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134. The diagonals of a parallelogram bisect each other. Let ABCD be a, and AD, B C, its diagonals.

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135. COR.-The diagonals of a rhombus bisect each other

=

Q. E. D.

at right angles.

THEOREM XLV.

136. The diagonals of a rectangle are equal.

Let ABCD be a rectangle, and AD, B C, its diagonals.

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137. COR.-The diagonals of a square are equal and bisect each other at right angles.

THEOREM XLVI.

138. If a parallel to the base of a triangle bisects one of the sides, it bisects the other also; and the part of it intercepted between the sides equals half the base.

In the ABC, let DE be II to the base AB and bisect AC at D.

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139. COR.-The straight line which joins the middle points of two sides of a triangle is parallel to the third side and is equal to half that side.

THEOREM XLVII.

140. The parallel to the bases of a trapezoid, bisecting one of the non-parallel sides, bisects the other also.

Let ABCD be a trapezoid, AB and DC its bases, E the middle point of AD, and let EF be to AB and DC.

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Then in the ▲ DBC, BC is bisected at F;

EF bisects BC.

(138)

(138)

Q.E. D.

141. COR. The straight line joining the middle points of the non-parallel sides of a trapezoid, is parallel to the base and equals half their sum.

THEOREM XLVIII.

142. The straight line drawn from the vertex of a right angle of a right-angled triangle to the middle of the hypothenuse is equal to half the hypothenuse.

Let the RA ABC be right-angled at B, and let BD be drawn to the middle of AC.

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