Chapter focus
Congruent figures have the same shape and size. Matching corresponding sides and angles lets us establish when two triangles are exact copies, even if they are turned or flipped.
These original explanations support the named Ganita Prakash Part II chapter. Use your prescribed book for its complete construction and investigation sequence.
1. What does congruent mean?
Two figures are congruent if one can be placed over the other so that they match exactly. Their corresponding side lengths and angle measures are equal. Orientation does not matter: a rotated or reflected copy can still be congruent.
Use matching marks to show equal sides and angles. The order of letters in a congruence statement matters because it identifies corresponding vertices. If △ABC ≅ △PQR, then A matches P, B matches Q and C matches R.
2. Triangle congruence conditions
- SSS: all three corresponding sides are equal.
- SAS: two corresponding sides and the included angle between them are equal.
- ASA: two corresponding angles and the included side are equal.
- RHS: for right triangles, the hypotenuse and one corresponding leg are equal, along with the right angle.
These conditions determine a unique triangle up to movement and reflection. Equal angles alone establish similarity, not necessarily congruence; the triangles may have different sizes.
3. Match corresponding parts carefully
Worked example
If △ABC ≅ △DEF in that order, then AB corresponds to DE, BC to EF, CA to FD; ∠A to ∠D, ∠B to ∠E, and ∠C to ∠F. If AB = 7 cm, then DE = 7 cm.
When writing the order, start at a pair of matching vertices and preserve the order around the triangle. A correct condition with incorrectly matched vertices can lead to a wrong conclusion.
4. Test a claim with a counterexample
Knowing only three angles does not prove congruence: triangles with angles 30°, 60° and 90° can be drawn at different scales. Knowing two sides and a non-included angle is also not a general congruence condition; more than one triangle may satisfy it.
Ask: do the given facts fix both the shape and the size? If the data leave room to stretch or scale the figure, congruence has not been established.
Practice: name the condition
1. Two triangles have corresponding side lengths 4 cm, 5 cm and 6 cm. Which condition applies?
Answer: SSS; all three corresponding sides are equal.
2. Two sides and the angle between them match in two triangles. Which condition applies?
Answer: SAS. The angle must be the included angle between the two known sides.
3. Are triangles with angles 40°, 60° and 80° necessarily congruent?
Answer: No. They have the same shape but can have different side lengths and sizes.
4. If △ABC ≅ △XYZ, which side corresponds to BC?
Answer: YZ, because B matches Y and C matches Z.
5. For two right triangles, what additional side information can establish congruence using RHS?
Answer: Equal hypotenuses and one pair of corresponding legs.
Chapter summary
Congruence means same shape and size, regardless of orientation. Use a valid side-angle condition and match corresponding vertices in the stated order. Equal angles by themselves do not establish congruence because size may differ.