Geometry Questions for SSC CHSL

Lines and Angles

When two parallel lines are cut by a transversal, several angle pairs are formed. Corresponding angles are equal, alternate interior angles are equal, and co-interior (same-side interior) angles add up to 180 degrees. Vertically opposite angles are always equal. These are the building blocks for almost every geometry problem.

• Corresponding angles: equal (same position at each intersection)
• Alternate interior angles: equal (Z-shape between the parallel lines)
• Co-interior angles: add to 180 degrees (C-shape or U-shape)
• Vertically opposite angles: equal (X-shape at any intersection)
• Sum of angles on a straight line = 180 degrees
• Sum of angles around a point = 360 degrees

Triangles — Key Theorems

Triangles are the most tested shape. The angle sum is always 180 degrees. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles. For similar triangles, corresponding sides are proportional and corresponding angles are equal. For congruent triangles, all sides and angles are identical. Pythagoras theorem applies only to right-angled triangles.

• Angle sum = 180 degrees
• Exterior angle = sum of two opposite interior angles
• Similar triangles: ratio of sides equal, ratio of areas = square of ratio of sides
• Pythagoras: a^2 + b^2 = c^2 (c is hypotenuse)
• Mid-point theorem: line joining midpoints of two sides is parallel to third side and half its length
• Basic Proportionality Theorem (BPT): if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally

Special Triangle Centres

SSC CHSL frequently asks about the four triangle centres. Knowing what each centre does and its key property is enough to solve these questions quickly without full calculation.

• Centroid (G): intersection of medians — divides each median in 2:1 from vertex
• Circumcentre (O): intersection of perpendicular bisectors — equidistant from all vertices (circumradius R)
• Incentre (I): intersection of angle bisectors — equidistant from all sides (inradius r)
• Orthocentre (H): intersection of altitudes — no simple distance property but angle tricks apply
• For equilateral triangle: all four centres coincide at the same point
• Centroid divides median: vertex-to-centroid = 2 parts, centroid-to-midpoint = 1 part

Circles — Properties and Theorems

Circles account for a large share of geometry questions in SSC CHSL. The most tested properties are: angle subtended by an arc at the centre is double the angle at any point on the remaining arc, angles in the same segment are equal, the angle in a semicircle is 90 degrees, and tangent-related rules. Tangent from an external point: both tangents are equal in length and the line from centre to external point bisects the angle between tangents.

• Angle at centre = 2 x angle at circumference (same arc)
• Angles in the same segment are equal
• Angle in semicircle = 90 degrees (angle in a half-circle is always a right angle)
• Tangent is perpendicular to radius at point of contact
• Two tangents from external point are equal in length
• Chord bisector from centre is perpendicular to the chord

Quadrilaterals and Polygons

For SSC CHSL, you need properties of square, rectangle, rhombus, parallelogram, and trapezium. The diagonals of a parallelogram bisect each other. Diagonals of a rhombus bisect each other at 90 degrees. In a rectangle, diagonals are equal. Sum of interior angles of any polygon = (n-2) x 180, where n is the number of sides.

• Parallelogram: opposite sides equal and parallel, diagonals bisect each other
• Rectangle: all angles 90 degrees, diagonals equal and bisect each other
• Rhombus: all sides equal, diagonals bisect at 90 degrees (but not equal)
• Square: all sides equal, all angles 90 degrees, diagonals equal and bisect at 90 degrees
• Trapezium: one pair of parallel sides, area = (1/2)(sum of parallel sides) x height
• Sum of interior angles of n-sided polygon = (n-2) x 180 degrees

• Confusing centroid ratio — many students say centroid divides median 1:2 from vertex, but it is actually 2:1 (vertex gets the bigger part, 2 units, midpoint side gets 1 unit).
• Applying Pythagoras to non-right triangles — always check if the triangle has a 90-degree angle before using a^2 + b^2 = c^2.
• Forgetting that the angle at centre is DOUBLE the angle at circumference, not the other way around — flipping this gives a wrong answer immediately.
• Using area ratio = side ratio for similar triangles instead of squaring — if sides are in ratio 3:4, areas are in ratio 9:16, not 3:4.
• Mixing up inradius and circumradius formulas — inradius uses Area/semi-perimeter, circumradius uses (abc)/(4 x Area). Do not swap these.

• CIAO for triangle centres: Centroid (medians), Incentre (angle bisectors), circumcircle centre = Circumcentre (perpendicular bisectors), Orthocentre (altitudes). Remember CIAO like the Italian goodbye — you wave from the vertex.
• Tangent trick: Think of a tangent as standing straight up on a road (radius) — it is always perpendicular. 90 degrees, no exception.
• Circle angle rule: Centre is the Boss — the angle at the centre is DOUBLE what the employee (circumference point) sees for the same arc.
• Rhombus diagonals are a CROSS — they cut each other at 90 degrees. Rectangle diagonals are EQUAL — like twin siblings same length but they do not cross at 90 degrees.

• SSC CHSL typically places 1 circle question, 1-2 triangle questions, and 1 polygon or quadrilateral question per paper — do not skip any of these sub-topics.
• Angle-in-semicircle = 90 degrees is a free mark — if you see a triangle inscribed in a circle with one side as diameter, the opposite angle is immediately 90 degrees.
• Tangent length from external point is almost always tested with a 3-4-5 or 5-12-13 Pythagorean triple hidden inside — recognise these quickly to avoid full calculation.
• For similar triangle questions, the examiner often gives you area of one triangle and ratio of sides — directly square the ratio and set up proportion. Takes under 30 seconds.
• SSC CHSL rarely asks proof-based questions — it is always find-the-angle or find-the-length type. Draw a rough figure on your rough sheet, mark known values, and apply the most relevant theorem directly.