Chapter VI.

LISTS OF PROPOSITIONS IN PLANE AND SOLID GEOMETRY.

General basis of the selection of material.-The subcommittee appointed to prepare a list of basal propositions made a careful study of a number of widely used textbooks on geometry. The bases of selection of the propositions were two: (1) The extent to which the propositions and corollaries were used in subsequent proofs of important propositions and exercises; (2) the value of the propositions in completing important pieces of theory. Although the list of theorems and problems is substantially the same in nearly all textbooks in general use in this country, the wording, the sequence, and the methods of proof vary to such an extent as to render difficult a definite statement as to the number of times a proposition is used in the several books examined. A tentative table showed, however, less variation than might have been anticipated.

Classification of propositions. The classification of propositions is not the same in plane geometry as in solid geometry. This is partly due to the fact that it is generally felt that the student should limit his construction work to figures in a plane and in which the compasses and straight edge are sufficient. The propositions have been divided as follows:

Plane geometry: I. Assumptions and theorems for informal treatment; II. Fundamental theorems and constructions: A. Theorems, B. Constructions; III. Subsidiary theorems.

Solid geometry: I. Fundamental theorems; II. Fundamental propositions in mensuration; III. Subsidiary theorems; IV. Subsidiary propositions in mensuration.

PLANE GEOMETRY.

I. Assumptions and theorems for informal treatment. This list contains propositions which may be assumed without proof (postulates), and theorems which it is permissible to treat informally. Some of these propositions will appear as definitions in certain methods of treatment. Moreover, teachers should feel free to require formal proofs in certain cases, if they desire to do so. The precise wording given is not essential, nor is the order in which the propositions are here listed. The list should be taken as representative of

the type of propositions which may be assumed, or treated informally, rather than as exhaustive.

1. Through two distinct points it is possible to draw one straight line, and only one. 2. A line segment may be produced to any desired length.

3. The shortest path between two points is the line segment joining them.

4. One and only one perpendicular can be drawn through a given point to a given straight line.

5. The shortest distance from a point to a line is the perpendicular distance from the point to the line.

6. From a given center and with a given radius one and only one circle can be described in a plane.

7. A straight line intersects a circle in at most two points.

8. Any figure may be moved from one place to another without changing its shape or size.

9. All right angles are equal,

10. If the sum of two adjacent angles equals a straight angle, their exterior sides form a straight line.

11. Equal angles have equal complements and equal supplements.

12. Vertical angles are equal.

13. Two lines perpendicular to the same line are parallel.

14. Through a given point not on a given straight line, one straight line, and only one, can be drawn parallel to the given line.

15. Two lines parallel to the same line are parallel to each other.

16. The area of a rectangle is equal to its base times its altitude.

II. Fundamental theorems and constructions.-It is recommended that theorems and constructions (other than originals) to be proved on college entrance examinations be chosen from the following list. Originals and other exercises should be capable of solution by direct reference to one or more of these propositions and constructions. It should be obvious that any course in geometry that is capable of giving adequate training must include considerable additional material. The order here given is not intended to signify anything as to the order of presentation. It should be clearly understood that certain of the statements contain two or more theorems, and that the precise wording is not essential. The committee favors entire freedom in statement and sequence.

A. THEOREMS.

1. Two triangles are congruent if (a) two sides and the included angle of one are equal, respectively, to two sides and the included angle of the other; (b) two angles and a side of one are equal, respectively, to two angles and the corresponding side of the other; (c) the three sides of one are equal, respectively, to the three sides of the other.

2. Two right triangles are congruent if the hypotenuse and one other side of one are equal, respectively, to the hypotenuse and another side of the other.

3. If two sides of a triangle are equal, the angles opposite these sides are equal; and conversely.2

4. The locus of a point (in a plane) equidistant from two given points is the perpendicular bisector of the line segment joining them.

1 Teachers should feel free to separate this theorem into three distinct theorems and to use other phraseology for any such proposition. For example, in 1, "Two triangles are equal if" * * * "a triangle is determined by * * *," etc. Similarly in 2, the statement might read: "Two right triangles are congruent if, beside the right angles, any two parts (not both angles) in the one are equal to corresponding parts of the other."

2 It should be understood that the converse of a theorem need not be treated in connection with the theorem itself, it being sometimes better to treat it later. Furthermore a converse may occasionally be accepted as true in an elementary course, if the necessity for proof is made clear. The proof may then bo given later.

5. The locus of a point equidistant from two given intersecting lines is the pair of lines bisecting the angles formed by these lines.

6. When a transversal cuts two parallel lines, the alternate interior angles are equal; and conversely.

7. The sum of the angles of a triangle is two right angles.

8. A parallelogram is divided into congruent triangles by either diagonal.

9. Any (convex) quadrilateral is a parallelogram (a) if the opposite sides are equal; (b) if two sides are equal and parallel.

