Point
An exact location with no length, breadth or thickness; named by a capital letter such as \(A\).
13 SSC CGL · Mathematics
Elementary Geometry
Handwritten-style notes for SSC CGL Tier I and Tier II.
01MAP IT!
The Big Picture — Graphic Mind Map
Geometry converts visible relationships—angle, length, parallelism, perpendicularity and proportion—into invariant rules.
ELEMENTARY
GEOMETRY
FACTS
GEOMETRY
FACTS
1. Lines & Anglesrays, pairs, parallel lines and transversals
2. Trianglestypes, inequalities, lines and centres
3. Congruencesame shape and same size
4. Similaritysame shape with proportional scale
5. Polygonsquadrilaterals and regular-angle sums
6. Circleschords, angles, tangents and common tangents
Mark what is equal before calculating
Parallel arrows, tick marks, right-angle boxes and equal-radius lines are the grammar of a diagram. They expose the theorem that applies.
\[\text{mark}\to\text{identify relation}\to\text{apply theorem}\to\text{verify}\]
\[\text{diagram suggests a relation; markings and stated facts justify it}\]
02ZERO TO EXPERT
The Foundation (Basics) — Complete Concept Build
Move from line and angle language to triangles, polygons, circles and their strongest theorems.
APoints, lines, rays and segments
Line
A straight path extending endlessly in both directions. Two distinct points determine one line.
Ray
Begins at one endpoint and extends endlessly in one direction.
Line segment
A finite part of a line between two endpoints. Its measurable length is \(AB\).
Parallel lines
Coplanar lines that never meet; written \(l\parallel m\).
Perpendicular lines
Lines meeting at a right angle; written \(l\perp m\).
\[\text{right angle}=90^\circ\]
BAngle types and angle pairs
By measure
- Acute: \(0^\circ\lt\theta\lt90^\circ\)
- Right: \(\theta=90^\circ\)
- Obtuse: \(90^\circ\lt\theta\lt180^\circ\)
- Straight: \(\theta=180^\circ\)
- Reflex: \(180^\circ\lt\theta\lt360^\circ\)
Complementary
\[\alpha+\beta=90^\circ\]
Supplementary
\[\alpha+\beta=180^\circ\]
Linear pair
Adjacent angles whose non-common arms form a straight line; their sum is \(180^\circ\).
Vertically opposite
When two lines intersect, opposite angles are equal.
Angles around a point
\[\sum\theta_i=360^\circ\]
An angle records rotation
Use a straight line as \(180^\circ\) and a full turn as \(360^\circ\). Linear-pair and around-a-point questions are subtraction from these totals.
CParallel lines cut by a transversal
Corresponding angles
Matching-corner angles are equal when lines are parallel.
Alternate interior angles
Interior angles on opposite sides of the transversal are equal.
Co-interior angles
\[\alpha+\beta=180^\circ\]
Interior angles on the same side are supplementary.
Converse tests
If a corresponding pair is equal, an alternate-interior pair is equal, or a co-interior pair is supplementary, the two lines are parallel.
Vertical equality
At each intersection, vertically opposite angles are equal even without parallel lines.
Angle transfer
Combine one parallel-line equality with a vertical or linear-pair relation to propagate angles across the diagram.
The transversal copies and supplements angles
Mark one acute angle \(x\). Every acute angle becomes \(x\); every obtuse partner becomes \(180^\circ-x\).
\[\text{two parallel-line intersections use only }x\text{ and }180^\circ-x\]
DTriangles — classification and universal facts
By sides
- Scalene: all sides unequal.
- Isosceles: two equal sides; opposite base angles equal.
- Equilateral: three equal sides; each angle \(60^\circ\).
By angles
- Acute: all angles acute.
- Right: one angle \(90^\circ\).
- Obtuse: one angle obtuse.
Angle sum
\[A+B+C=180^\circ\]
Exterior angle
\[\text{exterior angle}=\text{sum of two remote interior angles}\]
Side–angle order
\[a\gt b\iff A\gt B\]
The largest side lies opposite the largest angle.
Triangle inequality
\[|a-b|\lt c\lt a+b\]
For every choice of the third side \(c\).
Side marks predict angle marks
Equal sides face equal angles. Conversely, equal angles face equal sides. This two-way fact powers many isosceles questions.
