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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 4. The parts that they maust have respectively equal, are-1. Two sides and the between them. (IV.) 2. Three sides. (VIII.) 3. Two <s and one side similarly placed with respect to the equal <s. (XXVI.)
If two As 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 48 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 Z8 ACB and DFE are also equal, the A8 are equal in every respect. If not, let 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 AC, CB an < ACG, equal to the ZDFE, and meeting A B in G. The As ACG and DFE will be equal to each other, in every respect (according to the 26th proposition). Consequently, the Z AGC is equal to the ZDEF. But CG is equal to F E, and is therefore equal to C B. Consequently the Z8 CG B and C BG are equal to each other.
But CG A and CGB together make up two right Zs; therefore C B A and FED together make up two right Zs.
* Two Zs are said to be supplementary when they are together equal to two right Zs. They are said to be complementary, when they are together equal to one right Z.
If anything enables us to affirm that the Z8 CB A and FED cannot be supplementary to each other, it will follow that they are equal, and therefore that the As are equal in all respects. If we know that they are both obtuse, they cannot be supplementary, because the supplement of an obtuse < must pe acute.
If we know that the <s which are given equal are 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 the Zs which must be either equal or supplementary are opposite sides which are shorter than the other pair of equal sides, these Zs cannot be supplementary; for if they were, one of them would be obtuse, and an obtuse < in a A must be the largest of the three Z8 (Prop. XXXII.), and therefore must have the longest side opposite to it. (Prop. XIX.)
If we know that one of the two _8 cpposite the second pair of equal sides is a right Z, the other must also be a right Z, 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.
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.
To prove this proposition, we must know, that if
one side of a 4 be produced, the exterior _ is greater than either of the two interior and opposite Zs. (Prop. XVI.)
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, Aand the alternate s A LH and LHD are equal to another. We have to prove that the lines A B 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 H K, with the line LH, would then form a ALKH.
The _ALH would be the exterior < of this A, 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 Zs 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 C D 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 E F.
In other words (see Def. XXVI.) the straight lines A B and C D are parallel.
* The Zs BLH and LHC are also alternate Zs. If we know either pair of Zs to be equal, it will follow that the other pair are also equal; for the ZS ALH and B LH are together equal to two right Z8, as are also the <s LHD and LHC, and consequently the <8 BLH and ALH are together equal to L H D 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 80 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).
To prove this we must know,-
are equal to one another. (Ax. I.)
the vertically opposite s 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.) The straight lines A B 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 CD are parallel.
Proof. The L8 EL B and A LH, being the vertically opposite Zs formed by the intersection of the lines A B and EF are equal. (Prop. XV.)
But the < EL B is equal to the <LHD.
Therefore the < ALH is also equal to the < LHD.
But ALH and LHD are alternate <s. Consequently, as the line EF intersects the lines A B and CD, and the alternate <s are equal, the lines A B and C D are parallel. (Prop. XXVII.)
Similarly, if, instead of the Zs EL B and LHD, it were known that the Z8 EL A and LHC were equal, the proposition would be proved by showing that the < EL A is equal to the vertically opposite < BLH, and that consequently the alternate Zs BLH and LHC would be equal.
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 :-
line, the adjacent <s so formed are together
equal to two right Z8. (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 <8 so formed are equal, those straight lines are parallel. (Prop. XXVII.)