Congruence Of Triangles: Class 7 Maths NCERT

Have you ever seen two identical photos, two stamps from the same sheet, or two coins of the same denomination? They look exactly the same – same size, same shape! In mathematics, when figures have exactly the same size and the same shape, we say they are 'congruent'. This concept is super important, especially when we talk about triangles. In Class 7 Maths, you'll dive deep into the fascinating world of congruence, specifically focusing on triangles. You'll learn how to identify if two triangles are mirror images of each other, not just visually, but using precise mathematical rules. By the end of this chapter, you'll be able to confidently determine if two triangles are congruent using different criteria, which will be a powerful tool for solving many geometry problems. Get ready to match shapes like a pro!

What is Congruence?

Imagine you have two pieces of paper, and you cut out two identical circles. If you place one circle exactly on top of the other, they will perfectly cover each other, right? This perfect match is what we call congruence in geometry. Two figures are said to be congruent if they have the exact same shape and the exact same size. Think of it like a photocopy – the original and the copy are congruent. If you have two coins of ₹5, they are congruent. If you have two biscuit packets of the same brand and size, they are congruent. Congruence isn't just about looking similar; it means every part corresponds perfectly. For example, if two figures are congruent, then their corresponding sides are equal in length, and their corresponding angles are equal in measure. This fundamental idea forms the basis for understanding many geometric properties and relationships, allowing us to compare and relate different shapes in a precise mathematical way. It helps us determine if two structures are identical, which is crucial in engineering and design.

Congruence of Plane Figures, Line Segments, and Angles

The idea of congruence isn't limited to just triangles. It applies to any plane figure. For example, two squares are congruent if their side lengths are equal. Two circles are congruent if their radii are equal.

Let's look at some simpler figures first:

  1. Congruence of Line Segments: Two line segments are congruent if they have the same length. For example, if line segment AB is 5 cm long and line segment CD is also 5 cm long, then AB is congruent to CD. We write this as AB ≅ CD. The symbol '≅' means 'is congruent to'.
  1. Congruence of Angles: Two angles are congruent if they have the same measure. If angle PQR measures 60° and angle XYZ also measures 60°, then ∠PQR is congruent to ∠XYZ. We write this as ∠PQR ≅ ∠XYZ. The orientation or position of the angles doesn't matter, only their measure.

These basic ideas extend to more complex shapes. When we say two triangles are congruent, it means all their corresponding parts – sides and angles – are individually congruent.

Understanding Congruence of Triangles

Now let's focus on our main topic: Congruence of Triangles. Two triangles are congruent if they can be made to superimpose (fit perfectly) on each other. This means that all corresponding sides and all corresponding angles are equal.

If triangle ABC is congruent to triangle PQR (written as ΔABC ≅ ΔPQR), it implies the following corresponding parts are equal:

  • Corresponding Sides:
  • Side AB = Side PQ
  • Side BC = Side QR
  • Side CA = Side RP
  • Corresponding Angles:
  • Angle A = Angle P
  • Angle B = Angle Q
  • Angle C = Angle R

It's very important to note the order of the vertices when writing a congruence statement. If ΔABC ≅ ΔPQR, it means that vertex A corresponds to P, B to Q, and C to R. If you write ΔABC ≅ ΔQPR, it would imply a different correspondence (A to Q, B to P, C to R), which might be incorrect unless that's how they truly correspond. Understanding this correspondence is key to correctly applying congruence criteria and solving problems.