10. If a series of parallel lines cut off equal segments on one transversal, they cut off equal segments on any transversal.

11. (a) The area of a parallelogram is equal to the base times the altitude.

(b) The area of a triangle is equal to one-half the base times the altitude.

(c) The area of a trapezoid is equal to half the sum of its bases times its altitude. (₫) The area of a regular polygon is equal to half the product of its apothem and perimeter.

12. (a) If a straight line is drawn through two sides of a triangle parallel to the third side, it divides these sides proportionally.

(b) If a line divides two sides of a triangle proportionally, it is parallel to the third side. (Proofs for commensurable cases only.)

(c) The segments cut off on two transversals by a series of parallels are proportional. 13. Two triangles are similar if (a) they have two angles of one equal, respectively, to two angles of the other; (b) they have an angle of one equal to an angle of the other and the including sides are proportional; (c) their sides are respectively proportional, 14. If two chords intersect in a circle, the product of the segments of one is equal to the product of the segments of the other.

15. The perimeters of two similar polygons have the same ratio as any two corresponding sides.

16. Polygons are similar, if they can be decomposed into triangles which are similar and similarly placed; and conversely.

17. The bisector of an (interior or exterior) angle of a triangle divides the opposite side (produced if necessary) into segments proportional to the adjacent sides.

18. The areas of two similar triangles (or polygons) are to each other as the squares of any two corresponding sides.

19. In any right triangle the perpendicular from the vertex of the right angle on the hypotenuse divides the triangle into two triangles each similar to the given triangle.

20. In a right triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

21. In the same circle, or in equal circles, if two arcs are equal, their central angles are equal; and conversely.

22. In any circle angles at the center are proportional to their intercepted arcs. (Proof for commensurable case only.)

23. In the same circle or in equal circles, if two chords are equal their corresponding arcs are equal; and conversely.

24. (a) A diameter perpendicular to a chord bisects the chord and the arcs of the chord. (b) A diameter which bisects a chord (that is not a diameter) is perpendicular to it.

25. The tangent to a circle at a given point is perpendicular to the radius at that point; and conversely.

26. In the same circle or in equal circles, equal chords are equally distant from the center; and conversely.

27. An angle inscribed in a circle is equal to half the central angle having the

same arc.

28. Angles inscribed in the same segment are equal.

29. If a circle is divided into equal arcs, the chords of these arcs form a regular inscribed polygon and tangents at the points of division form a regular circumscribed polygon.

30. The circumference of a circle is equal to 2πr. (Informal proof only.) 31.3 The area of a circle is equal to r2. (Informal proof only.)

The treatment of the mensuration of the circle should be based upon related theorems concerning regular polygons, but it should be informal as to the limiting processes involved. The aim should be an understanding of the concepts involved, so far as the capacity of the pupil permits.

* The total number of theorems given in this list when separated, as will probably be found advantageous in teaching this number including the converses indicated, is 52.

B. CONSTRUCTIONS.

1. Bisect a line segment and draw the perpendicular bisector.

2. Bisect an angle.

3. Construct a perpendicular to a given line through a given point.

4. Construct an angle equal to a given angle.

5. Through a given point draw a straight line parallel to a given straight line. 6. Construct a triangle, given (a) the three sides; (b) two sides and the included angle; (c) two angles and the included side.

7. Divide a line segment into parts proportional to given segments.

8. Given an arc of a circle, find its center.

9. Circumscribe a circle about a triangle.

10. Inscribe a circle in a triangle.

11. Construct a tangent to a circle through a given point.

12. Construct the fourth proportional to three given line segments.

13. Construct the mean proportional between two given line segments. 14. Construct a triangle (polygon) similar to a given triangle (polygon). 15. Construct a triangle equal to a given polygon.

16. Inscribe a square in a circle.

17. Inscribe a regular hexagon in a circle.

III. Subsidiary list of propositions.-The following list of propositions is intended to suggest some of the additional material referred to in the introductory paragraph of Section II. It is not intended, however, to be exhaustive; indeed, the committee feels that teachers should be allowed considerable freedom in the selection of such additional material, theorems, corollaries, originals, exercises, etc., in the hope that opportunity will thus be afforded for constructive work in the development of courses in geometry.

1. When two lines are cut by a transversal, if the corresponding angles are equal, or if the interior angles on the same side of the transversal are supplementary, the lines are parallel.

2. When a transversal cuts two parallel lines, the corresponding angles are equal, and the interior angles on the same side of the transversal are supplementary. 3. A line perpendicular to one of two parallels is perpendicular to the other also. 4. If two angles have their sides respectively parallel or respectively perpendicular to each other, they are either equal or supplementary.

