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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.

В

Let ABC E be a parallelogram, and A C its diagonal.

We are to prove A ABC=A A EC.
In the A A B C and A EC
AC=AC,

Iden.
LACB= 2 CAE,

(being alt.-int. 4).
Z CAB=LACE,

§ 68 .. A ABC=A AEC,

§ 107 (having a side and two adj. As of the one equal respectively to a side and two adj. s of the other).

Q. E. D.

§ 68 PROPOSITION XXXIX. THEOREM.

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

Let the figure A B C E be a parallelogram.

and

We are to prove BC= A E, and A B = EC,
also, ZB= E, and Z BAE= LBCE.

Draw A C.
A ABC = A A EC,

§ 133 (the diagonal of a divides the figure into two equal A).

..BC=A E

AB=CE,
(being homologous sides of equal o ).

ZB=LE,
(being homologous & of equal A).

Z BAC = LACE,
and

Z EAC = L ACB,

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

Z BAC + < EAC = LACE + ZACB;
or,
ZBA E = LBCE.

Q. E. D. 135. COROLLARY. Parallel lines comprehended between parallel lines are equal.

PROPOSITION XL. THEOREM.

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.

Hyp.

Let the figure A BCE be a quadrilateral, having the

side A E equal and parallel to BC.
We are to prove A B equal and Il to E C.

Draw A C.
In the A A B C and A EC

BC = A E,
AC = AC,

Iden.
Z BCA = Z CA E,

$ 68 (being alt.-int. E). .. A ABC=A ACE,

§ 106 Chaving two sides and the included L of the one equal respectively to two sides

and the included L of the other).

. AB= EC,

(being homologous sides of equal A ).
Also,

< BAC = ZAC E,
(being homologous & of cqual A);
.. A B is | to EC,

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

equal the lines are parallel).
.. the figure A B C E is a 0,

§ 125 (the opposite sides being parallel).

Q. E. D.

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

В

Hyp.

Let the figure A B C E be a quadrilateral having

BC= AE and AB= EC.
We are to prove figure ABC E a O.

Draw A C.
In the A ABC and A EC

B = 4 E,
A B=CE,

Hyp.
AC=AC,

Iden. ..A ABC=A A EC,

$ 108 (having three sides of the one equal respectively to three sides of the other).

.. ZACB=LCA E,

ZBAC=LACE,
(being homologous É of equal A).

..BC is to A E,
and

A B is I to EC, (when two straight lines lying in the same plane are cut by a third straight

line, if the alt.-int. E be equal, the lines are parallel).
.. the figure A B C E is a o,

$ 125 (having its opposite sides parallel).

Q. E. D.

and

§ 69

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