Concept 1
Build figures from precise definitions
A point marks a position. A line extends without end in both directions; a ray has one endpoint; a segment has two. Perpendicular lines meet at 90°, while parallel coplanar lines never meet.
| Angle | Measure | Relationship |
|---|---|---|
| Acute / right / obtuse | <90° / 90° / between 90° and 180° | Classifies one angle |
| Complementary | Sum 90° | Need not be adjacent |
| Supplementary | Sum 180° | A linear pair is an adjacent example |
| Vertically opposite | Equal | Formed by two intersecting lines |
Concept 2
A transversal transfers angle information
When a transversal crosses two parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior angles are supplementary. Their converses can prove that two lines are parallel.
Fast pattern: once one angle is known, vertical and equal-angle relations keep its measure; adjacent linear-pair and co-interior relations replace it with 180° minus that measure.
Concept 3
Classify triangles, then apply universal rules
By sides, a triangle is scalene, isosceles or equilateral; by angles, it is acute, right or obtuse. Every triangle has interior-angle sum 180°. An exterior angle equals the sum of the two remote interior angles.
In an isosceles triangle, equal sides face equal angles; the converse also holds. An equilateral triangle has three 60° angles. The triangle inequality requires the sum of any two sides to be greater than the third.
Concept 4
Congruent triangles match in size and shape
Valid criteria are SSS, SAS, ASA (or AAS), and RHS for right triangles. Once congruence is established, corresponding parts are equal (CPCT).
AAA is not congruence: it fixes shape but not size. SSA is generally insufficient: the same data can sometimes make two different triangles.
Concept 5
Similarity preserves shape and creates ratios
Triangles are similar by AA, SAS similarity (included angle equal and surrounding sides proportional), or SSS similarity. Corresponding sides share one scale factor; corresponding angles are equal. If the side ratio is k, the area ratio is k².
If a line is parallel to one side of a triangle, it divides the other two sides proportionally. In a right triangle with legs a and b and hypotenuse c, a²+b²=c². Its converse identifies a right angle when the longest side satisfies that equality.
Concept 6
Know what creates each triangle centre
| Centre | Intersection of | Key fact |
|---|---|---|
| Centroid | Medians | Divides each median 2:1 from the vertex; always inside |
| Incentre | Internal angle bisectors | Equidistant from sides; centre of incircle; always inside |
| Circumcentre | Perpendicular bisectors | Equidistant from vertices; centre of circumcircle |
| Orthocentre | Altitudes | Inside an acute, at the right-angle vertex in a right, outside an obtuse triangle |
Concept 7
Use defining properties, not appearances
A parallelogram has opposite sides parallel and equal, opposite angles equal, and diagonals that bisect each other. A rectangle adds four right angles and equal diagonals. A rhombus adds four equal sides and perpendicular diagonals. A square has all rectangle and rhombus properties. A kite has two pairs of adjacent equal sides; a trapezium has one pair of parallel sides.
For an n-sided polygon, interior-angle sum is (n−2)×180°. The exterior angles, one at each vertex in the same direction, total 360°. Each interior angle of a regular n-gon is ((n−2)×180°)/n.
Concept 8
Connect radii, chords, arcs and tangents
A radius joins centre to circle; a chord joins two points on the circle; a diameter is the longest chord and equals 2r. Equal chords subtend equal central angles and lie equally far from the centre. A perpendicular from the centre to a chord bisects it.
- The angle at the centre is twice the angle at the circumference standing on the same arc.
- Angles in the same segment are equal; the angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral sum to 180°.
- A tangent is perpendicular to the radius at contact; tangents from the same external point are equal.
- The tangent–chord angle equals the angle in the alternate segment.
Worked methods
Examples with reasoning
Parallel-line angle
A transversal forms a 68° angle. Its alternate interior angle is 68°; the adjacent angle is 180°−68°=112°.
Exterior angle
Two remote interior angles are 47° and 63°. The exterior angle is 47°+63°=110°; the adjacent interior angle is 70°.
Similarity scale
Similar triangles have corresponding sides 6 and 9. The scale factor is 3/2, so a corresponding side of 10 becomes 15 and the area ratio is 4:9.
Pythagoras
Sides 9, 12 and 15 form a right triangle because 9²+12²=81+144=225=15².
Regular polygon
A regular octagon has interior sum (8−2)×180°=1080°, so each interior angle is 1080°/8=135°.
Circle angle
If a chord subtends 124° at the centre, it subtends 62° at any point on the remaining circumference because the central angle is twice the inscribed angle.
Exam traps
Check the condition before using the rule
Trusting the sketch
Not-to-scale drawings do not prove equality, parallelism or a right angle.
Using transversal rules early
Corresponding or alternate equality needs parallel lines, unless used as a converse to prove them.
Matching wrong vertices
Write corresponding triangle names in the same order before forming ratios.
Calling AAA congruence
AAA proves similarity only; size may differ.
Applying Pythagoras anywhere
It applies to right triangles, with c as the longest side opposite 90°.
Confusing radius and tangent
The 90° rule is at the point of contact, not at every radius or chord.
Original practice
Check understanding after learning
Original syllabus-aligned questions—not claimed previous-year questions.
Rapid revision
The one-minute sheet
- At an intersection: vertical angles are equal; a linear pair totals 180°.
- With parallel lines: corresponding and alternate angles equal; co-interior angles total 180°.
- Triangle angles total 180°; an exterior angle equals two remote interior angles.
- Congruence: SSS, SAS, ASA/AAS, RHS. Similarity: AA, proportional SAS, proportional SSS.
- For similar figures, side ratio k gives area ratio k².
- Right triangle: a²+b²=c², where c is the hypotenuse.
- Centroid—medians; incentre—angle bisectors; circumcentre—perpendicular bisectors; orthocentre—altitudes.
- Polygon interior sum=(n−2)180°; exterior sum=360°.
- Central angle is twice the inscribed angle on the same arc.
- A tangent is perpendicular to the contact radius; two tangents from one external point are equal.
Source ledger
Official foundations
- SSC CGL official syllabus: Geometry scope for Mathematical Abilities
- NCERT Mathematics, reprint 2026–27: Triangles and similarity
- NCERT official exemplar directory: lines, angles, triangles, quadrilaterals and circles
Explanations, worked examples and practice are original. Official references are credited and not reproduced. Drafted 26 September 2026.
Mensuration.