SECTION VI.

QUADRILATERAL FIgures.

M.-We have, now, become acquainted with the most general truths respecting triangles. Write them on your slates, that I may see whether you remember them all.

This having been done satisfactorily, let the pupils be called-upon to demonstrate any one of the problems or rather, let each pupil, in turn, assign a problem, for solution, to the class. After this useful exercise, the master may proceed thus:

M.-What, do you think, should be our next step, after the investigation of trilateral figures? P. To investigate quadrilateral figures.

M.-State all you know of quadrilateral figures. (Lesson V. Introduction.)

Let the pupils repeat what they remember respecting them.

M.-Into what two groups may all quadrilateral figures be classed?

P.-Into parallelograms and trapeziums.

M.-We shall begin with parallelograms, and, first, consider the manner in which a parallelogram is constructed. Draw a parallelogram, and give a definition of it.

α

b

P.-If a b is parallel to cd,

and a c is parallel to b d,—

the figure ab de is a parallelogram.

M.-Hence, a parallelogram is

P.-A four-sided figure whose opposite sides are parallel.

M.-Well, one relation between the sides of parallelograms being known, you may be able to discover another try.

P.-The opposite sides must be equal.

M.-Demonstrate this.

a

b

P-Let ab d'e be a parallelogram,

its opposite sides shall be equal; namely, a b c d, and a cbd. Join cb:

ab is parallel to cd,

..abc alternate dcb;

and...ac is parallel to bd,

...Lacb alternate dbc:

now, cb is common to the triangles acb and d bc; ..A acbdbc,

abcd, and a cbd.

M.-Why are the triangles acb and bde equal to each other?

P.-Because they have two angles, and the side adjacent to them, of the one, equal to two angles, and the side adjacent to them, of the other, each to each.

M.-And why, then, is a b c d, and a c =

bd?

P.-Because these are the sides which are opposite to the equal angles in the triangles.

M.-Then, conversely, if a four-sided figure has its opposite sides equal, what will you conclude as to the figure?

P.-That the figure is a parallelogram.

M.-Demonstrate this.

P.-Let ab de be a four-sided figure whose op

posite sides are equal: the figure is a parallelo

gram.

Join cb:

then, ·.· a b = cd, and a c = b d,

and cb is common to the triangles a cb, b dc,

and, hence, the figure ab dc is a parallelogram. M.-Why are the triangles a b c and d be equal to each other?

P.-Because they have three sides of the one equal to three sides of the other, each to each.

M.-And, why is the angle abc equal to the angle dbc, and the angle acb equal to the angle dbc? P.-Because these are opposite to the equal sides in the equal triangles.

M.-Again, if, in the foursided figure a b cd, you knew that a b is equal and parallel to the opposite side c d, what would you conclude the figure to be? P.-A parallelogram.

M.-Demonstrate this.

α

P.-Let ab dc be a four-sided figure of which the opposite sides ab, cd are equal and parallel: the figure shall be a parallelogram.

Join c d :

then.. a b is parallel to c d,

...abc alternate dc b,

and abcd:

and cb is common to the triangles a b c, d c b—

... A abc Adcb,

anda cbdbc.

Now, these are alternate angles;

.. ac is parallel to db,

and.. the figure ab dc is a parallelogram.

M.-Why is the triangle a b c equal to the triangle

dcb?

P.-Because these triangles have two sides of the one, ab and bc, equal to two sides of the other, cd and bc, each to each, and have likewise the angles a b c and dc b, contained by them, equal to each other. M.-What else is known of the lines a c and b d, besides their parallelism?

P.-a c is equal to b d.

M.-Well, how have a c and b d been drawn? P.-Joining the extremities a, c, and b, d, of the parallel and equal straight lines a b, c d.

M.-Hence, the two straight lines which join the extremities of two equal and parallel straight linesFinish the sentence.

P.-Are equal and parallel.

M. That is not quite correct. Repeat what has been said, and see if it be true in every case.

and yet these two lines have been drawn so as to join the extremities of two equal and parallel straight lines.

M.-Now, alter the preceding statement in conformity with this finding.

P. The two straight lines which join the extremities of two equal and parallel straight lines in the same direction, or toward the same parts, are equal and parallel.

M.-What, then, is the general truth respecting the sides of parallelograms?