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ON QUADRILATERALS.

122. DEF. A Quadrilateral is a plane figure bounded by four straight lines.

123. DEF. A Trapezium is a quadrilateral which has no two sides parallel.

124. DEF. A Trapezoid is a quadrilateral which has two sides parallel.

125. DEF. A Parallelogram is a quadrilateral which has its opposite sides parallel.

TRAPEZIUM,

TRAPEZOID.

PARALLELOGRAM.

126. DEF. A Rectangle is a parallelogram which has its angles right angles.

127. DEF. A Square is a parallelogram which has its angles right angles, and its sides equal.

128. DEF. A Rhombus is a parallelogram which has its sides equal, but its angles oblique angles.

129. DEF. A Rhomboid is a parallelogram which has its angles oblique angles.

The figure marked parallelogram is also a rhomboid.

RECTANGLE.

SQUARE.

RHOMBUS

130. DEF. The side upon which a parallelogram stands, and the opposite side, are called its lower and upper bases; and the parallel sides of a trapezoid are called its bases.

131. DEF. The Altitude of a parallelogram or trapezoid is the perpendicular distance between its bases.

132. DEF. The Diagonal of a quadrilateral is a straight line joining any two opposite vertices.

PROPOSITION XXXVIII. THEOREM.

133. The diagonal of a parallelogram divides the figure into two equal triangles.

[blocks in formation]

Let ABC E be a parallelogram, and AC its diagonal.

[blocks in formation]

..A ABC=A A EC,

$ 107 (having a side and two adj. As of the one equal respectively to a side and two

adj. & of the other).

Q. E. D.

[blocks in formation]

134. In a parallelogram the opposite sides are equal, and the opposite angles are equal.

B

A

E
Let the figure ABC E be a parallelogram.

We are to prove BC= A E, and A B = EC,

also, ZB= Z E, and Z BA E= ZBCE.

Draw A C.

§ 133

A ABC=A A EC,
(the diagonal of a o divides the figure into two equal A).

... B C = AE,

and

AB=CE
(being homologous sides of equal $).

LB=LE,
(being homologous é of equal A ).

LBAC=LACE,

and

LEAC= LACB,

(being homologous & of equal A). Add these last two equalities, and we have

ZBAC + LEAC= LACE + LACB; or,

ZBA E = LBC E.

Q. E. D.

135. COROLLARY. Parallel lines comprehended between parallel lines are equal.

[blocks in formation]

136. If a quadrilateral have two sides equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram.

[blocks in formation]

Let the figure A B CE be a quadrilateral, having the

side A E equal and parallel to BC.

[blocks in formation]

§ 68

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Draw A C.
In the A A B C and A EC
BC= A E

Нур.
AC = AC,

Iden.
LBCA=LCAE,

(being alt.-int. E).
A ABC
ДА СЕ,

$ 106 (having two sides and the included 2 of the one equal respectively to two sides

and the included 2 of the other).

.. A B

(being homologous sides of equal A).
Also,

ZBAC=LACE,
(being homologous & of equal A);
... A B is || to EC,

$ 69 (when two straight lines are cut by a third straight line, if the alt.-int. Is be

equal the lines are parallel).
.. the figure A B C E is a O,

$ 125 (the opposite sides being parallel).

EC,

Q. E. D.

[blocks in formation]

137. If in a quadrilateral the opposite sides be equal, the figure is a parallelogram.

[blocks in formation]

Let the figure A B C E be a quadrilateral having

BC= A E and AB= EC.

[blocks in formation]

AC=AC,

Iden. ..A ABC=A AEC, (having three sides of the one equal respectively to three sides of the other).

.. ZACB=LCAE,

§ 108

LACE,

and

Z BAC
(being homologous ts of equal A ).

.. B C is || to A E,
and
A B is || to EC,

$ 69 (when two straight lines lying in the same plane are cut by a third straight

line, if the alt.-int. Es be equal, the lines are parallel).

§ 125

.. the figure A B C E is a o,
(having its opposite sides parallel).

Q. E. D.

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