ESpecial lines and the four triangle centres
| Line | Definition | Concurrency point | Key fact |
|---|---|---|---|
| Median | Vertex to midpoint of opposite side | Centroid \(G\) | \(VG:GM=2:1\) |
| Altitude | Perpendicular from vertex to opposite side/extension | Orthocentre \(H\) | May lie outside an obtuse triangle |
| Perpendicular bisector | Perpendicular through a side’s midpoint | Circumcentre \(O\) | \(OA=OB=OC\) |
| Angle bisector | Divides a vertex angle equally | Incentre \(I\) | Equal perpendicular distances to all sides |
Centroid \(G\)
Always inside. It is the centre of mass of a uniform triangular lamina and divides each median in \(2:1\) from the vertex.
Orthocentre \(H\)
Inside an acute triangle, at the right-angle vertex of a right triangle, and outside an obtuse triangle.
Circumcentre \(O\)
Inside acute, midpoint of hypotenuse in right, outside obtuse. It is the centre of the circumcircle.
Incentre \(I\)
Always inside. It is the centre of the incircle tangent to all three sides.
Euler line
\[O,G,H\text{ are collinear},\qquad OG:GH=1:2\]
For a non-equilateral triangle.
Equilateral coincidence
\[G=H=O=I\]
All four centres coincide by symmetry.
Each centre comes from a different “equal” idea
Centroid balances areas, orthocentre uses right angles, circumcentre is equally far from vertices, and incentre is equally far from sides.
FCongruence — same shape and same size
SSS
Three corresponding sides equal.
SAS
Two sides and the included angle equal.
ASA / AAS
Two angles and one corresponding side equal.
RHS
For right triangles: right angle, hypotenuse and one side equal.
Not sufficient
AAA proves similarity, not congruence. SSA is generally ambiguous.
CPCT
Once triangles are proved congruent, corresponding parts are equal.
Match corresponding order
If \(\triangle ABC\cong\triangle PQR\), then \(A\leftrightarrow P\), \(B\leftrightarrow Q\), \(C\leftrightarrow R\). Wrong order produces wrong CPCT pairs.
GSimilarity — same shape, scalable size
AA / AAA
Two corresponding angles equal; the third follows from the angle sum.
SAS similarity
Two corresponding sides proportional and included angles equal.
SSS similarity
All three corresponding side ratios equal.
Scale factor \(k\)
\[\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=k\]
Perimeter and area ratios
\[\frac{P_1}{P_2}=k,\qquad\frac{A_1}{A_2}=k^2\]
Corresponding lengths
Altitudes, medians, angle bisectors, inradii and circumradii scale by \(k\).
Similarity is a zoom, not a stretch
Every corresponding length uses the same scale factor. Areas therefore use the square of that factor.
\[\text{length}\times k\Rightarrow\text{area}\times k^2\]
Basic proportionality theorem
In \(\triangle ABC\), if \(DE\parallel BC\) with \(D\) on \(AB\), \(E\) on \(AC\), then
\[\frac{AD}{DB}=\frac{AE}{EC}\]
Also \(\triangle ADE\sim\triangle ABC\).
Angle-bisector theorem
If \(AD\) bisects \(\angle A\) and meets \(BC\) at \(D\):
\[\frac{BD}{DC}=\frac{AB}{AC}\]
HPythagorean relationship
The theorem
\[a^2+b^2=c^2\]
For a right triangle with hypotenuse \(c\).
Converse
If the longest side \(c\) satisfies \(a^2+b^2=c^2\), the triangle is right-angled opposite \(c\).
Angle type test
\[c^2\lt a^2+b^2:\text{ acute},\qquad c^2\gt a^2+b^2:\text{ obtuse}\]
Primitive triplets bank
\[(3,4,5),(5,12,13),(7,24,25),(8,15,17)\]
Triplet generator
\[(m^2-n^2,2mn,m^2+n^2)\quad(m\gt n)\]
Altitude to hypotenuse
If altitude \(h\) divides hypotenuse into \(p,q\):
\[h^2=pq,\qquad a^2=c p,\qquad b^2=c q\]
The two leg-squares equal the hypotenuse-square
The theorem applies only to right triangles. Identify the side opposite \(90^\circ\) as the hypotenuse.
