2. Euclid I. 1.

3. Euclid I. 6.

4. Euclid I. 9.

If D and E are points on AB and AC equidistant from A, show that the bisector of the angle bisects DE and is at right angles to it.

5. Euclid I. 13.

Two straight lines AC, AD are drawn from A in the line BAE, on one side of it; then the angles BAC, CAD, DAE together equal two right angles.

6. Euclid I. 18.

7. Either, Euclid I. 22.

Show how the construction would fail if two of the lines were not together greater than the third. [Illustrate your answer by a figure.]

Or, Euclid I. 26, Case 1.

111.

College of Preceptors, Midsummer 1890.

Second Class.

1. Define a point and a straight line.

Euclid I. 2. How would you proceed further to draw a second equal straight line from the given point in a given direction?

If the given point lies on the smaller of the two circles (in Euclid's construction), show that the vertex of the equilateral triangle employed lies also on this circle.

2. Define a triangle, and classify triangles according to the equality or inequality of their sides.

Prove in any way Euclid I. 8.

Hence show that, if the opposite sides of a quadrilateral are equal, the opposite angles are equal.

3. When is a straight line said to be at right angles to a given straight line?

Euclid I. 12.

4. What is meant by "the exterior angle of a triangle formed by producing a side of the triangle"? How many such angles are there in a triangle?

Euclid I. 16. Is an exterior angle of a triangle greater or less than the adjacent interior angle?

5. Can we form a triangle with any three given lengths? Enunciate the Proposition on which you ground your answer.

The sum of the sides of a convex four-sided figure is greater than the sum of its two diagonals. Prove this.

6. Euclid I. 26, Case 1.

7. Euclid I. 38. What is meant by equal in the enunciation ?

ABCD is a square; BC, CD are bisected in E, F, respectively; and AE, EF, AF are drawn. Prove that AAEF is three-eighths of the square.

8. Euclid I. 42.

9. ABC is a right-angled triangle, A the right angle; squares BDEC, ABFG are described externally on BC, BA respectively; and AL is drawn perpendicular to DE to meet it in L. Prove that the rectangle BL equals the square AF. Prove also that AD is perpendicular to FC.

10. (i.) ABC is an equilateral triangle; on BC is described the square BDEC, and on DE the equilateral triangle DEF. Prove that EF is parallel to AB.

Or, (ii.) Bisect a parallelogram by a straight line

drawn through a given point in the plane of the

parallelogram.

IV.

College of Preceptors, Christmas 1890.

Second Class.

1. Define line, obtuse angle, rhombus.

Name as many different kinds of triangles as you can, with a picture of each.

2. Show how, with a plain ruler and a pair of compasses, you can produce a straight line, so as to be three times its original length.

3. PQ is a straight line, and R a point. From R draw a straight line equal to PQ.

Write out the Postulates and Axioms used in the construction.

4. Either, Euclid I. 8.

How many parts, at least, of one triangle must be equal to the corresponding parts of another triangle, so that the triangles may be equal in every respect? Draw figures to illustrate your answer.

Or, Euclid I. 13.

If one of the four angles which two intersecting straight lines make with one another, be a right angle, all the others are right angles.

5. Euclid I. 19.

Prove that the hypothenuse of a right-angled triangle is greater than either of the other sides.

6. Define parallel straight lines. Write down any Axiom you have learned bearing on the doctrine of parallels.

Prove Euclid I. 27 (after the method of superposition by preference).

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

7. Euclid I. 39.

The sides AB, AC of a triangle ABC are bisected at the points E and F. Prove that EF is parallel to BC. Thence show that if a perpendicular is drawn from A to the opposite side meeting it at D, the angle FDE is equal to the angle BAC. Also show that the figure AFDE is equal to half the triangle ABC.

8. On the base of an equilateral triangle, construct an oblong (or rectangle) equal in area to the triangle.

V.

College of Preceptors, Midsummer 1890.

First Class.

1. Define line, superficies, polygon, proposition, hypothesis.

Euclid I. 5.

Prove that a triangle is isosceles, if the bisector of any angle is perpendicular to the opposite side.

2. Euclid I. 14.

3. Euclid I. 21, Part I.

The four sides of any quadrilateral figure are together greater than the two diagonals together.

4. Show that any angle of a triangle is obtuse, right, or acute, according as it is greater than, equal to, or less than the other two angles of the triangle taken together. Construct an isosceles triangle which shall have the vertical angle four times each of the angles at the base. 5. Euclid II. 6.

(Questions 6, 7, 8, 9 and 10 were set on Books III. and IV.)

VI.

College of Preceptors, Christmas 1890.

1. Euclid I. 6.

2. Euclid I. 20.

First Class.

ABCD is a square: for what position of a point X is the sum of the straight lines XA, XB, XC and XD the least possible? Prove your answer.

3. Euclid I. 32.

The angle contained by one side of a regular polygon and an adjacent side produced is equal to half an angle at the base of an isosceles right-angled triangle. How many sides has the polygon? Explain how you get your result.

4. Euclid I. 48.

5. Euclid II. 5.

Enunciate this Proposition as one about (i.) the rectangle contained by two lines, (ii.) the rectangle contained by the sum and difference of two lines.

6. Divide a straight line AB at the point C, so that the rectangle contained by AB, BC may be equal to the square on AC.

Produce AB to D, making BD equal to BC; then the square on AD is equal to five times the square on AC.

(Questions 7, 8, and 9 were set on Books III. and IV.)

VII.

Oxford Local Examinations, July 1889.

Junior Candidates.

1. Define a circle, an obtuse-angled triangle, parallel straight lines.