138. The diagonals of a parallelogram bisect each other.

Let the figure ABCE be a parallelogram, and let the diagonals AC and BE cut each other at 0.

(having a side and two adj. of the one equal respectively to a side and two

PROPOSITION XLIII. THEOREM.

139. The diagonals of a rhombus bisect each other at right angles.

Let the figure A B C E be a rhombus, having the diagonals AC and BE bisecting each other at 0.

(having three sides of the one equal respectively to three sides of the other);

ZAOE

=

LAOB,

(being homologous & of equal ▲);

.LAOE and ZA OB are rt. .

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(When one straight line meets another straight line so as to make the adj.

equal, each is a rt. 4).

Q. E. D.

PROPOSITION XLIV. THEOREM.

140. Two parallelograms, having two sides and the included angle of the one equal respectively to two sides and the included angle of the other, are equal in all respects.

In the parallelograms A B C D and A'B'C' D', let ABA'B', A D = A' D', and

We are to prove that the [] are equal.

ALA'.

Apply A B C D to □ A'B'C' D', so that AD will fall on and coincide with A' D'.

Then A B will fall on A' B',

(for LA=L A', by hyp.),

and the point B will fall on B',
(for A B = A' B', by hyp.).

Now, BC and B'C' are both to A' D' and are drawn through point B';

.. the lines B C and B'C' coincide,

and C falls on B'C' or B'C' produced.

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In like manner D C and D'C' are to A'B' and are drawn

through the point D'.

.. DC and D'C' coincide;

.. the point C falls on D' C', or D' C' produced ;

.. C falls on both B'C' and D' C';

.. C must fall on a point common to both, namely, C'.
.. the two coincide, and are equal in all respects.

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Q. E. D.

141. COROLLARY. Two rectangles having the same base and altitude are equal; for they may be applied to each other and will coincide.

PROPOSITION XLV. THEOREM.

142. The straight line which connects the middle points of the non-parallel sides of a trapezoid is parallel to the parallel sides, and is equal to half their sum.

Let SO be the straight line joining the middle points of the non-parallel sides of the trapezoid ABCE.

( lines comprehended between || lines are equal);

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Ax. 7.

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Substitute for AF and BH their equals, A E – F E and BC+CH,