Book I.  Propositions 35 and 36

Problems

Back to Propositions 35 and 36.

1.   What does it mean to say that two figures --two triangles, two
1.   parallelograms, etc. -- are equal.

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It means that the space enclosed by each boundary
is exactly the same.  See the Introduction.

2.   a)  State the hypothesis of Proposition 35.

Parallelograms are on the same base and in the same parallels.

2.  b)  State the conclusion.

Those parallelograms are equal.

2.  c)  Practice Proposition 35.

3.   Prove Proposition 35 for the case where EB does not intersect DC.

3.   Prove parallelograms ABCD, EBCF equal.

3.   (Hint:   Triangles EAB, FDC are again equal.)

Triangles EAB, FDC are again equal for the same reasons.
To each of those triangles add the quadrilateral EBCD.
Therefore the whole parallelogram ABCD is equal to the whole parallelogram EBCF.

4.   Prove that if the base of a parallelogram is doubled, then its area is
4.   also doubled.

4.   (Hint:  I. 36.)

5.   Prove that if the height of a parallelogram (that is, the distance
4.   between the parallels) is doubled, then its area is also doubled.

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