IQuadrilaterals and regular polygons
| Figure | Key side/angle facts | Diagonal facts |
|---|---|---|
| Parallelogram | Opposite sides parallel/equal; opposite angles equal; adjacent angles supplementary | Diagonals bisect each other |
| Rectangle | Parallelogram with four right angles | Equal and bisect each other |
| Rhombus | Parallelogram with four equal sides | Perpendicular bisectors; bisect vertex angles |
| Square | Rectangle and rhombus | Equal, perpendicular, bisect each other and angles |
| Kite | Two pairs of adjacent equal sides | Perpendicular; one bisects the other |
| Trapezium | At least one pair of opposite sides parallel | No universal bisection/equality rule |
Quadrilateral angle sum
\[\sum\text{interior angles}=360^\circ\]
Polygon interior sum
\[(n-2)180^\circ\]
Regular polygon
\[\text{each exterior}=\frac{360^\circ}{n},\qquad\text{each interior}=180^\circ-\frac{360^\circ}{n}\]
Exterior-angle sum
\[\sum\text{one exterior at each vertex}=360^\circ\]
Number of diagonals
\[\frac{n(n-3)}2\]
Triangles from one vertex
\[n-2\]
Exterior angles complete one turn
Walking around any convex polygon turns through a total of \(360^\circ\). For a regular polygon, equal turns make finding \(n\) immediate.
JCircle language and chord facts
Core terms
Centre, radius, diameter, circumference, chord, arc, sector, segment, secant and tangent.
Diameter
\[d=2r\]
The longest chord; passes through the centre.
Perpendicular to a chord
A perpendicular from the centre to a chord bisects the chord; conversely, the line from the centre to the chord midpoint is perpendicular.
Equal chords
Equal chords subtend equal central angles, cut equal arcs and are equidistant from the centre—and conversely.
Closer chord is longer
Among chords of one circle, the chord nearer the centre is longer.
Chord length relation
If centre distance to chord is \(d\):
\[\text{chord}=2\sqrt{r^2-d^2}\]
A chord and the centre form right triangles
The centre-to-chord perpendicular cuts the chord equally. Pythagoras then relates half-chord, radius and centre distance.
KAngles in a circle and cyclic quadrilaterals
Central vs inscribed
\[\angle AOB=2\angle ACB\]
Both subtend the same arc \(AB\).
Same segment
Angles subtended by the same chord on the same arc/segment are equal.
Semicircle
\[\text{angle in a semicircle}=90^\circ\]
Cyclic quadrilateral
\[A+C=180^\circ,\qquad B+D=180^\circ\]
Exterior angle of cyclic quadrilateral
An exterior angle equals the opposite interior angle.
Equal angles imply concyclicity
If two points subtend the same segment at equal angles on the same side, the four relevant points are concyclic.
The centre sees twice the angle
When central and circumference angles stand on the same chord, the central angle is double. A diameter creates a \(180^\circ\) central angle, hence \(90^\circ\) at the circle.
LTangents and power of a point
Radius–tangent
\[OT\perp PT\]
A tangent is perpendicular to the radius at the contact point.
Two tangents from one point
\[PA=PB\]
Angle between tangents
\[\angle APB=180^\circ-\angle AOB\]
Tangent–chord theorem
The angle between a tangent and a chord equals the angle in the alternate segment subtended by that chord.
Tangent–secant power
\[PT^2=PA\cdot PB\]
\(A\) is nearer and \(B\) farther on the secant from external point \(P\).
Intersecting chords
\[PA\cdot PB=PC\cdot PD\]
For chords intersecting at internal point \(P\).
Equal tangents create symmetry
Joining the centre to the external point bisects the angle between the two tangents and the angle between the two contact radii.
Power of a point multiplies near × far
For secants from one external point, external segment × whole secant is constant. A tangent is the limiting case with equal factors.
\[PA\cdot PB=PC\cdot PD=PT^2\]
MCommon tangents to two circles
| Position of circles | Condition using centre distance \(d\) | Number of common tangents |
|---|---|---|
| Separate, neither touching | \(d\gt r_1+r_2\) | \(4\): two direct and two transverse |
| Externally touching | \(d=r_1+r_2\) | \(3\) |
| Intersecting | \(|r_1-r_2|\lt d\lt r_1+r_2\) | \(2\) direct tangents |
| Internally touching | \(d=|r_1-r_2|\), \(d\ne0\) | \(1\) |
| One strictly inside another | \(d\lt|r_1-r_2|\) | \(0\) |
| Concentric | \(d=0\) | \(0\) common tangents for unequal circles |
Direct common tangent length
\[L_d=\sqrt{d^2-(r_1-r_2)^2}\]
Transverse common tangent length
\[L_t=\sqrt{d^2-(r_1+r_2)^2}\]
Exists as a non-zero segment when \(d\gt r_1+r_2\).
Difference for direct, sum for transverse
The radii form right triangles with the common tangent and centre line. Direct tangents use \(|r_1-r_2|\); transverse tangents use \(r_1+r_2\).
