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DRILATERALS.

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 parallelogranı 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 A B C E be a parallelogram, and A C its diagonal.

[blocks in formation]

AC=AC,

Iden. ZACB= 2 CAE,

$ 68 (being alt.-int. £). Z CAB=LACE,

§ 68 ... A ABC=A A EC,

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

Q. E. D.

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

B .

and

Let the figure A B C E be a parallelogram.
We are to prove BC= A E, and A B = EC,
also, ZB= 2 E, and Z BA E= ZBC E.

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 A).

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

ZBAC = LACE,

Z EAC = Z A C B,

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

ZBAC + 2 E AC=LACE + LACB;
or,
ZBA E= BCE.

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

and

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.

B

Let the figure A BCE be a quadrilateral, having the

side A E equal and parallel to B C.

[blocks in formation]

Draw A C.
In the A ABC and A EC
BC= A E,

Hyp.
AC = AC,

Iden. ZBCA = LCA E,

§ 68 (being alt.-int. 4). .. A ABC = A ACE,

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

and the included 2 of the other).

.. A B = EC,

(being homologous sides of equal A).
Also,

< BAC = LACE,
(being homologous És of equal A);
.. A B is ll to EC,

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

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

§ 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.

Hyp.

Let the figure A B C E be a quadrilateral having

BC = A E and A B = EC.
We are to prove figure A B C E a O.

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

Hyp.
A B=CE,
AC = AC,

Iden. .:. A ABC=A AEC,

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

Z ACB= 2 CAE,
and

ZBAC=LACE,
(being homologous és of equal o ).

.. B C is I 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. Is be equal, the lines are parallel).
... the figure A B C E is a o,

§ 125 (having its opposite sides parallel).

Q. E. D.

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