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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 □, and BC a diagonal of the .
▲ BAC= ▲ CDB, and ▲ ABD= 4 ACD
For : AB is || to CD, and BC meets them,
.. ABC alternate ▲ DCB ;
and AC is to BD, and BC meets them,
** 2 ABC= 2 DCB, and ▲ ACB= 2 DBC,
and BC is common, a side adjacent to the equal ▲ s in each; .. AB=DC, and AC=DB, and ▲ BAC= 2 CDB,
and ▲ ABC= ▲ DCB.
ABC= ▲ DCB, and 4 DBC= ▲ ACB,
.. 48 ABC, DBC togethers DCB, ACB together,
L ABD= L ACD.
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.
Parallelograms on the same base and between the same parallels are equal.
Let the Os ABCD, EBCF be on the same base BC. and between the same s AF, BC.
Then must ☐ ABCD=□ EBCF.
CASE I. If AD, EF have no point common to both,
ABCD with ▲ FDC=□ EBCF with ▲ EAB;
:. ABCD=□ EBCF.
CASE II. If the sides AD, EF overlap one another,
the same method of proof applies.
CASE III. If the sides opposite to BC be terminated in the same point D,
the same method of proof is applicable,
Each of the Os is double of ▲ BDC;
::□ ABCD=□ DBCF.
Q. E. D.
Parallelograms on equal bases, and between the parallels, are equal to one another.
Let the Os ABCD, EFGH be on equal bases BC, FG, and between the same Is AH, BG.
Then must ☐ ABCD=☐ EFGH.
...they are on the same base BC and between the same ||s;
they are on the same base EH and between the same ||s; ..ABCDEFGH.
Q. E. D.
PROPOSITION XXXVII. THEOREM.
Triangles upon the same base, and between the same parallels, are equal to one another.
Let As ABC, DBC be on the same base BC and between the same s AD, BC.
Then must ▲ ABC= ▲ DBC.
From B draw BE to CA to meet DA produced in E.
Then EBCA and FCBD are parallelograms,
they are on the same base and between the same ||s.
Ex. 1. If P be a point in a side AB of a parallelogram ABCD, and PC, PD be joined, the triangles PAD, PBC are together equal to the triangle PDC.
Ex. 2. Two straight lines AB, CD intersect in E, and the triangle AEC is equal to the triangle BED. Shew that BC is parallel to AD.
Ex. 3. If A, B be points in one, and C, D points in another of two parallel straight lines, and the lines AD, BC intersect in E, then the triangles AEC, BED are equal.
PROPOSITION XXXVIII. THEOREM.
Triangles upon equal bases, and between the same parallels, are equal to one another.
Let As ABC, DEF be on equal bases, BC, EF, and between the same ||s BF, AD.
Then must A ABC= ▲ DEF.
From B draw BG || to CA to meet DA produced in G. From F draw FH || to ED to meet AD produced in H. Then CG and EH are parallelograms, and they are equal, they are on equal bases BC, EF, and between the same Is BF, GH.
: AABC= ADEF.
Q. E. D.
Ex. 1. Shew that a straight line, drawn from the vertex of a triangle to bisect the base, divides the triangle into two equal parts.
Ex. 2. If the triangles in the Proposition are not towards the same parts, shew that the straight line, joining the vertices of the triangles, is bisected by the line containing the bases.
Ex. 3. In the equal sides AB, AC of an isosceles triangle ABC points D, E are taken such that BD=AE. Shew that the triangles CBD, ABE are equal.