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Miscellaneous Exercises on Sections I. and II. 1. If two exterior angles of a triangle be bisected by straight lines which meet in 0; prove that the perpendiculars from 0 on the sides, or the sides produced, of the triangle are equal.
2. Trisect a right angle.
3. The bisectors of the three angles of a triangle meet in one point.
4. The perpendiculars to the three sides of a triangle drawn from the middle points of the sides meet in one point.
6. The angle between the bisector of the angle BAC of the triangle ABC and the perpendicular from A on BC, is equal to half the difference between the angles at B and C.
6. If the straight line AD bisect the angle at A of the triangle ABC, and BDE be drawn perpendicular to AD, and meeting AC, or AC produced, in E; shew that BD is equal to DE.
7. Divide a right-angled triangle into two isosceles triangles.
8. AB, CD are two given straight lines. Through a point E between them draw a straight line GEH, such that the in. tercepted portion GH shall be bisected in E.
9. The vertical angle O of a triangle OPQ is a right, acute, or obtuse angle, according as OR, the line bisecting PQ, is equal to, greater or less than the half of PQ.
10. Shew by means of Ex. 9 how to draw a perpendicular to a given straight line from its extremity without producing it.
On the Equality of Rectilinear Figures in respect of Area.
The amount of space enclosed by a Figure is called the Area of that figure.
Euclid calls two figures equal when they enclose the same amount of space. They may be dissimilar in shape, but if the areas contained within the boundaries of the figures be the same, then he calls the figures equal. He regards a triangle, for example, as a figure having sides and angles and area, and he proves
in this section that two triangles may have equality of area, though the sides and angles of each may be unequal.
Coincidence of their boundaries is a test of the equality of all geometrical magnitudes, as we explained in Note 1,
In the case of lines and angles it is the only test : in the case of figures it is a test, but not the only test ; as we shall shew in this Section.
The sign =, standing between the symbols denoting two figures, must be read is equal in area to.
Before we proceed to prove the Propositions included in this Section, we must complete the list of Definitions required in Book I., continuing the numbers prefixed to the definitions in page 6.
XXVII. A PARALLELOGRAM is a four-sided figure whose opposite sides are parallel.
For brevity we often designate a parallelogram by two letters only, which mark opposite angles. Thus we call the figure in the margin the parallelogram AC.
XXVIII. A Rectangle is a parallelogram, having one of its angles a right angle.
Hence by I. 29, all the angles of a rectangle are right angles.
XXIX. A RHOMBUS is a parallelogram, having its sides equal.
XXX. A SQUARE is a parallelogram, having its sides equal and one of its angles à right angle.
Hence, by I. 29, all the angles of a square are right angles.
XXXI. A TRAPEZIUM is a four-sided figure of which two sides only are parallel.
XXXII. A DIAGONAL of a four-sided figure is the straight line joining two of the opposite angular points.
XXXIII. The ALTITUDE of a Parallelogram is the perpendicular distance of one of its sides from the side opposite, regarded as the Base.
The altitude of a triangle is the perpendicular distance of one of its angular points from the side opposite, regarded as the base.
Thus if ABCD be a parallelogram, and AE a perpendicular let fall from A to CD, AE is the altitude of the parallelogram, and also of the triangle ACD.
If a perpendicular be let fall from B to DC produced, meeting DC in F, BF is the altitude of the parallelogram.
EXERCISES. Prove the following theorems :
1. The diagonals of a square make with each of the sides an angle equal to half a right angle.
2. If two straight lines bisect each other, the lines joining their extremities will form a parallelogram.
3. Straight lines bisecting two adjacent angles of a parallelogram intersect at right angles.
4. If the straight lines joining two opposite angular points of a parallelogram bisect the angles, the parallelogram is a rhombus.
5. If the opposite angles of a quadrilateral be equal, the quadrilateral is a parallelogram.
6. If two opposite sides of a quadrilateral figure be equal to one another, and the two remaining sides be also equal to one another, the figure is a parallelogram.
7. If one angle of a rhombus be equal to two-thirds of two right angles, the diagonal drawn from that angular point divides the rhombus into two equilateral triangles.
PROPOSITION XXXIV. THEOREM.
The opposite sides and angles of a parallelogram are equal to one another, and the diagonal bisects it.
Let ABDC be a O, and BC a diagonal of the O.
Then must AB=DC and AC=DB, and
= CDB, and LABD= ACD and
A ABC= ADCB. For :: AB is || to CD, and BC meets them, ...ABC=alternate 2 DCB ;
I. 29. and :: AC is || to BD, and BC meets them, ..LACB=alternate . DBC.
I. 29. Then in As ABC, DCB,
•ABC= . DCB, and ACB= 2 DBC, and BC is common, a side adjacent to the equal 2 s in each ; .. AB=DC, and AC=DB, and 2 BAC= 2 CDB, and A ABC= ADCB.
I. B. Also :::L ABC= _ DCB, and 2 DBC= . ACB, .: 28 ABC, DBC together= 28 DCB, ACB together, LABD=LACD.
Q. E. D. Ex. 1. Shew that the diagonals of a parallelogram bisect each other.
Ex. 2. Shew that the diagonals of a rectangle are equal.