Congruence of Triangles Ex 7.2 - Class 7 Maths NCERT

Welcome, Class 7 students! In our daily lives, we often encounter objects that are exactly the same in shape and size, like two identical ₹5 coins or two pages from the same book. In geometry, we call such figures "congruent." This chapter on Congruence of Triangles is super important because it teaches us how to determine if two triangles are identical without having to measure every single side and angle. Specifically, Exercise 7.2 introduces powerful rules, called congruence criteria (like SSS, SAS, ASA, and RHS), that act as shortcuts to prove if two triangles are congruent.

Mastering these criteria will build a strong foundation for advanced geometry. By the end of this page, you'll not only understand what each criterion means but also how to apply them step-by-step to solve problems, ensuring you're fully prepared for your exams and future mathematical challenges. Let's dive in and make geometry fun!

What is Congruence of Triangles?

Two geometric figures are said to be congruent if they have exactly the same shape and the same size. Imagine placing one figure perfectly on top of the other; if they match exactly, they are congruent. For triangles, this means that all six corresponding parts (three sides and three angles) of one triangle must be equal to the corresponding six parts of the other triangle. We use the symbol "≅" to denote congruence. So, if triangle ABC is congruent to triangle PQR, we write ΔABC ≅ ΔPQR.

It's absolutely crucial to understand corresponding parts. When ΔABC ≅ ΔPQR, it means:

  • Side AB corresponds to side PQ (AB = PQ)
  • Side BC corresponds to side QR (BC = QR)
  • Side CA corresponds to side RP (CA = RP)
  • Angle A corresponds to Angle P (∠A = ∠P)
  • Angle B corresponds to Angle Q (∠B = ∠Q)
  • Angle C corresponds to Angle R (∠C = ∠R)

Thinking about which part corresponds to which is the first step in correctly applying congruence criteria. Always try to visualise rotating or flipping one triangle to match the other. If you correctly identify corresponding vertices, the corresponding sides and angles will naturally follow. For instance, if A maps to P, B to Q, and C to R, then side AB corresponds to PQ, not QR.

The Four Main Congruence Criteria

Measuring all six parts of two triangles every time to check for congruence can be time-consuming. Thankfully, mathematicians have discovered shortcuts – specific sets of conditions, called congruence criteria, that are enough to prove two triangles are congruent. For Class 7, we focus on four key criteria:

  1. SSS (Side-Side-Side) Congruence Criterion:

If three sides of one triangle are 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).
  1. SAS (Side-Angle-Side) Congruence Criterion:

If two sides and the included angle (the angle formed between those two sides) of one triangle are 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). Notice that ∠B is between sides AB and BC, and ∠Q is between sides PQ and QR.
  1. ASA (Angle-Side-Angle) Congruence Criterion:

If two angles and the included side (the side common to both angles) of one triangle are 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). Here, side BC is included between ∠B and ∠C, and side QR is included between ∠Q and ∠R.
  1. RHS (Right-angle-Hypotenuse-Side) Congruence Criterion:

This criterion is specific to right-angled triangles. If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, then the two right-angled triangles are congruent.

  • Example: If in right-angled ΔABC (at B) and right-angled ΔPQR (at Q), AC = PR (hypotenuses) and BC = QR (a corresponding side), then ΔABC ≅ ΔPQR (by RHS).

It's important to remember that AAA (Angle-Angle-Angle) and SSA (Side-Side-Angle) are not congruence criteria. Triangles with equal angles can have different sizes (similar triangles), and SSA can result in two different triangles.

Applying Congruence Criteria: Step-by-Step Examples

  1. Example 1: Using SSS Criterion — Given two triangles, ΔABC and ΔDEF, where AB = 5 cm, BC = 6 cm, CA = 7 cm, and DE = 5 cm, EF = 6 cm, FD = 7 cm. Prove that ΔABC ≅ ΔDEF. Step 1: Identify corresponding sides. We are given: AB = 5 cm and DE = 5 cm. So, AB = DE. We are given: BC = 6 cm and EF = 6 cm. So, BC = EF. We are given: CA = 7 cm and FD = 7 cm. So, CA = FD. Step 2: Check which congruence criterion applies. Since all three corresponding sides of ΔABC are equal to the three corresponding sides of ΔDEF, the SSS criterion applies. Step 3: State the congruence. Therefore, ΔABC ≅ ΔDEF (by SSS congruence criterion).
  2. Example 2: Using SAS Criterion — In the figure, PQ = ST, ∠P = ∠S, and PR = SU. Are ΔPQR and ΔSTU congruent? If yes, by which criterion? Step 1: List the given equal parts. We are given: PQ = ST (Side) We are given: ∠P = ∠S (Angle) We are given: PR = SU (Side) Step 2: Check if the angle is included between the sides. In ΔPQR, the angle ∠P is included between sides PQ and PR. In ΔSTU, the angle ∠S is included between sides ST and SU. Since the angle is included between the two equal sides in both triangles, the SAS criterion applies. Step 3: State the congruence. Yes, ΔPQR ≅ ΔSTU (by SAS congruence criterion).
  3. Example 3: Using RHS Criterion — Given two right-angled triangles, ΔXYZ and ΔMNO, where ∠Y = ∠N = 90°, XZ = MO (hypotenuses), and XY = MN. Prove that ΔXYZ ≅ ΔMNO. Step 1: Identify the type of triangles and equal parts. Both are right-angled triangles, with ∠Y = ∠N = 90°. We are given: XZ = MO (Hypotenuse) We are given: XY = MN (Side) Step 2: Check if the RHS criterion applies. Since both are right-angled triangles, their hypotenuses are equal, and one pair of corresponding sides are equal, the RHS criterion applies. Step 3: State the congruence. Therefore, ΔXYZ ≅ ΔMNO (by RHS congruence criterion).

