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12. What do you understand by the equality of two angles ?
By what geometrical process would you ascertain which was the greater of two angles actually drawn upon a sheet of paper (their vertices not coinciding)?
7. Prove that a triangle cannot have two obtuse angles.
(8) 8. Prove that the two angles which one right line makes with another (on the same side of either line) are either two right angles, or are together equal to two right angles.
(8) 9. Prove that the complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another. (10)
10. ABCD is a trapezium, of which the side AB is parallel to the side CD, and its diagonals AC and BD meet in 0. Prove that the triangles AOD and BOC are equal.
(12) 11. Show how to construct a rectangle (of which one side is given) which shall be equal to a given rectangle, and prove your construction.
(14) 12. The side BC of a triangle ABC is bisected in D. DF is drawn parallel to CA to meet AB in F, and DE is drawn parallel to AB to meet CA in E. Join EF.
Show that the four triangles AEF, BFD, CDE, DEF are equal in all respects.
7. Define parallel straight lines. Show how to draw through a given point a straight line parallel to a given straight line, and prove your construction.
(8) 8. Prove that any two sides of a triangle, taken together, are greater than the third side.
If the distance between the centres of two circles exceeds their radii, taken together, prove that the circles will not meet one another. (8)
9. O is the point of intersection of the diagonals of a square, and through it are drawn any two straight lines at right angles to one another, meeting the sides in E, F, G, H, and the sides produced in P, Q, R, S. Prove that EFGH and PQRS each form a square.
(10) 10. ABC is a triangle right-angled at C, and CD is drawn at right angles to AB. Show that the square on AB exceeds the squares on AD and DB by twice the square on CD.
(10) 11. A frame consists of two equal bars, AB and CD, connected by two other equal bars, AD and BC, which cross one another, and which move freely on the pivots A, B, C, D. Show that the two triangles formed by the frame, in any position, are equal in all respects.
12. Show that, if one side of a triangle be produced, the exterior angle thus formed is equal to the two interior opposite angles.
Given one side of a triangle, one angle adjacent to it, and that the other adjacent angle is three times the third angle, construct the triangle. (14)
1880. 7. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal, show that the angle contained by the two sides of the one triangle is equal to that contained by the two sides of the other.
(16)* 8. If a quadrilateral figure is divided into two isosceles triangles by one diagonal, show that the diagonals are at right angles to each other. (8)
9. Two circles have the same centre C. Two points A, B are taken on one circle, and two points P, Q on the other, so that the angles ACP and BCQ are equal. Show that the triangles ACP, BCQ are equal in all respects.
(10) 10. Prove that triangles on equal bases and between the same parallels are equal.
ABCD is a parallelogram, of which AB and AD are adjacent sides. On AB, AP is cut off equal to one-third of AB, and on AD, AQ is cut off equal to one-third of AD; show that the straight lines joining CP and CQ divide the parallelogram into three equal parts.
(12) 11. Show how to construct a rectangle equal to a given rectilineal figure having five sides.
(13) 12. ABC is an isosceles triangle, whose equal sides are AB and AC; produce AB to D, so that BD may equal BC, and join DC. Show that the angle ACD is three times the angle ADC.
(13) 1881. 7. If three straight lines terminated at a point are such that no one of them falls within the angle made by the other two, show that the three angles just formed are together equal to four right angles.
(8) 8. Prove that the opposite sides of a parallelogram are equal to one another and the diameter divides it into two equal parts.
(9) 9. Show how to describe a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilinear angle, and prove your construction.
(10) 10. In a right-angled triangle, show that the distance of the middle point of the hypotenuse from the right angle is equal to half the hypote
(12) 11. ABC is an isosceles triangle, whose vertex is A, and ACD is an
* This number is given as in the actual examination paper, but it is clearly a misprint for (6).
isosceles triangle, whose vertex is C. Show that, if B, C, and D are in one right line, the angle at B is double the angle at D.
(13) 12. Show that the sum of the perpendiculars let fall from any point within a parallelogram upon the four sides is the same whatever be the point selected.
(13) 1882. 7. Prove that the diagonals of a parallelogram bisect one another, and that, if they intersect at right angles, the parallelogram must be either a square or a rhombus.
(10) 8. If a side of a triangle be produced, the exterior angle thus formed is equal to the two interior and opposite angles.
