Section 111.

QUADRILATERAL FIGURES.

DEFINITIONS.

1. A quadrilateral figure is one which is bounded by four straight

lines. Ex. ABCD

2. A parallelogram is a quadrilateral, of which the opposite sides

are parallel and equal. Ex. ABCD

Note.—Quadrilaterals fall into six classes, four of which are parallelograms—viz., the square, the rectangle, the rhombus, and the rhomboid.

3. A square is a parallelogram which has all its sides equal, and all

its angles right angles. Ex. ABCD.

4. A rectangle is a parallelogram which has only its opposite sides

equal, but all its angles right angles. Ex. ÅBCD

NOTE. — This kind of parallelogram is also termed an oblong. 5. A rhombus is a parallelogram which has all its sides equal, but

its angles are not right angles. Ex. ABCD

NOTE.—In each case its opposite angles are equal to each other. 6. A rhomboid is a parallelogram which has only its opposite sides

equal, but its angles are not right angles. Ex. ABCD

Note 1.-As in the preceding figure, its opposite angles are equal to each other.

NOTE 2.-The remaining classes of quadrilaterals are the trapezium and the trapezoid.

8. A trapezoid is a quadrilateral which has only two of its sides

parallel. E.. ABCD

Note 1.—Some of its sides and angles may be equal.

NOTE 2.-All quadrilateral figures are also called quadrangles, as they have also four angles.

9. A diagonal of a quadrilateral is a straight line which joins any

two of its opposite angles. Ex. AB

10. A diameter is a straight line drawn through its centre parallel

to two of its sides. Ex. AB

Problem 34.

To construct a square on a given base AB

1. At B in the given line AB erect a perpendicular BC

equal to AB (Pr. 2). 2. From the point A, with radius AB, describe an arc

above A.

3. From C, with the same radius, cut the arc in D.

4. Join AD and CD.

Then ABCD is the square required.

Problem 35.

To construct a square, the diagonal AB being given.

1. Bisect AB by the perpendicular CD (Pr. 1).

2. Cut off EF, and EG, equal to EA or EB.
3. Join AF, FB, BG, GA. Then AFBG is the square

required, having the given diagonal A B.

Problem 36.

To construct a rectangle, the lengths of two of the sides AB and BC being given.

1. From the point B in the given line AB, erect BC per

pendicular to AB (Pr. 2), and equal to the given line BC.

B

2. From A, with radius BC, describe an arc above A.
3. From C, with radius AB, cut the arc in the point D.
4. Join AD and CD. Then ABCD is the required rectangle.

Problem 37.

To construct a rectangle, one side AB and its diagonal AC being given.

1. Bisect the diagonal AC in E (Pr. 1), and from E as

centre, with the radius EA or EC, describe a circle. 2. From A and Cas centres, with the given line AB as radius,

describe arcs B and D. 3. Join AB, BC, CD, and DA. Then ABCD is the rectangle