NTheorem decision table
| Clue in diagram | Likely theorem | Immediate result |
|---|---|---|
| Parallel lines | Corresponding/alternate/co-interior angles | Equal or supplementary angle transfer |
| Two equal triangle sides | Isosceles theorem | Opposite angles equal |
| Matching triangle data | Congruence or similarity criterion | Equal parts or proportional sides |
| Right angle with three sides | Pythagorean theorem/converse | Square relation or angle type |
| Centre perpendicular to chord | Chord bisection | Two equal half-chords |
| Same circle chord/arc | Central/inscribed angle theorem | Double or equal angles |
| Radius to tangent point | Radius–tangent theorem | Right angle |
| External tangents/secants | Power of a point | Equal products or equal tangent lengths |
03SAVE TIME
Short Tricks & Magic Formulas
Mark equal parts first, then use angle totals, scale factors or right triangles before lengthy construction.
1Angle-chasing speed rules
Parallel-line palette
Once one angle is \(x\), every angle is either \(x\) or \(180^\circ-x\).
Triangle exterior
\[\text{exterior}=\text{remote angle}_1+\text{remote angle}_2\]
Cyclic shortcut
\[\text{opposite angles sum to }180^\circ\]
2Triangle ratio shortcuts
Centroid split
\[\text{vertex to }G:G\text{ to midpoint}=2:1\]
Similarity scale
\[\text{side ratio}=k\Rightarrow\text{area ratio}=k^2\]
Angle bisector
\[BD:DC=AB:AC\]
3Pythagorean and polygon bank
Triplet recognition
\[(3,4,5),(5,12,13),(8,15,17),(7,24,25)\]
Regular polygon sides
\[n=\frac{360^\circ}{\text{exterior angle}}\]
Polygon diagonals
\[N=\frac{n(n-3)}2\]
4Circle and tangent shortcuts
Semicircle sighting
If a triangle’s side is a diameter and the third vertex lies on the circle, the third angle is \(90^\circ\).
Equal tangents
\[PA=PB\]
Power product
\[PT^2=PA\cdot PB\]
5Common tangent memory formula
Direct
\[\sqrt{d^2-(r_1-r_2)^2}\]
Transverse
\[\sqrt{d^2-(r_1+r_2)^2}\]
Count from position
Separate \(4\), external touch \(3\), intersect \(2\), internal touch \(1\), nested \(0\).
SSC geometry speed dashboard
Read markings, name the figure, locate equal/supplementary relations, choose the theorem, then test the result against the diagram.
\[\text{marks}\to\text{figure}\to\text{relation}\to\text{theorem}\to\text{check}\]
04DON'T RUSH
The SSC / TCS Traps — Red Flags 🚩
Distractors rely on a diagram not drawn to scale, invalid congruence data, or mixing up radius, chord and tangent facts.
Trap 1: appearance treated as proof
- Visual equality, parallelism or perpendicularity is not evidence.
- Use only markings, measurements and stated facts.
- A rough sketch may deliberately mislead.
Trap 2: AAA or SSA claimed congruent
- AAA proves similarity only.
- SSA may produce two different triangles.
- Congruence needs SSS, SAS, ASA/AAS or RHS.
Trap 3: tangent/chord theorem misapplied
- Only the radius to the point of contact is perpendicular to the tangent.
- A random chord is not perpendicular to the tangent.
- Equal tangent lengths must start from the same external point.
Correspondence order lost
In congruent or similar triangles, write vertices in matching order before pairing sides or angles.
Area ratio not squared
If the side scale factor is \(k\), perimeter scales by \(k\) but area scales by \(k^2\).
Direct/transverse formulas swapped
Direct tangent uses radius difference; transverse tangent uses radius sum.
05LOCK IT IN
Memory Hooks & Mnemonics
Bind each theorem to a distinctive mark or picture.
“Z gives alternate angles”
With parallel lines, the Z-shape highlights equal alternate interior angles.
“M-A-P-A finds G-H-O-I”
Medians → centroid, Altitudes → orthocentre, Perpendicular bisectors → circumcentre, Angle bisectors → incentre.
“Zoom lengths by k, areas by k²”
Similar figures preserve angles and multiply every corresponding length by one scale factor.
“Right box unlocks squares”
Use Pythagoras only after locating or proving a right angle.
“Radius meets tangent at ninety”
The contact radius is perpendicular to the tangent.
“Direct differs; transverse totals”
Direct tangent length uses \(r_1-r_2\); transverse uses \(r_1+r_2\).
Parallel linesZ/F/C angle patterns
Centroidmedian split \(2:1\)
Similarityside \(k\), area \(k^2\)
Circle anglecentre is double
Tangentradius is perpendicular
Powernear × far = tangent²