Congruence of Triangles: Ex 7.1 NCERT Solutions Class 7 Maths

Hello, future geometry whiz! Welcome to the world of Congruence of Triangles. Have you ever noticed how two 5-rupee coins are exactly the same? Or how two pages from the same notebook fit perfectly over each other? That's exactly what congruence is! It means 'identical in every way'. In this chapter, and specifically in Exercise 7.1, we will learn the magic rules to check if two triangles are exact copies without measuring every single side and angle. It’s like a detective's shortcut in geometry! By the end of this lesson, you will understand what makes triangles congruent and you'll be able to solve the problems in your NCERT book with confidence. Let's begin this exciting journey with your YoLearn AI Tutor!

What Does 'Congruent' Really Mean?

In mathematics, when we say two figures are 'congruent', we mean they are perfect duplicates of each other. Think of them as identical twins. They have the same shape and the same size. If you could pick one up and place it on top of the other, it would cover it perfectly without any parts sticking out. For triangles, this has a very specific meaning. If triangle ABC is congruent to triangle PQR, it means:

  1. All corresponding sides are equal in length: The length of side AB is equal to the length of side PQ, BC is equal to QR, and AC is equal to PR.
  2. All corresponding angles are equal in measure: The measure of angle A is equal to the measure of angle P, angle B is equal to angle Q, and angle C is equal to angle R.

We use the symbol '≅' to show congruence. So, we would write ΔABC ≅ ΔPQR. The order of the letters is very important, as it tells us exactly which side corresponds to which side, and which angle corresponds to which angle.

The Four Rules of Congruence

SSS (Side-Side-Side)
If all three sides of one triangle are equal to the corresponding three sides of another triangle, then the two triangles are congruent.
SAS (Side-Angle-Side)
If two sides and the angle between them (the included angle) of one triangle are equal to the corresponding two sides and included angle of another triangle, then the two triangles are congruent.
ASA (Angle-Side-Angle)
If two angles and the side between them (the included side) of one triangle are equal to the corresponding two angles and included side of another triangle, then the two triangles are congruent.
RHS (Right-angle-Hypotenuse-Side)
This is a special case for right-angled triangles. If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, then the two triangles are congruent.

Worked Examples: How to Prove Congruence

  • Example using SSS Rule: In ΔABC, AB = 5 cm, BC = 6 cm, AC = 4 cm. In ΔPQR, PQ = 5 cm, QR = 6 cm, PR = 4 cm. Are the two triangles congruent? Solution: 1. Compare side AB with PQ. We see AB = PQ = 5 cm. 2. Compare side BC with QR. We see BC = QR = 6 cm. 3. Compare side AC with PR. We see AC = PR = 4 cm. Since all three corresponding sides are equal, we can say that ΔABC ≅ ΔPQR by the SSS (Side-Side-Side) congruence criterion.
  • Example using SAS Rule: In ΔXYZ, XY = 7 cm, ∠Y = 50°, YZ = 5 cm. In ΔLMN, LM = 7 cm, ∠M = 50°, MN = 5 cm. Check if they are congruent. Solution: 1. Compare side XY with LM. We find XY = LM = 7 cm. (Side) 2. Compare ∠Y with ∠M. We find ∠Y = ∠M = 50°. (Angle) 3. Compare side YZ with MN. We find YZ = MN = 5 cm. (Side) 4. Crucially, check if the angle is between the two sides. In ΔXYZ, ∠Y is between sides XY and YZ. In ΔLMN, ∠M is between sides LM and MN. Yes, it is. Therefore, ΔXYZ ≅ ΔLMN by the SAS (Side-Angle-Side) congruence criterion.

Exam Tip: The Importance of Order and CPCTC

One of the most common mistakes students make is writing the congruence relation in the wrong order. Writing ΔABC ≅ ΔPQR is not the same as writing ΔABC ≅ ΔQRP!

