Triangles — Class 9 Maharashtra SSC Mathematics revision notes
A 4-page visual revision sheet for Triangles, written for Class 9 Maharashtra SSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Triangles in one look
the angle sum, the tests for congruence, and the inequalities that decide what can be built
The two angle results everything starts from
- the three interior angles of any triangle add to 180°, whatever its shape or size
- the proof draws a line through one vertex parallel to the opposite side, then uses alternate angles to move the two base angles up to that vertex
- an exterior angle is formed by producing one side, and it equals the sum of the two interior opposite angles
- that follows at once from the angle sum, since the exterior angle and its own interior angle form a linear pair
- so an exterior angle is greater than either of the two remote interior angles taken alone
- a triangle can have at most one angle that is 90° or more, since two such angles would already reach 180°
Naming triangles two ways
- by sides: scalene has three unequal sides, isosceles has two equal, equilateral has all three equal
- by angles: acute has all three angles under 90°, right has exactly one at 90°, obtuse has one above 90°
- an equilateral triangle is also equiangular, every angle being 60°
- in an isosceles triangle the angles opposite the equal sides are equal, and the converse holds too
- the larger angle lies opposite the longer side, which is the ordering rule used constantly in proofs
- the two namings are independent, so a triangle can be right angled and isosceles at the same time
The tests for congruence
- SAS: two sides and the included angle. The angle must lie between the two sides, and an angle elsewhere does not prove congruence
- SSS: all three sides equal. No angle information is needed, because three fixed lengths can build only one triangle shape
- ASA and AAS: two angles and one corresponding side. Once two angles are known the third follows, so the side may be included or not, provided it corresponds
- RHS: for right angled triangles only: the hypotenuse and one other side. This is the one case where a non included side with an angle is enough
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Triangles sheet covers
- Angle sum property
- Exterior angle theorem
- The two angle results everything starts from
- Naming triangles two ways
- Congruence of triangles
- SAS test
- SSS test
- ASA test
- RHS test
- The tests for congruence
- SAS
- SSS
- ASA and AAS
- RHS
- Writing a congruence properly
- A short congruence proof
- Triangle inequality
- Side opposite larger angle
- The inequalities
- Using the exterior angle theorem
- Median
- Altitude
- Centroid
- Incentre
- The special lines inside a triangle
- Symbols in this chapter
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Triangles the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.