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culty. They are also often arranged in pairs, so that the solution of any Paper marked with an even number will be found easy after working the preceding Paper.

RUPERT DEAKIN.

KING EDWARD'S SCHOOL, STOURBRIDGE, February, 1891.

RIDER PAPERS.

PART I.

TO EUCLID I. 12.

(In Papers I. to VI. “Draw" means Draw without Proof.")

66

I.

1. Draw an isosceles triangle having each of the sides double of the base.

2. Draw a right-angled isosceles triangle, an obtuseangled isosceles triangle and an acute-angled isosceles triangle.

3. Take a straight line MABN divided into three equal parts at A and B. With centre A and radius AN describe the circle NCD. With centre B and radius BM describe the circle MCD, cutting the former circle in the points C and D. Join CA, CB, DA, DB. Prove that BC=AC, and AD=BD. What kind of figure is CADB?

4. In the figure of Question 3 join CD, and prove that the angle ACD is equal to the angle BCD.

5. In the figure of Prop. 2, let the given point A be on the circumference of the smaller circle.

Draw

Where does the point

6. Which of the twelve Axioms apply to magnitudes of all kinds, and which apply to geometrical magnitudes only?

the complete figure in this case. D come?

II.

1. Explain Proposition, Enunciation, Data and Quæsita.

2. Draw a right-angled scalene triangle, an obtuseangled scalene triangle and an acute-angled scalene triangle.

3. Take a straight line AB, and produce it both ways to M and N. Make MB-AN. Show how to describe an isosceles triangle on AB, having its two sides CA and CB each equal to MB or AN.

4. ABC is an isosceles triangle, having the side AB equal to the side AC, and the angle BAC is bisected by the straight line AD. Prove that AD also bisects the base BC.

5. In the figure of Prop. 2, let the given point A be joined to C instead of to B. Draw the complete figure in this case.

6. In Prop. 9, why is the equilateral triangle DEF described on the side remote from A? Draw figures to illustrate your answer.

III.

1. Explain Problem, Theorem, Q.E.D. and Q.E.F.

2. There are seven kinds of triangle. Draw one triangle of each kind and give its name.

3. In Prop. 9, prove that AF bisects the angle DFE.

4. ABC is an isosceles triangle, having the side AB equal to the side AC, and the angle BAC is bisected by the straight line AD. Prove that AD is perpendicular to the base BC.

5. The straight line AB is bisected at the point C; and from C the straight line CD is drawn at right angles to AB. In CD take any point E, and join AE and BE. Prove that AE=BE.

6. Write out all the Definitions, Axioms and Postulates that Euclid employs in Prop. 2.

IV.

1. What is a Postulate? Euclid assumes a fourth Postulate in Prop. 4. What is it?

2. Explain, with examples, Reductio ad Absurdum, Converse and Corollary.

3. Draw a quadrilateral figure ABCD, having its opposite sides equal, viz. AB to CD and AD to BC. Join BD. Prove that the angle BAD is equal to the angle BCD.

4. ABC is an isosceles triangle, having AB equal to AC. The angle ABC is bisected by the straight line BD, and the angle ACB by the straight line CD. Prove that DB=DC.

5. In the figure of Prop. 10, take any point E in CA; and from CB cut off CF equal to CE. Join DE and DF. Prove that DE=DF.

6. Show by drawing triangles that two triangles may have all their angles equal, each to each, and yet not be equal in area.

V.

1. What is an Axiom? Why is the twelfth Axiom objectionable?

2. In the figure of Prop. 5, if FC and BG meet in H, prove that HB=HC.

3. ABCD is a rhombus. Join AC. angle BAC is equal to the angle DAC.

Prove that the

4. ABC and DBC are two isosceles triangles on the same base BC, on the same side of BC, the vertex A being within the triangle DBC. Join AD. Prove that the angle BDA is equal to the angle CDA.

5. A and B are two given points, and CD is a given line not passing through either A or B. Join AB, and bisect it at E. Through E draw EF at right angles to AB, and meeting CD in F. Join AF and BF. Prove that AF BF.

6. Take a right angle BAC, and draw a complete figure showing how it may be divided into four equal parts, as in Prop. 9.

VI.

1. Why is the given line of unlimited length in Prop. 12?

2. In the figure of Prop. 5, if FC and BG meet in H, prove that FH=GH.

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