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PROPOSITION ΧΧΙΙ.

Straight lines which are parallel to the same straight line. are parallel to one another.

A

B

C

Let the straight lines A, B be each of them || to C.
Then shall A, B be || to one another.

For if not they will meet if produced, and there will thus be drawn through one point two straight lines parallel to the same straight line, which is impossible. (Lemma.)

PROPOSITION XXIII.

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

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

and. in the △s ABC, DCB,

AB, BC, and the included ABC, are respectively = DC, CB, and DCB;

.. the os ABC, DCB are equal in all respects. (1. 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.

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and let one of its sides BC be produced to K. Then shall ∠ACK be equal to the angles 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, there

fore LACH is = the alternate L ВАС.

(I. 21)

Again, ... CH is || to AB, and BK falls upon them; ... the exterior 2 HCK is = the interior and opposite LABC; (Ι. 21)

... the whole LACK is = the two angles ABC, BAC together.

Now to each of these equals add the ACB ;

ABC;

... the two angles ACK, ACB are together = the three 2s of the but the two angles ACK, ACB are together = two right 2 s; (I. 9) ... the three s of △ABC are together = two right 4 s.

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 = two right 2 s; (I. 24)

... the three angles of the one are together = the three 2 s of the other;

but two 2 s of the one are = two 2 s of the other;

... the remaining 2 of the one is = the remaining 2 of the other.

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 remaining 2 s 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 As are equal in all respects.

(1. 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,

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and

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AB is || to DC, .·. 1 ABD is = L BDC, (1.21) • AD is || to BC, .·. 1 ADB is = 1 DBC, (Ι. 21) ... the whole ABC is = the whole L ADC. Also ... LABD, LADB, and the side BD, are respectively = L CDB, 4 CBD, and side BD; (Ι. 3)

.. AS ABD, CDB are equal in all respects; ... AB, AD, and L BAD are respectively = CD, CB, and LBCD.

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