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· Then because BA, AE, BE are respectively parallel to DC, CF, DF,

(Hyp.) therefore the angle BAE = the angle DCF, and the angle BEA = the angle DFC, (Th. 27. Cor. 2.)

and the hypotenuse AB = the hypotenuse CD; (Hyp.) therefore AE = CF:

(Th. 17.) but AE = ab, and CF=cd,

(Th. 28.) and therefore ab = cd.

Q. E. D. Similarly the converse propositions may be proved.

THEOREM 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.

Let the three parallel straight lines AD, BE, CF make equal intercepts on the straight line AC, that is, let AB=BC.

Then shall the intercepts on any other line DEF be equal, that is, DE shall be equal to EF.

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4. The perpendiculars let fall from the extremities of the base of an isosceles triangle on the opposite sides will include an angle supplementary to the vertical angle of the triangle.

5. Shew that the angles of an equiangular triangle are equal to two-thirds of a right angle. 6. Find the magnitude of the angle of a regular octagon.

(Th. 26.) 7. How many equiangular triangles can be placed so as to have one common angular point, and fill up the space round it?

8. Shew that three regular hexagons can be placed so as to have a common point, and fill up the space round that point.

9. Shew that two regular octagons and one square have the same property.

Draw a pattern consisting of octagons and squares.

10. Shew that the angle of a regular pentagon is to the angle of a regular decagon as 3 to 4.

II. If a line is perpendicular to another it will be perpendicular to every line parallel to it.

12. If a polygon is equilateral, does it follow that it is equiangular, and conversely?

13. How many diagonals can be drawn in a pentagon ? How many in a decagon? How many in a polygon of n sides.

14. Shew that a square, a hexagon and a dodecagon will fill up the space round a point; and make a pattern of these polygons.

15. Examine whether a square, a pentagon and an icosagon have the same property; and also whether a pattern can be constructed of pentagons and decagons.

16. The exterior angle of a regular polygon is one-third of a right angle: find the number of sides in the polygon,

17. Two lines intersecting in A are respectively perpendicular to two lines intersecting in B: prove that any angle at A is equal or supplementary to any angle at B.

18. Shew that a trapezium may be divided into a parallelogram and a triangle.

19. The diagonals of any parallelogram will bisect one another.

20. The diagonals of a rhombus will bisect one another at right angles.

21. If two straight lines be drawn bisecting one another, and their extremities be joined, the figure so formed will be a parallelogram.

22. Given that a four-sided figure has its opposite sides equal, prove that it must be a parallelogram.

23. Prove that the diagonals of a rectangle are equal to one another.

24. Shew that if one element (a side) is given, a square is determined; if two elements (a side and angle), a rhombus is determined ; also that if two elements (two sides) are given, a rectangle is determined : and find the number of elements required to determine a parallelogram, a trapezium, a quadrilateral, a pentagon, and a polygon of any number (n) of sides.

QUESTIONS ON SECTION III.

1. Give the derivation of the words parallel, parallelogram, trapezium.

2. What is indirectly ascertained in Theorem 21? Would it be possible to ascertain it directly?

3. Prove Th. 24 by drawing a straight line to intersect A, B and X, and using Theorems 21, 22.

4. Given two angles of a triangle to be respectively 72o. 15:47" and 83o. 51'. 16", find the third angle.

5. If one angle of a triangle is equal to the other two, prove that it must be a right angle.

6. Isosceles triangles having equal vertical angles must have equal base angles.

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