Triangles, Congruence and the Pythagoras Theorem
Triangles, Congruence and the Pythagoras Theorem
Triangles alone can supply 3 questions in the Part-III maths paper. The examiner favours congruence criteria and the Pythagoras theorem, because both can be tested unambiguously in a multiple-choice format. Learn which criteria actually exist, and which ones look valid but are not.
Classification and Basic Properties
| By sides | By angles |
|---|---|
| Scalene — no equal sides | Acute — all angles below 90° |
| Isosceles — two equal sides | Right — one angle 90° |
| Equilateral — all sides equal | Obtuse — one angle above 90° |
- Angles opposite equal sides are equal, and the converse holds.
- Each angle of an equilateral triangle is 60 degrees.
- The exterior angle equals the sum of the two interior opposite angles.
- The sum of any two sides is greater than the third — the triangle inequality.
- The side opposite the largest angle is the longest side.
The triangle inequality is asked as a can-these-form-a-triangle question. Sides of 3, 4 and 8 cannot, because 3 + 4 is less than 8.
Congruence Criteria
| Criterion | Meaning |
|---|---|
| SSS | Three sides equal |
| SAS | Two sides and the included angle |
| ASA | Two angles and the included side |
| AAS | Two angles and a non-included side |
| RHS | Right angle, hypotenuse and one side |
Two criteria are not valid and both are offered as options. AAA gives similar triangles, not congruent ones, because the size is unconstrained. SSA is not a criterion either, except in the special right-angled case that is already covered by RHS.
✗ AAA proves two triangles are congruent | ✓ AAA proves similarity only; the triangles may differ in size
Similarity
Similar triangles have equal corresponding angles and proportional corresponding sides.
If two triangles are similar, ratio of areas = (ratio of corresponding sides)²
So triangles with sides in the ratio 2 : 3 have areas in the ratio 4 : 9. Candidates who give 2 : 3 for the areas find that value among the options. This squared relationship also holds for any pair of similar figures, not only triangles.
The Pythagoras Theorem
In a right triangle: hypotenuse² = base² + perpendicular²
The converse is equally examinable: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled.
Learn the common triples so you recognise them instantly: 3-4-5, 5-12-13, 8-15-17, 7-24-25, and any multiple of these such as 6-8-10.
Also worth knowing: the four centres of a triangle. The centroid is where the medians meet and divides each median in the ratio 2 : 1; the incentre is where the angle bisectors meet; the circumcentre is where the perpendicular bisectors meet; and the orthocentre is where the altitudes meet.
AAA से समरूपता सिद्ध होती है, सर्वांगसमता नहीं।
TRE pointer: The AAA option is planted in congruence questions almost every year, and the area-ratio question is planted in similarity ones. Both reward a candidate who knows the exact statement rather than the general idea. The four triangle centres, especially the centroid's 2 : 1 division of a median, appear as a direct one-line question. Since you are being hired to teach this, expect the paper to test the precise criterion rather than a rough justification.
60-Second Recap
- Exterior angle equals the sum of the two interior opposite angles.
- Triangle inequality: any two sides must together exceed the third.
- Valid congruence criteria are SSS, SAS, ASA, AAS and RHS.
- AAA gives similarity only; SSA is not a criterion.
- For similar figures, area ratio is the square of the side ratio.
- Centroid divides each median 2 : 1; triples to know are 3-4-5, 5-12-13, 8-15-17, 7-24-25.
Frequently Asked Questions
Is AAA a valid congruence criterion?
No. Equal angles fix the shape but not the size, so AAA establishes similarity only. Two triangles with the same three angles can be any size relative to each other. The valid congruence criteria are SSS, SAS, ASA, AAS and RHS, and AAA is offered as a distractor.
If two similar triangles have sides in the ratio 2 to 3, what is the ratio of their areas?
Four to nine, because the ratio of areas equals the square of the ratio of corresponding sides. Answering two to three is the standard error and that value is always among the options. The same squared rule applies to any pair of similar figures.
Can sides of 3, 4 and 8 form a triangle?
No. The triangle inequality requires the sum of any two sides to exceed the third, and three plus four is seven, which is less than eight. Whenever a question lists three lengths, apply this check to the two smallest sides against the largest.
What is the centroid of a triangle?
The point where the three medians meet. It divides each median in the ratio two to one, measured from the vertex. It is distinct from the incentre where angle bisectors meet, the circumcentre where perpendicular bisectors meet, and the orthocentre where altitudes meet.