Criteria for Congruence of Triangles

  1. 1. SSS Congruence Criterion (Side-Side-Side) — If three sides of one triangle are respectively equal to the three corresponding sides of another triangle, then the two triangles are congruent. Example: If in ΔABC and ΔPQR: AB = PQ, BC = QR, and CA = RP, then ΔABC ≅ ΔPQR by SSS criterion.
  2. 2. SAS Congruence Criterion (Side-Angle-Side) — If two sides and the included angle (the angle between those two sides) of one triangle are respectively equal to two corresponding sides and the included angle of another triangle, then the two triangles are congruent. Example: If in ΔABC and ΔPQR: AB = PQ, ∠B = ∠Q, and BC = QR, then ΔABC ≅ ΔPQR by SAS criterion. Notice ∠B is between AB and BC.
  3. 3. ASA Congruence Criterion (Angle-Side-Angle) — If two angles and the included side (the side between those two angles) of one triangle are respectively equal to two corresponding angles and the included side of another triangle, then the two triangles are congruent. Example: If in ΔABC and ΔPQR: ∠B = ∠Q, BC = QR, and ∠C = ∠R, then ΔABC ≅ ΔPQR by ASA criterion. Notice BC is between ∠B and ∠C.
  4. 4. RHS Congruence Criterion (Right-angle-Hypotenuse-Side) — If in two right-angled triangles, the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one corresponding side of the other triangle, then the two triangles are congruent. Example: If in right-angled ΔABC (with ∠B = 90°) and right-angled ΔPQR (with ∠Q = 90°): Hypotenuse AC = Hypotenuse PR, and side BC = side QR, then ΔABC ≅ ΔPQR by RHS criterion. (Or AB = PQ instead of BC = QR).

Solved Examples on Congruence

  • Example 1: Using SSS In ΔABC, AB = 4 cm, BC = 5 cm, AC = 6 cm. In ΔPQR, PQ = 4 cm, QR = 5 cm, PR = 6 cm. Are the triangles congruent? Solution: We are given: AB = PQ (4 cm) BC = QR (5 cm) AC = PR (6 cm) Since all three corresponding sides are equal, by the SSS Congruence Criterion, ΔABC ≅ ΔPQR.
  • Example 2: Using SAS In ΔXYZ and ΔDEF, XY = 7 cm, ∠Y = 70°, YZ = 5 cm. In ΔDEF, DE = 7 cm, ∠E = 70°, EF = 5 cm. Are they congruent? Solution: We are given: XY = DE (7 cm) ∠Y = ∠E (70°) YZ = EF (5 cm) Here, two sides and the included angle are equal. So, by the SAS Congruence Criterion, ΔXYZ ≅ ΔDEF.
  • Example 3: Using ASA In ΔMNO and ΔPQR, ∠N = 50°, NO = 8 cm, ∠O = 60°. In ΔPQR, ∠Q = 50°, QR = 8 cm, ∠R = 60°. Are they congruent? Solution: We are given: ∠N = ∠Q (50°) NO = QR (8 cm) ∠O = ∠R (60°) Here, two angles and the included side are equal. So, by the ASA Congruence Criterion, ΔMNO ≅ ΔPQR.
  • Example 4: Using RHS In right-angled ΔGHI (at H) and ΔJKL (at K), GI = 10 cm, HI = 6 cm. In ΔJKL, JL = 10 cm, KL = 6 cm. Are they congruent? Solution: We are given: ∠H = ∠K = 90° (Both are right-angled triangles) Hypotenuse GI = Hypotenuse JL (10 cm) Side HI = Side KL (6 cm) Since it's a right-angled triangle, the hypotenuse and one corresponding side are equal. So, by the RHS Congruence Criterion, ΔGHI ≅ ΔJKL.

YoLearn Exam Tip: Avoid Common Mistakes!

When proving congruence, make sure you always state the specific congruence criterion (SSS, SAS, ASA, or RHS) you are using. Also, pay very close attention to the correspondence of vertices. If you say ΔABC ≅ ΔPQR, it explicitly means A corresponds to P, B to Q, and C to R. Mixing up the order can lead to incorrect conclusions, especially when identifying corresponding parts (CPCTC - Corresponding Parts of Congruent Triangles are Congruent). Double-check which side is included between two angles (for ASA) or which angle is included between two sides (for SAS) to avoid common errors. Practice drawing figures to visualize the given information clearly.

