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Triangles

Similar figures and congruent figures may appear to be closely related concepts, but there is an important difference between them. Go through the given video to understand the difference so that you can identify similar figures from congruent figures.

Congruency of line segments:

Two line segments are congruent to each other if their lengths are equal”.

Consider the following line segments.

Here, the line segments AB and PQ will be congruent to each other, if they are of equal length.

Conversely, we can say that, “Two line segments are of equal length if they are congruent to each other”.

i.e. if, then AB = PQ.

Congruency of angles:

Two angles are said to be congruent to each other if they have the same measure”.

The angles shown in the following figures are congruent to each other as both the angles are of the same measure 45°.

Thus, we can write ∠BAC ≅ ∠QPR.

Its converse is also true.

If two angles are congruent to each other, then their measures are also equal”.

There is one special thing about congruent figures that their corresponding parts are always equal.

For example, if two triangles are congruent then their corresponding sides will be equal. Also, their corresponding angles will be equal.

Look at the following triangles.

Here, ΔABC ΔDEF under the correspondence ΔABC ↔ ΔDEF. This correspondence rule represents that in given triangles, AB ↔ DE (AB corresponds to DE), BC ↔ EF, CA ↔ FD, ∠A ↔ ∠D, ∠B ↔ ∠E, ∠C ↔ ∠F. These are corresponding parts of congruent triangles (CPCT), ΔABC and ΔDEF.

Since ΔABC and ΔDEF are congruent, their corresponding parts are equal.

Therefore, AB = DE, BC = EF, CA = FD

And, ∠A = ∠D, ∠B = ∠E, ∠C = ∠F

Similarly, we can apply the method of CPCT on other congruent triangles also.

Let us now try and apply what we have just learnt in some examples.

Example 1:

Find which of the pairs of line segments are congruent.

 (i) (ii)

Solution:

(i) Lengths of the two line segments are not same. Therefore, they are not congruent.

(ii) Each of the line segments is of length 3.1 cm, i.e. they are equal. Therefore, they are congruent.

Example 2:

If and = 9 cm, then find the length of.

Solution:

Since, i.e. line segment AB is congruent to line segment PQ, therefore, and are of equal length.

= 9 cm

Example 3:

If ∠ABC ≅∠PQR and ∠PQR = 75o, then find the measure of ∠ABC.

Solution:

If two angles are congruent, then their measures are equal.

Since ∠ABC ≅ ∠PQR,

∴ ∠ABC = ∠PQR

Therefore, ∠ABC = 75o

Example 4:

Which of the following pairs of angles are congruent?

(i)

(ii)

Solution:

(i) The measure of both the angles is the same. Therefore, they are congruent.

(ii) The measures of the two angles are different. Therefore, they are not congruent.

Example 5:

Identify the pairs of similar and congruent figures from the following.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Solution:

Figures (i) and (iii) are similar because their corresponding angles are equal and their corresponding sides are in the same ratio. However, these figures are not congruent as they are of different sizes.

Figures (ii) and (viii) are congruent as they are of the same shape and size (circles with radius 1 cm each).

Example 6:

Are the following figures similar or congruent?

Solution:

The two given...

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