The proofs or solutions of the following propositions are to be worked out by the student, by the use of those previously demonstrated. In some cases the lines necessary to the construction are drawn, and the propositions which may be needed are referred to. In others, parts of the solution are given. In all cases they are intended to be so clear that a student may, unaided, complete the proof by the exercise of thought and skill. It will often assist him in obtaining a construction to suppose the figure formed, and to note the properties involved; then, by using these properties, he may form the construction.

1. Upon a given straight line to describe an isosceles triangle having each of the equal sides equal to a given straight line.

2. If the angles at the base of an isosceles triangle be bisected, the portions of the bisecting lines between their intersection and the angles bisected, will be equal (I. 14).

3. If a point be taken without a given straight line, the perpendicular is the shortest line that can be drawn from the point to the line (I. 19), (I. 21).

4. The diagonal of a rhombus bisects its angles (I. 9).

5. The four sides of any quadrilateral are together greater than the two diagonals (I. 2).

6. Construct a triangle having given two angles and a side opposite one of them.

7. Why is the problem, to construct a triangle when we have X only the three angles, indefinite?-what ambiguity is there when we have two sides and an angle opposite one of them?

8. A straight line parallel to the base of an isosceles triangle makes equal angles with the sides (I. 27).

9. If a line joining two parallels is bisected, show that any line through the point of bisection and terminating in the parallels is also bisected (I. 27), (I. 5).

10. If AB and CD be parallel and FG = EF, show that any other line GH is bisected at K.

Draw FL parallel to GH, and work in the triangles FLE, GKF.

triangle ABC. B by AD, BD. AC, CB.


11. Trisect a given straight line.

On the line AB construct an equilateral
Bisect the angles at A and
Draw DE, DF parallel to
Prove that AE = EF = FB.

12. Trisect a right angle.

Let ABC be the right angle. On AB construct an equilateral triangle ABD. Bisect the angle ABD by BE. Prove that ABE - EBD-DBC (I. 30).


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13. What is the magnitude of an angle of a regular decagon (I. 30 Cor.)?

14. What is the magnitude of an exterior angle of a regular hexagon (I. 30 Cor.)?

15. If the opposite sides of a quadrilateral be equal, it is a parallelogram.

16. If the opposite angles of a quadrilateral be equal, it is a parallelogram.

17. The four triangles into which a parallelogram is divided by its diagonals, are equal in area.

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18. If the middle points of the three sides of a triangle be joined by straight lines, the four triangles formed will be equal in area.

Draw CD parallel to AB and produce FE to D. Show now the equality of the triangles AEF, DEC. Hence DC-AF FB; .. (I. 31) EF is parallel to BC. Similarly, FG is parallel to AC, and EG to AB. be deduced.



21. The square on a side of a triangle subtending an acute angle, is less than the squares on the other sides.

Let BAC be an acute angle, then BCAC+ AB2.

Make AD AC, and at right angles to AB (I. 8).


Hence the proposition may

19. Given the middle points of the three sides of a triangle, to construct the triangle.

20. To construct a parallelogram, the middle points of the sides of which are the angles of a given parallelogram.


24. Two angles are equal if their sides be parallel, each to each, and lying in the same. direction (I. 27), (Ax. 3).




NOTE.-The angle in a semicircle is a right angle.


22. The square on the side subtending the obtuse angle, is greater than the squares on the other sides.


23. Any side of a triangle is greater than the difference between the other two.

25. To find the side of a square equal in area to two given squares (I. 42).

26. To find the side of a square equal in area to the differ ence between two given squares.

27. If each of the equal angles of an isosceles triangle be double the third angle, then the exterior angle formed by producing one of the equal sides beyond the base is three times the third angle.

28. In the last example how many degrees in the various angles of the figure?

29. If the equal sides of an isosceles triangle be produced beyond the base, the angles on the other side of the base will be equal.

30. If one angle of a parallelogram contain 40°, what is the value of each of the others?

31. If a line be drawn bisecting an angle, any point of it is equally distant from the sides of the angle.

32. The lines bisecting the three angles of a triangle all intersect in the same point.

33. One of the angles of a parallelogram is three halves of a right angle. What are the values of the others in parts of a right angle? in degrees?

34. One of the exterior angles of an equiangular figure is of a right angle. How many sides has the figure?

35. Construct a five-sided figure, four of whose sides are 3, 4, 5, 6, and whose angles in the same order are 1, 3, 1, right angles.

36. Trace back all the references contained in (I. 42) to the fundamental axioms, postulates, and definitions.

37. Construct a triangle equal to a given octagon.




1. Every right-angled parallelogram, or rectangle, is said to be contained by any two of the straight lines which contain one of the right angles.

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By rectangle we mean the figure formed, and not the product of the lines. We will show hereafter that the area of the rectangle is equal to the product of the number of units in the two sides.

2. In every parallelogram the figure formed by either of the parallelograms about the diagonal, together with the two complements, is called a gnomon.

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