Exam Tips and Common Mistakes

To score full marks in congruence problems, avoid these common pitfalls and follow these tips:

  1. Correctly Identifying Corresponding Parts: This is the most frequent mistake. If ΔABC ≅ ΔPQR, it means A corresponds to P, B to Q, and C to R. Ensure that the order of vertices in your congruence statement matches the corresponding parts. For example, if AB = QR, you might need to write ΔABC ≅ ΔRQP instead of ΔABC ≅ ΔPQR.
  1. "Included" Angle/Side for SAS and ASA: Always double-check if the given angle is between the two given sides for SAS, or if the given side is between the two given angles for ASA. If it's not the included part, these criteria cannot be used directly.
  1. Don't Use AAA or SSA: Remember, Angle-Angle-Angle (AAA) only proves similarity (same shape, different size), not congruence. Side-Side-Angle (SSA) is generally not a valid congruence criterion because it can lead to ambiguous cases where two different triangles can be formed with the same given SSA measurements.
  1. Specify the Criterion: When writing your proof, always clearly state which congruence criterion (SSS, SAS, ASA, or RHS) you are using to conclude congruence. For example, "ΔABC ≅ ΔPQR (by SAS criterion)."
  1. RHS is ONLY for Right-Angled Triangles: Do not attempt to use the RHS criterion if the triangles are not explicitly stated or shown to be right-angled. The 'R' stands for 'Right-angle' for a reason!

Practice Questions with Solutions

  • Q: In ΔABC, AB = AC. D is the midpoint of BC. Show that ΔABD ≅ ΔACD. A: Step 1: List the given information. We are given: AB = AC (Given) We are given: D is the midpoint of BC, which implies BD = CD (Definition of midpoint) We also have: AD = AD (Common side to both triangles) Step 2: Identify the congruence criterion. Since all three sides of ΔABD are equal to the corresponding three sides of ΔACD (AB=AC, BD=CD, AD=AD), the SSS congruence criterion applies. Step 3: State the conclusion. Therefore, ΔABD ≅ ΔACD (by SSS congruence criterion).
  • Q: Given that ΔXYZ and ΔPQR are two triangles such that XY = PQ, ∠Y = ∠Q, and YZ = QR. Are the triangles congruent? If so, by which criterion? A: Step 1: List the given equal parts. We are given: XY = PQ (Side) We are given: ∠Y = ∠Q (Angle) We are given: YZ = QR (Side) Step 2: Check if the angle is included. In ΔXYZ, ∠Y is the angle included between sides XY and YZ. In ΔPQR, ∠Q is the angle included between sides PQ and QR. Since the included angles are equal and the two corresponding sides are equal, the SAS criterion applies. Step 3: State the conclusion. Yes, ΔXYZ ≅ ΔPQR (by SAS congruence criterion).
  • Q: In triangles DEF and GHI, it is given that ∠E = ∠H, EF = HI, and ∠F = ∠I. Are these triangles congruent? If yes, by what rule? A: Step 1: List the given equal parts. We are given: ∠E = ∠H (Angle) We are given: EF = HI (Side) We are given: ∠F = ∠I (Angle) Step 2: Check if the side is included. In ΔDEF, side EF is included between angles ∠E and ∠F. In ΔGHI, side HI is included between angles ∠H and ∠I. Since the included sides are equal and the two corresponding angles are equal, the ASA criterion applies. Step 3: State the conclusion. Yes, ΔDEF ≅ ΔGHI (by ASA congruence criterion).
  • Q: In right-angled triangles ABC (right-angled at B) and PQR (right-angled at Q), it is given that AC = PR and BC = QR. Are ΔABC and ΔPQR congruent? A: Step 1: List the given information. Both triangles are right-angled at B and Q, respectively (∠B = ∠Q = 90°). We are given: AC = PR (Hypotenuse) We are given: BC = QR (Side) Step 2: Identify the congruence criterion. Since both are right-angled triangles, their hypotenuses are equal (AC=PR), and one pair of corresponding sides are equal (BC=QR), the RHS congruence criterion applies. Step 3: State the conclusion. Therefore, ΔABC ≅ ΔPQR (by RHS congruence criterion).

Frequently Asked Questions

What does it mean for two triangles to be congruent?

Two triangles are congruent if they are exactly the same in both shape and size. This means that if you could pick up one triangle, you could place it perfectly on top of the other, and all their corresponding parts (sides and angles) would match exactly.

Why are SSS, SAS, ASA, and RHS called congruence criteria?

These are called congruence criteria because they provide specific, minimal sets of conditions that are sufficient to prove that two triangles are congruent. Instead of checking all six parts, checking just three specific parts (like two sides and the included angle for SAS) is enough to guarantee congruence, saving time and effort.

Can I use AAA (Angle-Angle-Angle) to prove that two triangles are congruent?

No, AAA is not a congruence criterion. If three angles of one triangle are equal to three angles of another triangle, the triangles are only guaranteed to be similar (same shape, but possibly different sizes). For example, a small equilateral triangle and a large equilateral triangle both have 60-degree angles, but they are not congruent.

What is the most important thing to remember when using SAS or ASA congruence criteria?

The most important thing to remember is the concept of the "included" part. For SAS, the angle MUST be between the two given sides. For ASA, the side MUST be between the two given angles. If the angle or side is not included, the criterion cannot be applied directly.