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ON SIMILAR POLYGONS.

PROPOSITION IV. THEOREM. 279. I'wo triangles which are mutually equiangular are similar.

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B4
In the A ABC and A' B' C' let 4 A, B, C be equal to

A', B', C' respectively.
We are to prove A B : A' B' = AC : A' C' = BC : B'C'.

Apply the A A' B' C' to the A A BC,

so that < A shall coincide with 2 A.
Then the A A' B' Co will take the position of A A E H.
Now LA EH (same as 2 B') = 2 B.
.. E H is ll to BC,

$ 69 (when two straight lines, lying in the same plane, are cut by a third straight line, if the ext. int. be equal the lines are parallel). . AB : AE = AC : A H,

$ 276 (one side of a A is to either part cut off by a line II to the base, as the other

side is to the corresponding part).
Substitute for A E and A H their equals A' B' and A' C'.
Then AB : A' B' = AC : A'C'.
In like manner we may prove

AB : A' B' = BC : B'C'.
in the two A are similar.

§ 278

Q. E. D. 280. Cor. 1. Two triangles are similar when two angles of the one are equal respectively to two angles of the other.

281. Cor. 2. Two right triangles are similar when an acute angle of the one is equal to an acute angle of the other.

PROPOSITION V. THEOREM. 282. Two triangles are similar when their homologous sides are proportional.

In the triangles A B C and A' B'C' let

A B AC BO

A' B' A' C - B'C'.
We are to prove
& A, B, and C equal respectively to A A', B', and C'.

Take on A B, A E equal to A' B',
and on A C, A H equal to A' C'. Draw E H.

АВА С
A' B' A'C'

Hyp.
Substitute in this equality, for A' B' and A' C' their equals
A E and A H.
Then

AB AC

A E AH
.. E H is II to BC,

$ 277 (if a line divide two sides of a A proportionally, it is Il to the third side). Now in the A A BC and A EH ZA BC= LA EH,

$ 70 (being ext. int. angles). ZA CB= L A HE,

· § 70 ZA= LA.

Iden. .. A A B C and A E H are similar, $ 279 (two mutually equiangular A are similar). : A B A E

$ 278 ''BO - EH' (homologous sides of similar A are proportiona?).

Hyp.

AB A' B'
But

BC - B'C'
A E A' B'

Ax. 1
EH - B'C
Since
A E = A'B',

Cons.
EH= B'C'.
Now in the A A E H and A' B'C',
EH = B'C', A E= A' B', and A H = A'C',
. A A E H= A A' B'C',

$ 108 (having three sides of the one equal respectively to three sides of the other). But Δ AEH is similar to Δ ΑΒC. .. A A' B'C' is similar to A ABC.

Q. E. D. 283. SCHOLIUM. The primary idea of similarity is likeness of form ; and the two conditions necessary to similarity are :

I. For every angle in one of the figures there must be an equal angle in the other, and

II. the homologous sides must be in proportion.

In the case of triangles either condition involves the other, but in the case of other polygons, it does not follow that if one condition exist the other does also.

Thus in the quadrilaterals Q and Q', the homologous sides are proportional, but the homologous angles are not equal and the figures are not similar.

In the quadrilaterals R and R', the homologous angles are equal, but the sides are not proportional, and the figures are not similar.

PROPOSITION VI. THEOREM. 284. Two triangles having an angle of the one equal to an angle of the other, and the including sides proportional, are similar.

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In the triangles A B C and A' B'C' let
ZA = LA', and -

, А В А С

A B A'C
We are to prove A A B C and A' B' C' similar.

Apply the A A' B' C' to the A A B C so that < A' shall coincide with Z A.

Then the point B' will fall somewhere upon A B, as at E,

the point C' will fall somewhere upon A C, as at H, and B C upon EH. Now

AB AC
A'B' = A' C

Hyp.
Substitute for A' B' and A' C' their equals A E and A H.

A B A C
Then

A E = A H ... the line EH divides the sides A B and AC proportionally ; .. E H is II to BC,

$ 277 (if a line divide two sides of a A proportionally, it is il to the third side). ... the A A B C and A E H are mutually equiangular and similar. ..A A' B' C' is similar to A ABC.

Q. E. D.

PROPOSITION VII. THEOREM. 285. Two triungles which have their sides respectively parallel are similar.

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In the triangles A B C and A' B' C' let A B, AC, and

BC be parallel respectively to A' B', A'C', and
B'C'.
We are to prove A ABC and A' B' C' similar.
The corresponding s are either equal,

§ 77 (two whose sides are II, two and two, and lie in the same direction, or

opposite directions, from their vertices are equal). or supplements of each other,

$ 78 (if two es have two sides II and lying in the same direction from their vertices,

while the other two sides are II and lie in opposite directions, the É are supplements of each other).

Hence we may make three suppositions: 1st. A+ A' = 2 rt. A, B+B'=2 rt. A, C + C' = 2 rt. A. 2d. A = A', B+ B'=2 rt. Ls, C + C'=2 rt. Is. 3d. A = A',

B=
B

C =C'. Since the sum of the s of the two A cannot exceed four right angles, the 3d supposition only is admissible. § 98

.. the two A ABC and A' B'C' are similar, $ 279 (two mutually equiangular S are similar).

Q. E. D.

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