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with one exception, which will be mentioned presently. It will be seen that three parts of the one must be equal to three parts of the other respectively, one at least of these three parts being a side of the A. The parts that they must have respectively equal, are-1. Two sides and the between them. (IV.) 2. Three sides. (VIII.) 3. Two Zs and one side similarly placed with respect to the equal <8. (XXVI.)

If two ▲s have two sides of the one equal to two sides of the other, each to each, and one in the one equal to one in the other, opposite corresponding equal sides, the As are not necessarily equal in all respects. But the s opposite the other pair of equal sides will be either equal or supplementary.*

In the As A B C and DEF, let the side A. C be equal to the side DF, the side CB to the side FE, and the CAB to the FDE. If the 28 ACB and DFE are also equal, the ▲s are equal in every respect. If not, let

A A

one of them, A CB, be larger than the other, and from C draw the line CG, making with the longer of the two sides A C, CB an AC G, equal to the DFE, and meeting A B in G. The As A CG and DFE will be equal to each other, in every respect (according to the 26th proposition). Consequently, the AGC is equal to the DEF. But CG is equal to F E, and is therefore equal to C B. Consequently the 8 CGB and CBG are equal to each other.

But CGA and CGB together make up two rights; therefore C BA and FED together make up two right 28.

Twos are said to be supplementary when they are together equal to two rights. They are said to be complementary, when they are together equal to one right Z.

If anything enables us to affirm that the s CB A and FED cannot be supplementary to each other, it will follow that they are equal, and therefore that the ▲s are equal in all respects. If we know that they are both obtuse, they cannot be supplementary, because the supplement of an obtuse must be acute.

s which are given equal are

If we know that the either right or obtuse, the s opposite the other pair of equal sides must be acute (Prop. XXXII.), and therefore cannot be supplementary to each other.

If thes which must be either equal or supplementary are opposite sides which are shorter than the other pair of equal sides, these s cannot be supplementary; for if they were, one of them would be obtuse, and an obtusein a A must be the largest of the threes (Prop. XXXII.), and therefore must have the longest side opposite to it. (Prop. XIX.)

If we know that one of the two s opposite the second pair of equal sides is a right, the other must also be a right, because a right is the supplement of a right Z; therefore it comes to the same thing whether they are considered as equal or as supplementary.

PROPOSITION XXVII.

If a straight line meets two other straight lines, and the alternate angles so formed are equal, those straight lines are parallel,—that is, will never meet one another, however far they may be produced.

is

To prove this proposition, we must know, that if one side of a ▲ be produced, the exterior greater than either of the two interior and opposites. (Prop. XVI.)

E

The proof of this proposition is of the indirect kind. A B and C D are two straight lines intersected by the straight line EF,

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L

B

K

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to prove that the lines AB and CD when produced will not meet.

Proof. Suppose it were possible that they should meet at a point K. The two lines L K and HK, with the line LH, would then form a ▲ LKH.

The ALH would be the exterior of this ▲, formed by the production of the line K L, and would (Prop. XVI.) be greater than the interior and opposite LHD.

But this is impossible, for the s ALH and LHD are equal.

Therefore the supposition that would lead to this impossibility must be itself impossible. That is, it is impossible that the straight lines A B and CD should meet, when produced, in the point K.

In a similar way it might be shown that they could not meet, when produced, on the other side of the line EF.

In other words (see Def. XXVI.; the straight lines A B and C D are parallel.

* The s BLH and LHC are also alternates. If we know either pair of s to be equal, it will follow that the other pair are also equal; for the Дs ALH and B LH are together equal to two rights, as are also the s LHD and LHC, and consequently the s BLH and ALH are together equal to L HD and LHC. If then either` pair of alternate s be equal, and be taken away from these equal sums, the remainders will be equal.

PROPOSITION XXVIII.

If a straight line intersects two other straight lines, and the exterior angle so formed is equal to the interior and opposite angle on the same side of the intersecting line, or if the two interior angles on either side of the intersecting line are together equal to two right angles, then those two straight lines are parallel.

The above enunciation really includes two separate propositions, which will be more clearly understood if taken separately.

1. If a straight line intersects two other straight lines, and the exterior angle so formed is equal to the interior and opposite angle on the same side of the intersecting line, then those two straight lines are parallel (i.e., will not meet, however far they may be produced).

A

To prove this we must know,

1. That magnitudes which are equal to the same are equal to one another. (Ax. I.)

2. That if two straight lines intersect one another, the vertically opposites are equal. (Prop. XV.) 3. That if one straight line meet two other straight lines, and make the alternate s equal, those straight lines are parallel. (Prop. XXVII.)

L

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The straight lines AB and CD are intersected by the line EF, and the exterior ZELB is equal to the interior -B and opposite LHD on the same side of the intersecting D line. We have to prove that the straight lines A B and C D are parallel.

A

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ELB and ALH, being the s formed by the intersection of (Prop. XV.)

the lines A B and EF are equal.

But the EL B is equal to the LHD. Therefore the ALH is also equal to the LHD.

But ALH and LHD are alternate 8. Consequently, as the line EF intersects the lines A B and CD, and the alternates are equal, the lines A B and CD are parallel. (Prop. XXVII.)

Similarly, if, instead of the s ELB and LHD, it were known that the 8 ELA and LHC were equal, the proposition would be proved by showing that the ELA is equal to the vertically opposite BLH, and that consequently the alternate 8 BLH and LHC would be equal.

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2. If a right line meets two other right lines, and the two interior angles together equal to two right angles, those right lines are parallel.

To prove this proposition we must know:

1. That if one straight line meets another straight line, the adjacent s so formed are together equal to two rights. (Prop. XIII.)

2. That things that are equal to the same are equal to one another. (Ax. I.)

3. That if the same quantity be taken from two equal quantities the remainders are equal. (Ax. III.)

4. That if one straight line meets two other straight lines, and the alternate s so formed are equal, those straight lines are parallel. (Prop. XXVII.)

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