CBSE Class 7 Mathematics Chapter 7: Congruence of Triangles NCERT Solutions

Two geometric figures are considered congruent if they have the same size and the same shape. This means one figure can be perfectly superimposed onto the other.

(a) Two line segments are congruent if they have the same length. This ensures they can be perfectly aligned one over the other.

(b) If two angles are congruent, they must have the same measure. Therefore, if one angle measures 70°, the other congruent angle must also measure 70°.

(c) The statement $\backslash angle\; A\; =\; \backslash angle\; B$ is a shorthand notation. When we write this, we actually mean that the measures of angle A are equal to the measures of angle B, which can be written as $m\backslash angle\; A\; =\; m\backslash angle\; B$. This equality of measures is what makes the angles congruent.

Congruent shapes are identical in both size and form. Here are two examples from everyday life:

1. Two identical footballs: All standard footballs are manufactured to be the same size and shape, making them congruent to each other.
2. Two teacher's tables of the same model: If two tables are produced from the same design and specifications, they will have identical dimensions and form, thus being congruent.

Given the congruence statement $\backslash triangle\; ABC\; \backslash cong\; \backslash triangle\; FED$, the correspondence between the vertices is A $\backslash leftrightarrow$ F, B $\backslash leftrightarrow$ E, and C $\backslash leftrightarrow$ D. This means the corresponding parts (angles and sides) are equal in measure or length.

The corresponding congruent parts are:

1. Angles:
   • $\backslash angle\; A\; \backslash leftrightarrow\; \backslash angle\; F$
   • $\backslash angle\; B\; \backslash leftrightarrow\; \backslash angle\; E$
   • $\backslash angle\; C\; \backslash leftrightarrow\; \backslash angle\; D$
2. Sides:
   • $\backslash overline\{AB\}\; \backslash leftrightarrow\; \backslash overline\{FE\}$
   • $BC\; \backslash leftrightarrow\; \backslash overline\{ED\}$
   • $\backslash overline\{AC\}\; \backslash leftrightarrow\; \backslash overline\{FD\}$

We are given the congruence statement $\backslash Delta\; DEF\; \backslash cong\; \backslash Delta\; BCA$. This statement establishes a specific correspondence between the vertices of the two triangles: D corresponds to B, E corresponds to C, and F corresponds to A.

Using this correspondence, we can identify the parts of $\backslash Delta\; BCA$ that correspond to the given parts of $\backslash Delta\; DEF$:

(i) The angle $\backslash angle\; E$ in $\backslash Delta\; DEF$ corresponds to the angle $\backslash angle\; C$ in $\backslash Delta\; BCA$, because E is the second vertex in DEF and C is the second vertex in BCA.

(ii) The side $\backslash overline\{EF\}$ in $\backslash Delta\; DEF$ connects the second and third vertices. It corresponds to the side connecting the second and third vertices in $\backslash Delta\; BCA$, which is $\backslash overline\{CA\}$.

(iii) The angle $\backslash angle\; F$ in $\backslash Delta\; DEF$ corresponds to the angle $\backslash angle\; A$ in $\backslash Delta\; BCA$, as F is the third vertex in DEF and A is the third vertex in BCA.

(iv) The side $\backslash overline\{DF\}$ in $\backslash Delta\; DEF$ connects the first and third vertices. It corresponds to the side connecting the first and third vertices in $\backslash Delta\; BCA$, which is $\backslash overline\{BA\}$.