Class 7 Mathematics Notes Chapter 7 (Congruence of Triangles) – Mathematics Book
Alright class, let's get straight into a very important chapter for your geometry foundation, especially for competitive exams: Chapter 7 - Congruence of Triangles. Understanding this well will help you solve many geometry problems later on.
Chapter 7: Congruence of Triangles - Detailed Notes
1. What is Congruence?
- Meaning: Two figures are congruent if they have exactly the same shape and the same size. Think of them as exact copies or clones of each other.
- Superposition: If you can place one figure exactly on top of the other so that they cover each other completely, they are congruent. Imagine two identical coins or two pages from the same book.
- Symbol: The symbol for congruence is ≅. So, if figure F1 is congruent to figure F2, we write F1 ≅ F2.
2. Congruence among Plane Figures:
- Any two plane figures (shapes drawn on a flat surface) are congruent if they are identical in shape and size.
3. Congruence among Line Segments:
- Two line segments are congruent if they have the same length.
- If line segment AB has length 5 cm and line segment PQ also has length 5 cm, then AB ≅ PQ.
4. Congruence among Angles:
- Two angles are congruent if they have the same measure (in degrees).
- If ∠ABC = 60° and ∠XYZ = 60°, then ∠ABC ≅ ∠XYZ. The length of the arms of the angles does not matter for congruence.
5. Congruence of Triangles (The Main Focus):
- Definition: Two triangles are congruent if all their corresponding sides are equal in length, and all their corresponding angles are equal in measure.
- Correspondence: When we say two triangles are congruent, there's a specific one-to-one matching (correspondence) between their vertices, sides, and angles.
- If we write ΔABC ≅ ΔPQR, this specific notation implies the following correspondence:
- Vertices: A ↔ P, B ↔ Q, C ↔ R (A corresponds to P, B to Q, C to R)
- Sides: AB ↔ PQ, BC ↔ QR, CA ↔ RP (Side AB corresponds to PQ, etc.)
- Angles: ∠A ↔ ∠P, ∠B ↔ ∠Q, ∠C ↔ ∠R (Angle A corresponds to P, etc.)
- If we write ΔABC ≅ ΔPQR, this specific notation implies the following correspondence:
- CPCTC: This is a very important abbreviation: Corresponding Parts of Congruent Triangles are Congruent (or equal). If you have proved that two triangles are congruent, you can then state that any pair of their corresponding sides or angles are equal.
6. Criteria for Congruence of Triangles:
We don't need to check all 6 pairs of corresponding parts (3 sides, 3 angles) to prove congruence. There are shortcut rules or criteria:
-
(i) SSS (Side-Side-Side) Congruence Criterion:
- Rule: If the 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).
- Diagram:
A P / \ / \ / \ / \ B-----C Q-----R (AB=PQ, BC=QR, AC=PR)
-
(ii) SAS (Side-Angle-Side) Congruence Criterion:
- Rule: If two sides and the included angle (the angle between the two sides) of one triangle are equal to the two corresponding sides and the included angle of another triangle, then the two triangles are congruent.
- Important: The angle must be the one formed by the two sides being considered. SSA or ASS is not a congruence criterion.
- Example: If in ΔLMN and ΔXYZ, LM = XY, MN = YZ, and ∠M = ∠Y (angle included between LM, MN and XY, YZ respectively), then ΔLMN ≅ ΔXYZ (by SAS).
- Diagram:
L X / \ / \ / M \ / Y \ /_____\ /_____\
N Z
(LM=XY, Angle M=Angle Y, MN=YZ)
``` -
(iii) ASA (Angle-Side-Angle) Congruence Criterion:
- Rule: If two angles and the included side (the side between the two angles) of one triangle are equal to the two corresponding angles and the included side of another triangle, then the two triangles are congruent.
- Important: The side must be the one connecting the vertices of the two angles being considered.
- Example: If in ΔABC and ΔDEF, ∠B = ∠E, BC = EF, and ∠C = ∠F (side BC is included between ∠B, ∠C and side EF is included between ∠E, ∠F), then ΔABC ≅ ΔDEF (by ASA).
- Diagram:
A D / \ / \ / \ / \ B-----C E-----F (Angle B=Angle E, BC=EF, Angle C=Angle F) - Note on AAS: Sometimes you might see AAS (Angle-Angle-Side). If two angles and a non-included side are equal, the triangles are still congruent. This is because if two angles are equal, the third angle must also be equal (Angle Sum Property: 180°). Then you can use ASA. NCERT Class 7 focuses on ASA, but knowing AAS exists is helpful.
