Vardaan
Class 7 Maths • Chapter 07

Congruence of Triangles

Vardaan Learning Institute • Detailed Notes with Practice Questions

🔎 1. What is Congruence?

Two figures are congruent (≅) if they have the same shape AND same size. When superimposed (placed on top of each other), they match exactly.

Congruent vs. Similar:
• Congruent (≅): Same shape + SAME size. Example: Two identical stamps.
• Similar (~): Same shape, DIFFERENT size (covered in higher classes). Example: A photo and its enlargement.
Corresponding Parts: When two triangles are congruent, their CPCT hold true:
CPCT = Corresponding Parts of Congruent Triangles (are equal)
If △ABC ≅ △PQR, then: AB=PQ, BC=QR, AC=PR and ∠A=∠P, ∠B=∠Q, ∠C=∠R

📋 2. Criteria for Congruence of Triangles

SSS
Side-Side-Side

All three sides of one triangle equal to corresponding three sides of the other triangle. No angle needed!

SAS
Side-Angle-Side

Two sides AND the included angle (the angle between the two sides) are equal. Angle must be BETWEEN the two sides!

ASA
Angle-Side-Angle

Two angles AND the included side (side between the two angles) are equal.

AAS
Angle-Angle-Side

Two angles and a non-included side are equal. (Since 3rd angle can be derived, this works!)

RHS
Right Angle-Hypotenuse-Side

Only for RIGHT triangles: Right angle + Hypotenuse + one leg are equal.

⚠️ What does NOT prove congruence • AAA (Angle-Angle-Angle) — proves SIMILARITY only, not congruence! (Triangles could be different sizes)
• SSA (Side-Side-Angle) — DOES NOT work! The angle must be included (SAS) or hypotenuse (RHS) for right triangles.

✏️ 3. Worked Examples

Example 1: In △ABC and △DEF: AB=DE=5cm, BC=EF=7cm, AC=DF=9cm. Are they congruent?
All three sides are equal. So by SSS criterion, △ABC ≅ △DEF. By CPCT, ∠A=∠D, ∠B=∠E, ∠C=∠F.

Example 2: In △PQR and △XYZ: PQ=XY=6cm, ∠P=∠X=40°, PR=XZ=8cm. Are they congruent?
Two sides and the included angle (∠P is between PQ and PR). By SAS criterion, △PQR ≅ △XYZ.

Example 3: In △ABC, AB=AC (isosceles) and AD bisects ∠A. Prove △ABD ≅ △ACD.
In △ABD and △ACD: AB=AC (given), ∠BAD=∠CAD (AD bisects ∠A), AD=AD (common side).
→ By SAS, △ABD ≅ △ACD. So BD=CD (CPCT) — AD bisects BC! (Median and angle bisector coincide in isosceles triangle!)
Criterion What you need Key memory tip
SSS All 3 sides equal 3 sides → definitely congruent
SAS 2 sides + INCLUDED angle Sandwich angle between the 2 sides
ASA 2 angles + INCLUDED side Sandwich side between the 2 angles
AAS 2 angles + any one side If 2 angles known, 3rd can be found → then use ASA logic
RHS Right ∠ + hypotenuse + one side ONLY for right triangles
AAA 3 angles ❌ NOT sufficient (gives similar, not congruent)
SSA 2 sides + non-included angle ❌ NOT sufficient (ambiguous case)
✏️ Practice Questions — Congruence of Triangles (26 Questions)

Section A — Identify the Criterion Easy

  1. Q1. △ABC and △DEF have AB=DE, BC=EF, ∠B=∠E. Which criterion of congruence applies?
  2. Q2. △PQR and △XYZ have ∠P=∠X, ∠Q=∠Y, QR=YZ. Which criterion?
  3. Q3. Right △ABC (right angle at B) and right △PQR (right angle at Q), hypotenuse AC = PR, BC = QR. Criterion?
  4. Q4. △LMN has LM=LN=7cm and △PQR has PQ=PR=7cm, and MN=QR=10cm. Criterion?
  5. Q5. △ABC and △FGH: ∠A=∠F=60°, ∠B=∠G=70°, AB=FG. Criterion?
  6. Q6. Is it possible to conclude congruence from only: ∠A=∠P, ∠B=∠Q, ∠C=∠R? Why or why not?