The sides AB, BC, CA of a triangle are produced so as to form three exterior angles ; if two of these exterior angles are together double of the third, show that one of the angles of the triangle is two-thirds of a right angle.
(10) 9. Let ABC be an isosceles triangle, of which A is the vertex. Draw CF meeting AB produced in F, so that CF=AC, and draw BG meeting AC produced in G, so that BG=AB ; let BG and CF meet in (. Show that the angle BOC is three times the angle at A.
(10) 10. Construct the parallelogram whose diagonals and one side are equal to three given straight lines.
What condition must be fulfilled by the given lines that it may be possible to construct the parallelogram ?
(12) 11. Prove that the complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.
If straight lines be drawn from any point in the diagonal of a parallelogram to the two corners through which the diagonal does not pass, they form two pairs of equal triangles.
(14) 12. Show that the straight line which joins the middle points of two sides of a triangle divides the triangle into parts, of which one is three times as large as the other.
(14) 1883. 7. Draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it.
Let AB be drawn at right angles to CD and meet it in B; join AC, AD; if BC is greater than BD, show that AC is greater than AD. (8)
8. In any triangle (ABC), show that, if the angle A is greater than the angle B, the side BC will be greater than the side CA.
Let ABC be an equilateral triangle, and let a point Q be taken in BC produced in the direction B to C); show that Q is nearer to A than it is to B.
(14) 9. If a straight line falling on two other straight lines makes the interior angles on the same side together equal to two right angles, the two straight lines are parallel to each other.
If A, B, C, D are the angular points of a quadrilateral taken in order, and if the angles at A and B are together equal to those at C and D, show that two sides of the quadrilateral are parallel to one another.
(10) 10. Equal triangles on the same base and on the same side of it are between the same parallels.
If a quadrilateral is divided into four equal triangles by its diagonals, show that it is a parallelogram.
(12) 11. Draw a triangle ABC, and through A draw a line parallel to BC; show how to draw through B a line cutting AC in P and the abovementioned parallel in Q, so that BP shall be one-third of PQ. (14)
12. If one diagonal (AC) of a quadrilateral bisects the other diagonal (BD), show that AC also bisects the quadrilateral itself.
(10) 1884. 7. Two angles of a triangle are equal; show that the sides opposite to these angles are equal.
If the equal sides of an isosceles triangle are produced, and the exterior angles between the base and the sides produced are bisected ; show that the triangle contained by the base and the bisectors is isosceles.
(8) 8. Show how to construct an equilateral triangle, having given three lines severally equal to the distances from its angular points to a point within it.
State the conditions that must be satisfied by the three given lines in order that the construction may be possible.
(16) 9. If two straight lines cut one another, show that the vertically opposite angles are equal to one another.
On the sides of an equilateral triangle three equilateral triangles are drawn external to the given triangle; show that the whole figure is an equilateral triangle.
(10) 10. Show how to draw a straight line through a given point parallel to a given straight line.
Draw a line DE parallel to the base BC of a triangle ABC, cutting AB in D and AC in E, so that DE shall be equal to the sum of BD and CE.
(14) 11. Show that parallelograms on the same base and between the same parallels are equal.
Show how to construct an isosceles triangle equal to a given triangle.
(10) 12. ABC is a triangle right-angled at C; draw CD at right angles to AB; in CD take CE equal to BD, and in CA take CF equal to AD; join AE and BF; and show that AE and BF are equal.
Alternative proofs which are interesting for one reason or another are numerous. A short series of them is here introduced, comprising those only which have a distinct advantage over the ordinary proofs both in point of simplicity and of brevity, and which also fulfil the conditions of any of the ordinary examinations.
A triangle; pt. point; str. straight; adj. adjacent; ext". exterior;
1. SYMBOLS AND ABBREVIATIONS.
= is or are equal to; || parallel ;
I perpendicular to, or at
right angles to; sq. square;
om parallelogram ; circle ;
ce circumference ; rt. right;
equil. equilateral; opp. opposite;
intr. interior; < less than ;
> greater than.
2. PROPOSITION V.-THEOREM. The angles at the base of an isosceles triangle are equal to one another, and, if the equal sides be produced, the angles on the other side of the base are equal.
Let the AABC be isosceles, having AB, AC equal. And let AB, AC be produced to D and E.
Then LABC = LACB, and 2 DBC = ZECB.