PROPOSITION XXIII.

Straight lines which join the extremities of equal and parallel straight lines towards the same parts are themselves equal and parallel.

Then ... AB is | to CD; .. LABC is = L DCB, (1. 21)

and in the As ABC, DCB,

AB, BC, and the included

ABC, are respectively

.. the As ABC, DCB are equal in all respects. (I. 1)

Hence AC is = DB.

Also ACB is = L DBC,

and.. AC is | to BD.

(1. 20)

PROPOSITION XXIV.

If one side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; also the three interior angles of every triangle are together equal to two right angles.

and let one of its sides BC be produced to K. Then shallACK be equal to the anglès ABC, BAC together;

also the three interior angles of the ▲ ABC shall be together equal to two right angles.

Through C draw CH || to AB.

Then. CH is || to AB, and AC meets them, therefore ACH is the alternate 4 BAC.

(I. 21) Again, CH is | to AB, and BK falls upon them; .. the exterior LHCK is the interior and opposite LABC; (I. 21)

=

=

.. the whole ACK is the two angles ABC, BAC together.

Now to each of these equals add the ACB;

=

.. the two angles ACK, ACB are together the three Ls of the ▲ ABC';

but the two angles ACK, ACB are together = two rights;

(1.9)

.. the threes of ▲ ABC are together two rights.

PROPOSITION XXV.

If two angles of one triangle are equal to two angles of another, then shall the remaining angle of the one be equal to the remaining angle of the other.

For the three angles of each ▲ are together rights;

= two

(I. 24)

... the three angles of the one are together = the three < s of the other;

but two 4s of the one are two 4s of the other;

... the remaining

the other.

=

of the one is the remaining of

COR. If two angles of one triangle

=

are equal to two angles of another, and any side of one is the corresponding side of the other; then shall the s be equal in all respects.

For, by the proposition, the remainings are = one another.

Hence there are two angles and the side adjacent in the one triangle equal to two angles and the side adjacent in the other;

.. the ▲s are equal in all respects.

(I. 3)

QUADRILATERAL AND MULTILATERAL

FIGURES.

DEFINITIONS.

Rectilineal figures are those contained by straight

lines.

Quadrilateral figures by four straight lines.

Multilateral figures, or polygons, by more than four straight lines.

A parallelogram is a four-sided figure, of which the opposite sides are parallel.

The straight line joining the opposite angles of a quadrilateral figure is called a diagonal.

PROPOSITION XXVI.

The opposite sides and angles of a parallelogram are equal, and the diagonal bisects it.

... AB is | to DC, .. ▲ ABD is = L BDC, (1. 21)

and

... AD is to BC, ..
||

LADB is = L DBC, (I. 21)

.. the whole

Also

ABD, ▲ ADB, and the side BD, are re

spectively CDB, CBD, and side BD;

(1.3)

.. AS ABD, CDB are equal in all respects; .. AB,

AD, and BAD are respectively CD, CB, and BCD.

=

PROPOSITION XXVII.

If one angle of a parallelogram is a right angle, all its angles are right angles.

Р

R

Let PQRS be a, having the 4 PSR a right 4.
Then shall the other angles of PQRS be right 4 s.

. PS is to QR;

.. LS PSR, QRS are together = two right 4 s: (1. 21) but 4 PSR is a right 4;

.. also QRS is a right ▲ ;

.. the 4 s PQR, SPQ opposite to these are right ▲ s. (1.26)

DEFINITION.

A rectangle is a right-angled parallelogram, and is said to be contained by any two of its adjacent sides.