Similarity & Congruence
Congruent shapes are identical — same shape and same size. One can be mapped to the other by rotation, reflection, or translation.
Similar shapes have the same shape but different sizes. Corresponding angles are equal and corresponding sides are in the same ratio (the scale factor).
Scale factor k = corresponding side of image ÷ corresponding side of object
Area ratio = k² Volume ratio = k³
📖 Learn
Congruence Conditions for Triangles
Two triangles are congruent if any of these conditions hold:
SAS
Two sides and the included angle equal
ASA / AAS
Two angles and one corresponding side equal
RHS
Right angle, hypotenuse, and one other side equal
💡 AAA (all angles equal) does NOT prove congruence — it only proves similarity. The triangles could be different sizes.
Similar Triangles
Two triangles are similar if:
- All corresponding angles are equal (AA is sufficient — the third follows automatically), or
- All corresponding sides are in the same ratio (SSS similarity), or
- Two sides in the same ratio and the included angle equal (SAS similarity)
If triangles ABC and PQR are similar with scale factor k:
PQ/AB = QR/BC = PR/AC = k
All angles: ∠A=∠P, ∠B=∠Q, ∠C=∠R
Finding Missing Lengths in Similar Shapes
Method: find the scale factor, then multiply/divide.
Scale factor k = known image side ÷ known object side
Missing length = corresponding known length × k
Always match up corresponding sides — sides opposite equal angles, or sides in the same position.
💡 Write the sides as a ratio: small:large = 3:5. Then find unknown using equivalent fractions or cross-multiplication.
Area and Volume Scale Factors
Linear scale factor: k
Area scale factor: k²
Volume scale factor: k³
Example: Two similar cylinders have radii 4 cm and 12 cm. Scale factor k = 3.
- Ratio of curved surface areas = 3² = 9
- Ratio of volumes = 3³ = 27
💡 To find k from areas: k = √(area ratio). To find k from volumes: k = ∛(volume ratio).
Similar Shapes in Real Life
Similar shapes appear in: map scales, shadow lengths, mirror reflections, architectural models, photography.
Shadow problems: a vertical pole and its shadow form a triangle. A nearby object and its shadow form a similar triangle. Use ratios to find unknown heights or distances.
Parallel lines cutting transversals: when a line is parallel to one side of a triangle, it creates a smaller similar triangle inside.
✏️ Worked Examples
Example 1: Finding a missing length
Triangles ABC and PQR are similar. AB = 6, BC = 8, AC = 10. PQ = 9. Find QR and PR.
Scale factor k = PQ/AB = 9/6 = 1.5
QR = BC × k = 8 × 1.5 = 12
PR = AC × k = 10 × 1.5 = 15
Example 2: Area ratio
Two similar triangles have areas 16 cm² and 100 cm². Find the scale factor and ratio of corresponding sides.
Area ratio = 100/16 = 25/4
Linear scale factor k = √(25/4) = 5/2 = 2.5
Ratio of sides = 2:5
Example 3: Shadow problem
A 1.8 m person casts a 2.4 m shadow. A nearby tree casts a 16 m shadow. Find the tree's height.
Both person and tree stand vertically → two similar right triangles
Person height/shadow = tree height/shadow
1.8/2.4 = h/16
h = 1.8 × 16/2.4 = 12 m
Example 4: Volume ratio
Two similar cones have radii 2 cm and 6 cm. The smaller has volume 40 cm³. Find the larger cone's volume.
Scale factor k = 6/2 = 3
Volume ratio = k³ = 27
Larger volume = 40 × 27 = 1080 cm³