Congruence, similarity and Pythagoras in triangles
Two triangles are congruent when they have the same shape and size. The tests are SSS, SAS, ASA and RHS; for example, SAS means two sides and the angle between them are equal. Congruent triangles have all corresponding parts equal, which is how many geometry results are proved. Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio, so one is an enlargement of the other. The basic proportionality theorem says that a line drawn parallel to one side of a triangle divides the other two sides in the same ratio.
In a right-angled triangle, Pythagoras' theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b². Sides of 3, 4, 5 and 5, 12, 13 are common Pythagorean triples. The converse also holds: if the sides of a triangle satisfy c² = a² + b², the angle opposite c is a right angle. For similar triangles, the ratio of their areas equals the square of the ratio of corresponding sides, so doubling every side multiplies the area by four.
Why it matters
Similar triangles let surveyors find the height of a building or a tree from its shadow, and Pythagoras' theorem is used by carpenters, architects and navigators to check right angles and find straight-line distances. These results are the base for trigonometry, coordinate geometry and the distance formula, so they appear in many Class 10 board questions.
Key terms
- Congruent triangles
- Triangles that match exactly in all three sides and all three angles.
- Similar triangles
- Triangles with equal corresponding angles and proportional corresponding sides.
- Basic proportionality theorem
- A line parallel to one side of a triangle divides the other two sides in the same ratio.
- Hypotenuse
- The longest side of a right-angled triangle, opposite the right angle.