Section B — Prove Congruence Medium

  1. Q7. In quadrilateral ABCD, AB=AD and BC=DC. Prove △ABC ≅ △ADC.
  2. Q8. In isosceles △ABC where AB=AC, D is midpoint of BC. Prove △ABD ≅ △ACD. What can you conclude about AD?
  3. Q9. Two triangles share a common side PQ. △PQR and △PQS lie on opposite sides of PQ with PR=PS and QR=QS. Prove △PQR ≅ △PQS.
  4. Q10. In △ABC, ∠B=∠C and D is the midpoint of BC. Prove △ABD ≅ △ACD.
  5. Q11. In the figure, two lines PQ and RS bisect each other at O. Prove △POR ≅ △QOS.
  6. Q12. PA and PB are two tangents drawn from external point P to a circle with centre O. OA⊥PA and OB⊥PB, OA=OB. Prove △OAP ≅ △OBP.

Section C — Find Unknown Values Medium

Q13. △ABC ≅ △PQR. AB=3x+1, PQ=4x-2. Find x and AB.
Q14. △DEF ≅ △KLM. ∠E = 5y+10, ∠L = 3y+30. Find y and both angles.
Q15. If △ABC ≅ △DEF and perimeter of △ABC = 24 cm, find perimeter of △DEF.
Q16. △PQR ≅ △STU. PQ=5cm, QR=7cm, PR=9cm. Find ST+TU+SU.

Section D — Higher Order Hard

  1. Q17. In △ABC, altitudes BD and CE are drawn to sides AC and AB respectively. If BD=CE, prove that △BCD ≅ △CBE. Hence prove △ABC is isosceles.
  2. Q18. In a quadrilateral PQRS, PQ=RS and PS=QR. Prove that △PQS ≅ △RQS. What type of quadrilateral is PQRS?
  3. Q19. ABCD is a square. Prove that △ABD ≅ △CBD and hence show diagonal BD bisects ∠B and ∠D.
  4. Q20. In △PQR, PQ=PR. If S is any point on QR, prove that ∠PSQ + ∠PSR = 180°. (Hint: Use properties of isosceles triangle.)
  5. Q21. Two right triangles have hypotenuse 10 cm each and one leg 6 cm each. Are they necessarily congruent? Which criterion?
  6. Q22. In △ABC, AB=AC. A point D is on AC such that BD=BC. Prove △ABD ≅ △ABC? Will this work? Why or why not?
  7. Q23. ABCD is a rectangle. Diagonal AC = diagonal BD. Prove △ABC ≅ △DCB.
  8. Q24. In △XYZ, XY=XZ. Perpendicular XM is drawn to YZ. Prove YM=MZ using congruence. What do you conclude?
  9. Q25. △OAB and △OCD are congruent with OA=OC and OB=OD. Prove that AB=CD.
  10. Q26. ABC is a triangle right-angled at C. D is the midpoint of hypotenuse AB. Prove CD = (1/2)AB using coordinates or congruence.
✅ Key Answers: Q1: SAS | Q2: AAS | Q3: RHS | Q4: SSS | Q5: ASA | Q6: No — only AAA, not sufficient | Q13: x=3, AB=10cm | Q14: y=10, both 60° | Q15: 24cm | Q16: 21cm | Q21: Yes, by RHS criterion

📝 Quick Revision

  1. Congruent figures: same shape AND same size (≅ symbol); CPCT — corresponding parts of congruent triangles are equal
  2. SSS: 3 sides equal | SAS: 2 sides + included (between them) angle equal
  3. ASA: 2 angles + included (between them) side equal | AAS: 2 angles + any side
  4. RHS: Right angle + Hypotenuse + one Side (for right triangles only)
  5. ❌ AAA proves similarity, NOT congruence | ❌ SSA is not a congruence criterion
  6. In proof: State given, state what we need to prove, write "In △___ and △___:", list equal parts, state which criterion, conclude congruence, then use CPCT if needed