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Attitude D, in Fig. 6–2, is termed an external perspective of triangle ABC since it is outdoor the triangle and is shaped by way of one aspect, BC, and an extension of one other aspect, AC. With admire to attitude D, angles A and B are distant inside angles of triangle ABC, while perspective C is an adjoining inside perspective. therefore we need to turn out that attitude D is greater than perspective A and bigger than attitude B. allow us to turn out that attitude D is greater than perspective B. Fig. 6–3 the matter prior to us is a tantalizing one simply because, whereas it does appear visually visible that perspective D is bigger than perspective B, there's no obvious approach to facts. an concept is required, and this is often provided via Euclid. He tells us to bisect aspect BC (Fig. 6–3), to hitch the mid-point E of BC to A, and to increase AE to the purpose F, in order that AE = EF. He then proves that triangle AEB is congruent to triangle CEF, that's, that the perimeters and angles of 1 triangle are equivalent, respectively, to the edges and angles of the opposite. This congruence is straightforward to end up. Euclid had formerly proved that vertical angles are equivalent, and we see from Fig. 6–3 that angles 1 and a couple of are vertical angles. extra, the truth that E is the mid-point of BC signifies that BE = EC. in addition, we developed EF to equivalent AE. consequently, within the triangles in query, facets and the incorporated perspective of 1 triangle are equivalent to 2 facets and the incorporated perspective of the opposite. yet Euclid had formerly proved that triangles are congruent if in basic terms aspects and the incorporated perspective of 1 are equivalent to 2 aspects and the integrated perspective of the opposite. considering that those proof are precise of our triangles, the 2 triangles has to be congruent. simply because triangles AEB and CEF are congruent, attitude B of the 1st triangle equals attitude three of the second. we all know that perspective three is the perspective to settle on within the moment triangle because the perspective which corresponds to B, simply because attitude B is contrary AE, and attitude three is contrary the corresponding equivalent facet EF. The facts is essentially comprehensive. perspective D is greater than attitude three as the complete, attitude D in our case, is bigger than the half, attitude three. for this reason attitude D can be more than perspective B simply because perspective B has a similar dimension as attitude three. we've proved a big theorem, and we must always see sequence of easy deductive arguments ends up in an indubitable outcome. And now allow us to end up one other, both vital theorem in order to convey one or different positive factors of Euclid’s paintings: THEOREM 2. If traces are reduce by way of a transversal with the intention to make trade inside angles equivalent, then the strains are parallel. Fig. 6–4 back allow us to see what the concept ability earlier than we think about its evidence. In Fig. 6–4, AB and CD are traces reduce by way of the transversal EF. The angles 1 and a couple of are known as exchange inside angles, and we're advised that they're equivalent. the theory asserts that, below this situation, AB needs to be parallel to CD. As in relation to the previous theorem, the statement is apparently right, and but the tactic of facts is in no way obvious. the following Euclid makes use of what's frequently referred to as the oblique approach to facts; that's, he supposes that AB isn't really parallel to CD.