The order of the vertices tells you exactly which parts correspond. In ΔABC ≅ ΔPQR:

  • A corresponds to P (∠A = ∠P)
  • B corresponds to Q (∠B = ∠Q)
  • C corresponds to R (∠C = ∠R)
  • Side AB corresponds to side PQ (AB = PQ)
  • Side BC corresponds to side QR (BC = QR)
  • Side AC corresponds to side PR (AC = PR)

Once you have proven that two triangles are congruent, you can state that any of their corresponding parts are also equal. This reason is called 'Corresponding Parts of Congruent Triangles are Congruent', or CPCTC for short. This is a very powerful tool you will use a lot!

Practice Questions with Solutions

  • Q: Two line segments are congruent if ________. A: Step 1: Recall the definition of congruence for line segments. Congruence means they have the same size and shape. Step 2: For a line segment, its 'size' is its length. Final answer: Two line segments are congruent if they have the same length.
  • Q: In triangles ABC and PQR, AB = 3.5 cm, BC = 7.1 cm, AC = 5 cm, PQ = 7.1 cm, QR = 5 cm and PR = 3.5 cm. State the congruence relation between the two triangles. A: Step 1: Compare the sides of ΔABC with the sides of ΔPQR to find the corresponding equal sides. AB = 3.5 cm and PR = 3.5 cm. So, AB = PR. BC = 7.1 cm and PQ = 7.1 cm. So, BC = PQ. AC = 5 cm and QR = 5 cm. So, AC = QR. Step 2: Write down the correspondence of vertices based on the equal sides. A corresponds to R, B corresponds to P, and C corresponds to Q. Step 3: Write the congruence relation using the SSS criterion, making sure the vertices are in the correct order. Final answer: ΔABC ≅ ΔRPQ (by SSS congruence rule).
  • Q: You want to show that ΔART ≅ ΔPEN. If you have to use the SSS criterion, then you need to show (i) AR = ?, (ii) RT = ?, and (iii) AT = ? A: Step 1: Understand the SSS criterion. It requires all three corresponding sides to be equal. Step 2: Look at the congruence statement ΔART ≅ ΔPEN. The order of letters tells us the corresponding parts. Step 3: Match the corresponding sides. AR is the first two letters, RT is the last two, and AT is the first and last. (i) AR corresponds to PE. (ii) RT corresponds to EN. (iii) AT corresponds to PN. Final answer: (i) AR = PE, (ii) RT = EN, and (iii) AT = PN.
  • Q: In the given figure, AC = BD and AD = BC. Which of the following statements is meaningfully correct? (i) ΔABD ≅ ΔBAC (ii) ΔABD ≅ ΔABC A: Step 1: Analyze the given information. We have two triangles, ΔABD and ΔBAC, which share a common side AB. Step 2: List the known equal parts for ΔABD and ΔBAC. Given: AD = BC (Side) Given: BD = AC (Side) Common side: AB = BA (Side) Step 3: Check which congruence criterion applies. Since three corresponding sides are equal, the SSS criterion applies. Step 4: Write the correct congruence relation by matching vertices. A corresponds to B, B corresponds to A, and D corresponds to C. Step 5: Compare this with the given options. The correct relation is ΔABD ≅ ΔBAC. Final answer: The statement (i) ΔABD ≅ ΔBAC is meaningfully correct.

Frequently Asked Questions

What is the difference between congruent and similar triangles?

Congruent triangles are identical in both shape and size. Similar triangles have the same shape but can have different sizes. You can think of a photograph and its enlargement as being similar, while two identical passport photos are congruent.

If two triangles have all three angles equal (AAA), are they congruent?

No, not necessarily. Two triangles with equal corresponding angles are 'similar', but not always congruent. One could be a larger or smaller version of the other. You need at least one side length to be equal to prove congruence.

Why is the order of letters so important when we write ΔABC ≅ ΔPQR?

The order acts as a map. It tells you exactly which angle in the first triangle is equal to which angle in the second, and which side is equal to which side. Getting the order wrong leads to incorrect conclusions about the corresponding parts.

Can we use SSA (Side-Side-Angle) to prove congruence?

No, SSA is not a valid congruence criterion. Knowing two sides and a non-included angle is not enough to guarantee that two triangles are congruent. This is a very common point of confusion, so remember to avoid it!