-
(iv) RHS (Right angle-Hypotenuse-Side) Congruence Criterion:
- Rule: 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 triangles are congruent.
- Components:
- R: Both triangles must have a Right angle (90°).
- H: The Hypotenuses (side opposite the right angle) must be equal.
- S: One pair of corresponding sides (other than the hypotenuse) must be equal.
- Example: If ΔPQR and ΔXYZ are right-angled at Q and Y respectively, and hypotenuse PR = XZ, and side QR = YZ, then ΔPQR ≅ ΔXYZ (by RHS).
- Diagram:
P X |\ |\ | \ | \ | \ | \ |___\ |___\ Q R Y Z (Angle Q = Angle Y = 90 deg, Hyp PR = Hyp XZ, Side QR = Side YZ)
Key Takeaways for Exams:
- Understand the meaning of Congruence (Same Shape, Same Size).
- Memorize the four criteria: SSS, SAS, ASA, RHS.
- Pay close attention to the conditions for SAS (included angle) and ASA (included side).
- Remember RHS is only for right-angled triangles.
- Understand the importance of correct correspondence when writing congruence statements (ΔABC ≅ ΔPQR).
- Know how to use CPCTC after proving congruence.
Multiple Choice Questions (MCQs)
-
Two plane figures are congruent if they have:
a) Same shape
b) Same size
c) Same shape and same size
d) Same perimeter -
If ΔABC ≅ ΔPQR, which of the following is NOT true?
a) BC = QR
b) AC = PR
c) AB = PQ
d) BC = PQ -
In ΔPQR and ΔXYZ, PQ = XY, QR = YZ, RP = ZX. By which congruence criterion are the two triangles congruent?
a) SAS
b) ASA
c) SSS
d) RHS -
In ΔDEF and ΔLMN, DE = LM, EF = MN, and ∠E = ∠M. Which congruence criterion applies?
a) SSS
b) SAS
c) ASA
d) RHS -
For ΔABC and ΔPQR to be congruent by ASA criterion, if ∠B = ∠Q and ∠C = ∠R, which sides must be equal?
a) AB = PQ
b) AC = PR
c) BC = QR
d) AB = QR -
Which congruence criterion is applicable only for right-angled triangles?
a) SSS
b) SAS
c) ASA
d) RHS -
If ΔGHI ≅ ΔJKL by RHS criterion, which angle must be 90°?
a) ∠G or ∠J
b) ∠H or ∠K
c) ∠I or ∠L
d) Cannot be determined without knowing the correspondence of right angles. -
Given ΔXYZ ≅ ΔRST. If XY = 5 cm, ∠Y = 60°, and YZ = 4 cm, what are the values for RS and ∠S?
a) RS = 5 cm, ∠S = 60°
b) RS = 4 cm, ∠S = 60°
c) RS = 5 cm, ∠S cannot be determined
d) RS = 4 cm, ∠S cannot be determined -
In ΔABC, AB = AC and AD is the angle bisector of ∠A, meeting BC at D. Which criterion can be used to prove ΔABD ≅ ΔACD?
a) SSS
b) SAS
c) ASA
d) RHS -
Two triangles are congruent if two angles and the side included between them in one triangle are equal to the two corresponding angles and the side included between them of the other triangle. This is the:
a) SSS criterion
b) SAS criterion
c) ASA criterion
d) RHS criterion
Answer Key for MCQs:
- c) Same shape and same size
- d) BC = PQ (The correspondence is B↔Q, C↔R, so BC must correspond to QR)
- c) SSS
- b) SAS (Angle E is between sides DE and EF; Angle M is between sides LM and MN)
- c) BC = QR (The side must be included between the angles B, C and Q, R)
- d) RHS
- d) Cannot be determined without knowing the correspondence of right angles. (We need to know which vertex corresponds to the right angle, e.g., if H and K are right angles).
- a) RS = 5 cm, ∠S = 60° (XY corresponds to RS, ∠Y corresponds to ∠S by CPCTC)
- b) SAS (AB=AC (given), ∠BAD = ∠CAD (AD bisects ∠A), AD=AD (common side). The angle is included between the sides.)
- c) ASA criterion
Study these notes carefully, practice identifying the criteria in different diagrams, and you'll master this topic. Good luck with your preparation!