Practice Questions with Solutions

  • Q: In ΔXYZ and ΔPQR, XY = 5 cm, YZ = 6 cm, ZX = 7 cm. If PQ = 5 cm, QR = 6 cm, RP = 7 cm, are ΔXYZ and ΔPQR congruent? State the criterion. A: Step 1: Compare the corresponding sides of ΔXYZ and ΔPQR. We have XY = PQ = 5 cm. YZ = QR = 6 cm. ZX = RP = 7 cm. Step 2: Identify the congruence criterion. Since all three corresponding sides are equal, the SSS (Side-Side-Side) congruence criterion applies. Final answer: Yes, ΔXYZ ≅ ΔPQR by the SSS Congruence Criterion.
  • Q: Given ΔABC and ΔDEF. If AB = DE, ∠B = ∠E, and BC = EF, state the congruence criterion that proves ΔABC ≅ ΔDEF. A: Step 1: Analyze the given equal parts. We have two sides (AB, BC) and the angle (∠B) included between them for ΔABC. Similarly, for ΔDEF, we have two sides (DE, EF) and the angle (∠E) included between them. Step 2: Identify the congruence criterion. Since two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, the SAS (Side-Angle-Side) congruence criterion is used. Final answer: ΔABC ≅ ΔDEF by the SAS Congruence Criterion.
  • Q: In ΔLMN and ΔSTU, ∠L = 40°, LM = 8 cm, ∠M = 70°. In ΔSTU, ∠S = 40°, ST = 8 cm, ∠T = 70°. Are the triangles congruent? Explain. A: Step 1: Check the given information for ΔLMN: ∠L = 40°, LM = 8 cm, ∠M = 70°. Step 2: Check the given information for ΔSTU: ∠S = 40°, ST = 8 cm, ∠T = 70°. Step 3: Compare corresponding parts. We have ∠L = ∠S (40°), LM = ST (8 cm), and ∠M = ∠T (70°). Here, the side LM (or ST) is included between the angles ∠L and ∠M (or ∠S and ∠T). Step 4: Identify the congruence criterion. Since two angles and the included side are equal, the ASA (Angle-Side-Angle) congruence criterion applies. Final answer: Yes, ΔLMN ≅ ΔSTU by the ASA Congruence Criterion.
  • Q: ΔPQR is a right-angled triangle with ∠Q = 90°. ΔXYZ is a right-angled triangle with ∠Y = 90°. If hypotenuse PR = hypotenuse XZ and side QR = side YZ, prove that ΔPQR ≅ ΔXYZ. A: Step 1: Identify the type of triangles. Both are right-angled triangles. Step 2: List the given equal parts: ∠Q = ∠Y = 90° (Right angles) PR = XZ (Hypotenuses) QR = YZ (One corresponding side) Step 3: Identify the congruence criterion for right-angled triangles. Since the right angle, the hypotenuse, and one side of ΔPQR are equal to the corresponding parts of ΔXYZ, the RHS (Right-angle-Hypotenuse-Side) congruence criterion applies. Final answer: ΔPQR ≅ ΔXYZ by the RHS Congruence Criterion.

Frequently Asked Questions

What does "congruent" mean in geometry?

Congruent means having exactly the same size and the same shape. If two figures are congruent, one can be perfectly superimposed on the other, meaning they would completely cover each other without any part sticking out.

Why is the order of vertices important when writing congruence statements (e.g., ΔABC ≅ ΔPQR)?

The order of vertices indicates the correspondence between the parts of the two triangles. If ΔABC ≅ ΔPQR, it means vertex A corresponds to P, B to Q, and C to R, and therefore, corresponding sides and angles are equal in that specific order (e.g., AB = PQ, ∠A = ∠P).

Can two triangles be congruent if only their angles are equal?

No, just having equal angles (AAA similarity criterion) only guarantees that the triangles are similar, meaning they have the same shape but not necessarily the same size. For congruence, both shape and size must be identical, which requires at least one corresponding side to be equal as well (like in ASA or AAS criteria).

Is there an SSA congruence criterion?

No, there is no general SSA (Side-Side-Angle) congruence criterion. This is because if you have two sides and a non-included angle, it's possible to construct two different triangles that fit the criteria but are not congruent. The only exception is for right-angled triangles, which is covered by the RHS criterion.