SSC Notes / Quantitative Aptitude / Chapter 20
Chapter 20 of 27

Triangles

Congruence and similarity rules, the four centres of a triangle, and the Pythagoras theorem — SSC's most frequently tested Geometry topic.

📖 ~14 min read 🔢 SSC Quantitative Aptitude

Introduction

Triangles is the single most heavily tested Geometry topic in SSC exams — covering congruence/similarity rules, the four special centres, and the ever-present Pythagoras theorem.

Angle Sum Property

The sum of the interior angles of any triangle = 180°. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Types of Triangles

By SidesBy Angles
Equilateral — all sides equal, all angles 60°Acute — all angles < 90°
Isosceles — two sides equal, angles opposite them equalRight — one angle = 90°
Scalene — all sides differentObtuse — one angle > 90°

Congruence Rules (Two triangles are exactly identical)

RuleMeaning
SSSAll three sides equal
SASTwo sides and the included angle equal
ASATwo angles and the included side equal
RHSRight angle, hypotenuse, and one side equal (right triangles only)

Similarity Rules (Same shape, proportional sides)

RuleMeaning
AA (or AAA)Two (or three) corresponding angles equal
SSS (similarity)All corresponding sides proportional
SAS (similarity)Two sides proportional and included angle equal
💡 Key Result: If two triangles are similar with a side ratio of k, then the ratio of their areas is k², and the ratio of their perimeters is k.

Four Centres of a Triangle

CentrePoint of Intersection Of
CentroidMedians — divides each median in 2:1 ratio
IncentreAngle bisectors — centre of the inscribed circle
CircumcentrePerpendicular bisectors of sides — centre of the circumscribed circle
OrthocentreAltitudes

Pythagoras Theorem

Flowchart — Pythagoras Theorem (Right Triangles Only)
(Hypotenuse)² = (Base)² + (Height)²
Common Pythagorean Triplets (memorise these)
3, 4, 5  |  5, 12, 13  |  8, 15, 17  |  7, 24, 25  |  9, 40, 41
Q. The two shorter sides of a right triangle are 9 cm and 12 cm. Find the hypotenuse.
Recognise the triplet: 9,12,15 (3× the 3,4,5 triplet)
Hypotenuse = 15 cm

Basic Proportionality Theorem (Thales)

If a line is drawn parallel to one side of a triangle, intersecting the other two sides, it divides those two sides in the same ratio.

✅ Practice Focus: Congruence (SSS/SAS/ASA/RHS) vs Similarity (AA/SSS/SAS) rules · Area ratio = (side ratio)² for similar triangles · Four centres and their defining intersection · Pythagorean triplets memorised for instant recognition.

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