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EXERCISES ON PARALLELOGRAMS.
1. If one angle of a parallelogram is a right angle, all its angles are right angles.
2. If the opposite sides of a quadrilateral are equal, the figure is a parallelogram.
3. If the opposite angles of a quadrilateral are equal, the figure is a parallelogram.
4. If a quadrilateral has all its sides equal and one angle a right angle, all its angles are right angles.
5. The diagonals of a parallelogram bisect each other.
6. If the diagonals of a quadrilateral bisect each other, the figure is a parallelogram.
7. If two opposite angles of a parallelogram are bisected by the diagonal which joins them, the figure is equilateral.
8. If the diagonals of a parallelogram are equal, all its angles are right angles.
9. In a parallelogram which is not rectangular the diagonals are unequal.
10. Any straight line drawn through the middle point of a diagonal of a parallelogram and terminated by a pair of opposite sides, is bisected at that point.
11. If two parallelograms have two adjacent sides of one equal to two adjacent sides of the other, each to each, and one angle of one equal to one angle of the other, the parallelograms are equal in all respects.
12. Two rectangles are equal if two adjacent sides of one are equal to two adjacent sides of the other, each to each.
13. In a parallelogram the perpendiculars drawn from one pair of opposite angles to the diagonal which joins the other pair are equal.
14. If ABCD is a parallelogram, and X, Y respectively the middle points of the sides AD, BC; shew that the figure AYCX is a parallelogram.
MISCELLANEOUS EXERCISES ON SECTIONS I. AND II.
MISCELLANEOUS EXERCISES ON SECTIONS I. AND II.
1. Shew that the construction in Proposition 2 may generally be performed in eight different ways. Point out the exceptional case.
2. The bisectors of two vertically opposite angles are in the same straight line.
3. In the figure of Proposition 16, if AF is joined, shew (i) that AF is equal to BC;
(ii) that the triangle ABC is equal to the triangle CFA in all respects.
4. ABC is a triangle right-angled at B, and BC is produced to D: shew that the angle ACĎ is obtuse.
5. Shew that in any regular polygon of n sides each angle contains 2(n-2)
6. The angle contained by the bisectors of the angles at the base of any triangle is equal to the vertical angle together with half the sum of the base angles.
7. The angle contained by the bisectors of two exterior angles of any triangle is equal to half the sum of the two corresponding interior angles.
8. If perpendiculars are drawn to two intersecting straight lines from any point between them, shew that the bisector of the angle between the perpendiculars is parallel to (or coincident with) the bisector of the angle between the given straight lines.
9. If two points P, Q be taken in the equal sides of an isosceles triangle ABC, so that BP is equal to CQ, shew that PQ is parallel to BC.
10. ABC and DEF are two triangles, such that AB, BC are equal and parallel to DE, EF, each to each ; shew that AC is equal and parallel to DF.
11. Prove the second Corollary to Prop. 32 by drawing through any angular point lines parallel to all the sides.
12. If two sides of a quadrilateral are parallel, and the remaining two sides equal but not parallel, shew that the opposite angles are supplementary ; also that the diagonals are equal.
THE AREAS OF PARALLELOGRAMS AND TRIANGLES.
Hitherto when two figures have been said to be equal, it has been implied that they are identically equal, that is, equal in all respects.
But figures may be equal in area without being equal in all respects, that is, without having the same shape.
The present section deals with parallelograms and triangles which are equal in area but not necessarily identically equal.
[The ultimate test of equality, as we have already seen, is afforded by Axiom 8, which asserts that magnitudes which may be made to coincide with one another are equal. Now figures which are not equal in all respects, cannot be made to coincide without first undergoing some change of form : hence the method of direct superposition is unsuited to the purposes of the present section.
We shall see however from Euclid's proof of Proposition 35, that two figures which are not identically equal, may nevertheless be so related to a third figure, that it is possible to infer the equality of their areas. ]
DEFINITIONS. 1. The Altitude of a parallelogram with reference to a given side as base, is the perpendicular distance between the base and the opposite side.
2. The Altitude of a triangle with reference to a given side as base, is the perpendicular distance of the opposite vertex from the base.
[From this point the following symbols will be introduced into the text:
= for is equal to ; :. for therefore. If it is thought desirable to shorten written work by the use of symbols and abbreviations, it is strongly recommended that only some well recognized system should be allowed, such, for example, as that given on page 11.]
PROPOSITION 35. THEOREM. Parallelograms on the same base, and between the same parallels, are equal in area. DE F A DE
A E DF
Let the parallelograms ABCD, EBCF be on the same base BC, and between the same parallels BC, AF.
Then shall the parallelogram ABCD be equal in area to the parallelogrum EBCF.
CASE I. If the sides AD, EF, opposite to the base BC, are terminated at the same point D: then each of the parallelograms ABCD, EBCF is double of the triangle BDC;
.. they are equal to one another. Ax. 6. CASE II. But if the sides AD, EF are not terminated at the same point:
then because ABCD is a parallelogram,
Ax. 1. ... the whole, or remainder, EA= the whole, or remainder, FD.
Then in the triangles FDC, EAB,
Proved. and the side DC = the opposite side AB,
I. 34. Because also the exterior angle FDC
= the interior opposite angle EAB,
I. 29. .. the triangle FDC = the triangle EAB. From the whole figure ABCF take the triangle FDC ; and from the same figure take the equal triangle EAB;
then the remainders are equal. Ax. 3. Therefore the parallelogram ABCD is equal to the parallelo
THEOREM. Parallelograms on equal bases, and between the same parallels, are equal in area. A
Let ABCD, EFGH be parallelograms on equal bases BC, FG, and between the same parallels AH, BG. Then shall the parallelogram ABCD be equal to the parallelogram EFGH. Construction. Join BE, CH. Proof. Then because BC= FG;
Hyp. and the side FG=the opposite side EH;
I. 34. .. BC=EH:
Ax. 1. and BC is parallel to EH;
Hyp. BE and CH are also equal and parallel.
I. 33. Therefore EBCH is a parallelogram.
Def. 36. Now the parallelograms ABCD, EBCH are on the same base BC, and between the same parallels BC, AH; .. the parallelogram ABCD= the parallelogram EBCH. I. 35.
Also the parallelograms EFGH, EBCH are on the same base EH, and between the same parallels EH, BG ; .. the parallelogram EFGH = the parallelogram EBCH. I. 35. Therefore the parallelogram ABCD is equal to the parallelogram EFGH.
Ax. 1. Q.E.D.
From the last two Propositions we infer that:
(i) A parallelogram is equal in area to a rectangle of equal base and equal altitude.
(ii) Parallelograms on equal bases and of equal altitudes are equal in area.