Triangles, Congruence and the Pythagoras Theorem: Class 7 Study Guide
The triangle is the simplest closed figure, and its properties are used in building and design.
The three angles of any triangle add up to 180°. An exterior angle is formed when one side is extended; it equals the sum of the two opposite interior angles. In any triangle the sum of two sides is greater than the third side (the triangle inequality).
Two figures are congruent if they have exactly the same shape and size. Triangles are congruent if one of these holds: SSS (three sides equal), SAS (two sides and the angle between them), ASA (two angles and the side between them) or RHS (right angle, hypotenuse and one side).
Congruence conditions
| Rule | What must match | Notes |
|---|---|---|
| SSS | all three sides | no angles needed |
| SAS | two sides and the included angle | the angle is between the sides |
| ASA | two angles and the included side | side between the angles |
| RHS | right angle, hypotenuse and a side | right-angled triangles only |
The Pythagoras theorem
In a right-angled triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: c² = a² + b². For sides 3 and 4: c² = 9 + 16 = 25, so c = 5. Triples such as (3, 4, 5), (5, 12, 13) and (8, 15, 17) are right-angled. To check whether a triangle is right-angled, test whether the squares add up.
The longest side.
a² and b².
c² = a² + b².
Find c.
Using the theorem
A 10 m ladder leans against a wall with its foot 6 m from the wall. The height it reaches is √(10² − 6²) = √64 = 8 m. A rectangular field 40 m by 30 m has a diagonal of √(1600 + 900) = 50 m. Check that your answer is sensible: the hypotenuse must be longer than either other side.
Worked examples and common mistakes
Example: Angles of a triangle are x, 2x and 3x: 6x = 180, so x = 30° (angles 30°, 60°, 90°). Common mistakes: treating any side as the hypotenuse, using the theorem on triangles without a right angle and writing AAA as a congruence rule (equal angles do not guarantee equal size). Sketch and label before calculating.
Check your understanding: model answers
1. 70°. 2. 10. 3. No, equal angles can belong to triangles of different sizes.
✨ Key Takeaways
- Angles in a triangle add to 180°.
- Exterior angle = sum of two opposite interior angles.
- Congruence: SSS, SAS, ASA, RHS.
- c² = a² + b² in a right-angled triangle.
💬 Questions for Discussion & Reflection
- Two angles are 50° and 60°. Find the third.
- Right triangle with legs 6 and 8. Find the hypotenuse.
- Does AAA prove congruence?
Print graph paper for geometry →