal bases, that is the greater which has the greater altitude.

Book III. (with the exception of its last Section) is independent of Book II., and may be studied immediately after Book I.

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