NCERT Solutions for Class 7 Maths Chapter 7 Congruence of Triangles

Class 7 Maths Chapter 7 Includes:

Facts

• When two line segments have equal lengths then they are called Congruent.

• When the measures of two angles are equal then the angles are called Congruent.

• Two triangles are congruent if the three sides of the one triangle are equal respectively to the three corresponding sides of the other. This rule is called SSS congruence criterion.

• Two angles and the side included between them in one triangle are equal respectively to the corresponding two angles and the included side of another triangle. This rule is called the ASA congruence criterion.

• Two triangles are congruent if two angles and the included angle between them in one of the triangles are equal to the corresponding sides and the included angle of the other triangle. This rule is called the SAS congruence criterion.

• Two right-angled triangles are congruent if the hypotenuse and a leg of one of the triangles are equal to the hypotenuse and the corresponding leg of the other right triangle. This rule is called RHS congruence criterion.

To solve Exercise 7.1 of NCERT Solutions for Class 7 Chapter 7, you need to understand the facts given below.

Line Segment

The shortest distance between two points is called a line segment.

Angle

Two rays having the same vertex form an angle.

Measure

The opening between the arms of an angle is called its measure.

Some Facts of Angles

• Vertically opposite angles are equal.

• If the sum of two adjacent angles is 1800, then they are said to form a linear pair.

• When two parallel lines are intersected by a transversal, then alternate angles are equal and corresponding angles are equal.

• The sum of interior opposite angles on the same side of the transversal is 1800.

• In a right-angles triangle, the square of the hypotenuse is equal to the sum of squares of its legs.

Congruence of Plane Figures

When a plane figure covers the other exactly, then they are said to be of the same shape and same size. Such objects are said to be congruent. Congruent objects are exact replicas of each other. The relationship between two congruent objects is called Congruence.

When two figures are congruent to each other then we write their relationship as Fig.1, Fig.2. (Fig. is referred to as a figure).

Note: Two congruent objects are always similar but two similar figures need not be congruent.

For Example:

• Any two squares are similar but they are congruent only if they have the same length.

• Any two circles are always similar but they are congruent only if they have the same radius.

• Any two equilateral triangles are the same but they are congruent only if their side lengths are the same.

Note:

1. If any two line segments have the same length, then they are congruent. Also if two line segments are congruent, they are said to have the same length.

2. Any two angles having the same measure are congruent. Also, when two angles are congruent, their measures are the same.

3. The congruence of angles entirely depends on the equality of their measures.

4. Two congruent line segments \[\overline{AB}\] and \[\overline{CD}\] re-written as \[\overline{AB}\] ≅ \[\overline{CD}\].

5. Two congruent angles ∠ABC and ∠PQR are written as ∠ABC ≅∠PQR.

Congruence of Triangles

Two triangles are congruent if they are copies of each other and when superimposed, they cover each other exactly.

Suppose,  1st triangle is ΔABC and the 2nd triangle is ΔPQR, we can write ΔABC ≅ ΔPQR.

Since, congruent means “equal in all respects”, a triangle is said to be congruent to another triangle when both the triangles are identical or equal in all respects. In other words, if one figure is placed on the other, they should coincide with each other exactly to be called congruent to one another.

Thus, a triangle is said to be congruent to another if and only if all the six elements (three sides and three angles) of one triangle are respectively equal to the corresponding six elements of the other.

In ΔABC ≅ ΔPQR, the correspondence is:

B ↔P but B does not match (or correspond) Q or R

C ↔ Q but C does not match (or correspond) P or R

D ↔R but D does not match (or correspond) P or Q

(The symbol ↔ means to match or correspond)

Thus,

\[\overline{BC} = \overline{PQ}\]

\[\overline{CD} = \overline{QR}\]

\[\overline{DB} = \overline{RP}\]

and

∠B =  ∠P

∠C = ∠Q

∠D = ∠R

The above order is important. Thus, the triangles BCD and PQR are congruent under the correspondence BCD ↔ PQR.

To understand Exercise 7.2 in NCERT Solutions for Class 7 chapter 7, you need to understand the criteria for congruence of triangles.

Criteria For Congruence Of Triangles

So far you have learned that for two triangles to be congruent, six parts of one triangle should necessarily be congruent to the six corresponding parts of the other triangle. Now, you will study the rules of congruence, which will help you to understand that two triangles are congruent if three pairs of particular corresponding parts are congruent. The four rules of congruence as follows:

1. Side - Side - Side [SSS]   (Three sides).

2. Side - Angle - Side [SAS]   (Two sides and included angle).

3. Angle - Side - Angle [ASA]   (Two angles and included side).

4. Right Angle - Hypotenuse - Side [RHS]   (Hypotenuse and one side of a right triangle).