Congruence of Two Triangle

We shall say that two figures in the plane are similar whenever one is congruent to a dilate of the other. Therefore the two quadrilaterals are similar, since one is just an enlargement of the other. Any two circles are similar, if the two circles have the same radius. We simply take dilation by 1 to satisfy the definition. For the moment, we will study similar triangles, as illustrated below.

Figure 3.1      Dilation of a triangle

We can easily generate similar triangles by dilating a triangle with respect to one of its vertices or with respect to a point 0 not a vertex like shown below.

Figure 3.2      Process of dilating a triangle

Let T be a triangle whose sides have lengths a, b, c respectively. If we dilate T by a factor of r, we obtain a triangle which we denote by rT. The lengths of its sides will be ra, rb, rc, as we saw in the preceding section. Note that r can be any positive number. For instance in Figure below we have drawn triangles T, T, and 2T.

Figure 3.3      Sample of dilating process of a triangle

Denote by T' the dilation of T by r. Let a', b', c' be the lengths of the corresponding sides. Then we have

a' = ra, b' = rb, c' = rc.

Therefore the ratios of the corresponding sides are all equal, that is:

We have seen that if two triangles are similar, then the ratios of the lengths of corresponding sides are equal to a constant r. We now prove the converse.

Definition.
Two triangles which it’s all corresponding side have same length are similar and congruent.

Note:
If both of the triangles have same length side, then it is denoted by SSS.
Notation for congruent is 
Another ways to explain the congruency will be given soon.

Theorem 1.1.
Let T, T' be triangles. Let a, b, c be the lengths of the sides of T, and let a', b', c' be the lengths of the sides of T'. If there exists a positive number r such that

a' = ra, b' = rb, c' = rc,

then the triangles are similar.

Proof.
The dilation by of T' transforms T' into a triangle T’’ whose sides have lengths a, b, c, because

Therefore T, T’’ have corresponding sides of the same length. By condition SSS we conclude that T and T’’ are congruent. Therefore T is congruent to a dilation of T', and triangles T and T' are similar.

Theorem 1.2.
If two triangles are similar, then their corresponding angles have the same measure.

Theorem 1.3.
If two triangles have corresponding angles having the same measure, then the triangles are similar.


Proof.
Let T, T' be the triangles. Let A, B, C be the angles of T and let A', B', C' be the corresponding angles of T'. Let a, b, c and a', b', c' be the lengths of corresponding sides. Let

r = a'/a

be the ratio of the lengths of one pair of corresponding sides. Then a' = ra. Dilation by r transforms T into a triangle T" whose sides have lengths

a" = ra, b" = rb, c" = rc

respectively. The triangles T' and T" have one corresponding side having the same length, namely

a" = a' = ra.

The situation is shown below.

Figure 3.4      Condition of theorem 3.6 and 3.7

We have seen in the previous theorem that dilation preserves the measures of angles. Hence the angles adjacent to this side in T' and T" have the same measure, that is:

m(ÐB') = m(ÐB") and m(ÐC') = m(ÐC").

It follows from the ASA property that T', T" are congruent. Hence T' is congruent to a dilation of T, and hence T' is similar to T, as was to be shown.


Theorem 1.4. 

Two triangle are congruent if
a. there is similar side
b. angle in that side and angle which is opposite to the side is similar
then we called it congruent by SAA.

Theorem  1.5. 

Two triangles are congruent if both of triangles are right angle triangle and one leg and also its hypotenuse is same length.

We let the proof as exercises for the reader.