NCERT Solutions Class 7 Maths Chapter 7

NCERT Solutions for Class 7 Mathematics Chapter 7 Congruence of Triangles

Triangles that have the same size and shape are called congruent triangles. The symbol ≅ is used to indicate the congruence between them. Since this is an important and interesting  topic, Extramarks offers NCERT Solutions for Class 7 Mathematics Chapter 7 that will encourage  and guide you through the topic. These are detailed step-by-step solutions that cover all the questions covered in this chapter. Students will find these solutions very helpful for their exam preparation.

NCERT Solutions for Class 7 Mathematics Chapter 7 – Congruence of Triangles

NCERT Solutions for Class 7 Mathematics- 

Congruence is the term usually used in Mathematics to define an object and its mirror image. Two objects or shapes are said to be congruent to each other if they superimpose on each other. In other words, we can say that their shape and dimensions are the same. In the case of geometric figures, line segments are congruent if they are of the same length and angles are congruent if they are of the same measure.

In the case of triangles, the corresponding sides, as well as the angles of congruent triangles, are all equal. There are certain criterias to check  if two triangles are congruent or not. It is advisable to first find out the dimensions of the triangles before trying to prove them congruent. However, the evaluation of the congruence of triangles can be done by proving only three values out of these six. 

The chapter covers the following topics:

•  Congruence of Angles
• Congruence of Plane Figures
• Congruence Among Line Segments
• Congruence of Triangles
• Criteria for Congruence  of Triangles

Some Facts of Angles

• The vertically opposite angles are always equal.
• Two adjacent angles are said to form a linear pair only if their sum is 1800.
• The alternate angles and corresponding angles are equal when two parallel lines are intersected by a transversal.
• The sum of interior opposite angles on the same side of the transversal is always 1800.
• In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of its other two sides.

Congruence of Plane Figures

When a plane figure covers the other one completely, then they are said to be of the same shape and same size (such objects are said to be congruent). Thus, we can say that congruent objects are replicas of one another and the relationship between such congruent objects is termed Congruence.

Note: As per the rule of congruence, two congruent objects are always similar but two similar figures are not always congruent, they may or may not be congruent.

For Example:

1. Any two squares can be similar but they are congruent only if they are of the same length.
2. Any two circles are always similar but they are congruent only if they are of the same radius.
3. Any two equilateral triangles can be similar but they are congruent only if the length of their sides is the same.

Note:

If two line segments are of the same length, they are said to be congruent. Also, if two line segments are congruent, it implies that they have the same length.

If two angles are of the same measure, they are said to be congruent. Also, when two angles are congruent, it implies their measure is the same.

Thus, we can say that the congruence of angles depends on their measures i.e. if they are equal or not. 

To indicate that two line segments are congruent, the symbol ≅ is used. 

The same symbol is used for congruent angles. For example, if ∠ABC and ∠PQR are congruent, it will be written as ∠ABC ≅∠PQR.

Congruence of Triangles

Two triangles are said to be congruent if they are copies of each other and cover each other exactly if superimposed. 

For example, consider two triangles: ΔABC and ΔPQR. If both of them are congruent to each other, we can write them as ΔABC ≅ ΔPQR.

Since the literal meaning of congruent is “equal in all respects”, a triangle is congruent to another triangle when both of them are identical or equal to each other in all respects. In other words, when one of them is placed on the other, they should coincide with each other exactly. Only then they can be called congruent to one another.

Thus, any two triangles are said to be congruent if all of their six elements, including three sides and three angles of triangles are equal to the corresponding six elements of the other.

Criteria For Congruence Of Triangles

The congruence of triangles can be proved even without the actual measure of the sides and angles of the triangles. The different rules of congruency include:

1. SSS (Side-Side-Side)
2. SAS (Side-Angle-Side)
3. ASA (Angle-Side-Angle)
4. AAS (Angle-Angle-Side)
5. RHS (Right angle-Hypotenuse-Side)

They have been discussed in detail below: 

1. SSS (Side-Side-Side)- If all the three sides of a triangle are equal to the corresponding three sides of the other triangle, then the two of them are said to be congruent by the SSS rule.
2. SAS (Side-Angle-Side)- If any two sides of a triangle and an angle included between the sides of such a triangle are equal to the corresponding two sides and the angle between the sides of another triangle, then the two of them are said to be congruent by the SAS rule.
3. ASA (Angle-Side- Angle)- If any two angles and a side included between the angles of a triangle are equal to the corresponding two angles and side included between the angles of another triangle, then the two of them are said to be congruent by the ASA rule.
4. AAS (Angle-Angle-Side)- When two angles of a triangle and a non-included side of a triangle are equivalent to the corresponding angles and sides of another, then the triangles are said to be congruent by the AAS rule.

Here’s an example of how to prove congruence by the AAS rule. Suppose there are two triangles ABC and XYZ, where

∠B = ∠Y [Corresponding sides] ∠C = ∠Z [Corresponding sides] and

AC = XZ [Adjacent sides]

By angle sum property of triangle, we know that;

∠A + ∠B + ∠C = 180 degree 

∠X + ∠Y + ∠Z = 180 degree

From equations 1 and 2 we can say;

∠A + ∠B + ∠C = ∠X + ∠Y + ∠Z

∠A + ∠E + ∠F = ∠X + ∠Y + ∠Z [Since ∠B = ∠Y and ∠C = ∠Z] ∠A = ∠X

Hence, in triangles ABC and XYZ,

∠A = ∠X

AC = XZ

∠C = ∠Z

Hence, by the ASA rule of congruence,

Δ ABC ≅ Δ XYZ

1. RHS (Right angle-Hypotenuse-Side)- If the hypotenuse and a side of a right-angled triangle are equal to the hypotenuse and a side of the other right-angled triangle, then the two of them are said to be congruent by the RHS rule.

NCERT Solutions for Class 7 Mathematics

In case you found the solutions of NCERT Class 7 Mathematics Chapter 7 useful and want to continue studying Mathematics through Extramarks website, you can click on the link below to find solutions for the other chapters. 

NCERT Solutions for Class 7

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The NCERT Solutions of Extramarks for Class 7 cover all the topics of Class 7 in a detailed and comprehensive manner. With the assistance of these up-to-date solutions, studying for Class 7 is made easier.  So in case you are interested in solutions for all the subjects of Class 7, you can click on the link given below. 

NCERT Solutions Class 7 Maths Chapter-wise List