5. Any exterior angle of a triangle is equal to the sum of the two opposite interior angles.

6. The sum of the angles of a convex polygon of n sides is 2 (n-2) right angles. 7. In any parallelogram (a) the opposite sides are equal; (b) the opposite angles are equal; (c) the diagonals bisect each other.

8. Any (convex) quadrilateral is a parallelogram, if (a) the opposite angles are equal; (b) the diagonals bisect each other.

9. The medians of a triangle intersect in a point which is two-thirds of the distance from the vertex to the mid-point of the opposite side.

10. The altitudes of a triangle meet in a point.

11. The perpendicular bisectors of the sides of a triangle meet in a point.

12. The bisectors of the angles of a triangle meet in a point.

13. The tangents to a circle from an external point are equal.

14.4

(a) If two sides of a triangle are unequal, the greater side has the greater angle opposite it, and conversely.

(b) If two sides of one triangle are equal respectively to two sides of another triangle, but the included angle of the first is greater than the included angle of the second, then the third side of the first is greater than the third side of the second, and conversely.

(c) If two chords are unequal, the greater is at the less distance from the center, and conversely.

4 Such inequality theorems as these are of importance in developing the notion of dependence or functionality in geometry. The fact that they are placed in the "Subsidiary list of propositions" should not imply that they are considered of less educational value than those in List II. They are placed here because they are not "fundamental” in the same sense that the theorems of List II are fundamental.

(d) The greater of two minor arcs has the greater chord, and conversely. 15. An angle inscribed in a semicircle is a right angle.

16. Parallel lines tangent to or cutting a circle intercept equal arcs on the circle. 17. An angle formed by a tangent and a chord of a circle is measured by half the

intercepted arc.

18. An angle formed by two intersecting chords is measured by half the sum of the intercepted arcs.

19. An angle formed by two secants or by two tangents to a circle is measured by half the difference between the intercepted arcs.

20. If from a point without circle a secant and a tangent are drawn, the tangent is the mean proportional between the whole secant and its external segment. 21. Parallelograms or triangles of equal bases and altitudes are equal.

22. The perimeters of two regular polygons of the same number of sides are to each other as their radii and also as their apothems.

SOLID GEOMETRY.

In the following list the precise wording and the sequence are

not considered:

I. FUNDAMENTAL THEOREMS.

1. If two planes meet, they intersect in a straight line.

2. If a line is perpendicular to each of two intersecting lines at their point of intersection it is perpendicular to the plane of the two lines.

3. Every perpendicular to a given line at a given point lies in a plane perpendicular to the given line at the given point.

4. Through a given point (internal or external) there can pass one and only one perpendicular to a plane.

5. Two lines perpendicular to the same plane are parallel.

6. If two lines are parallel, every plane containing one of the lines and only one is parallel to the other.

7. Two planes perpendicular to the same line are parallel.

8. If two parallel planes are cut by a third plane, the lines of intersection are parallel. 9. If two angles not in the same plane have their sides respectively parallel in the same sense, they are equal and their planes are parallel.

10. If two planes are perpendicular to each other, a line drawn in one of them perpendicular to their intersection is perpendicular to the other.

11. If a line is perpendicular to a given plane, every plane which contains this line is perpendicular to the given plane.

12. If two intersecting planes are each perpendicular to a third plane, their intersection is also perpendicular to that plane.

13. The sections of a prism made by parallel planes cutting all the lateral edges are congruent polygons.

14. An oblique prism is equal to a right prism whose base is equal to a right section of the oblique prism and whose altitude is equal to a lateral edge of the oblique prism. 15. The opposite faces of a parallelopiped are congruent.

16. The plane passed through two diagonally opposite edges of a parallelopiped divides the parallelopiped into two equal triangular prisms.

17. If a pyramid or a cone is cut by a plane parallel to the base:

(a) The lateral edges and the altitude are divided proportionally;

(b) The section is a figure similar to the base;

(c) The area of the section is to the area of the base as the square of the distance from the vertex is to the square of the altitude of the pyramid or cone.

18. Two triangular pyramids having equal bases and equal altitudes are equal. 19. All points on a circle of a sphere are equidistant from either pole of the circle. 20. On any sphere a point which is at a quadrant's distance from each of two other points not the extremities of a diameter is a pole of the great circle passing through these two points.

21. If a plane is perpendicular to a radius at its extremity on a sphere, it is tangent to the sphere.

22. A sphere can be inscribed in or circumscribed about any tetrahedron.

23. If one spherical triangle is the polar of another, then reciprocally the second is the polar triangle of the first.

24. In two polar triangles each angle of either is the supplement of the opposite side of the other.

25. Two symmetric spherical triangles are equal.