Geometry Questions for CDS

Triangle Properties and Congruence

Triangles form the backbone of CDS geometry. You must know all congruence rules (SSS, SAS, ASA, AAS, RHS) and similarity rules (AA, SAS, SSS). The angle sum is always 180 degrees. An exterior angle equals the sum of the two non-adjacent interior angles — this single rule solves many CDS questions instantly.

• Angle sum of a triangle = 180 degrees
• Exterior angle = sum of two remote interior angles
• Pythagoras: In a right triangle, hypotenuse squared = sum of squares of other two sides
• For similar triangles, ratio of areas = square of ratio of corresponding sides
• Median divides triangle into two equal area parts
• Centroid divides each median in ratio 2:1 from vertex

Circle Theorems

Circle questions appear in almost every CDS paper. The key insight: the angle subtended at the centre is double the angle subtended at any point on the remaining arc. Angles in the same segment are equal. Angle in a semicircle is always 90 degrees. Tangent is always perpendicular to the radius at the point of contact.

• Central angle = 2 x inscribed angle on same arc
• Angle in a semicircle = 90 degrees always
• Angles in the same segment of a circle are equal
• Tangent-radius angle = 90 degrees
• Two tangents from external point are equal in length
• Tangent-chord angle = inscribed angle in alternate segment (Alternate Segment Theorem)

Quadrilaterals and Polygons

CDS tests properties of parallelograms, rhombuses, rectangles, squares, and trapeziums. The key rule: in a parallelogram, opposite sides are equal and parallel, opposite angles are equal, and diagonals bisect each other. For a rhombus, diagonals bisect each other at 90 degrees. Sum of interior angles of any polygon = (n-2) x 180 degrees.

• Parallelogram: opposite sides equal and parallel, diagonals bisect each other
• Rhombus: all sides equal, diagonals bisect at 90 degrees
• Rectangle: all angles 90 degrees, diagonals are equal
• Square: all sides equal, all angles 90 degrees, diagonals equal and bisect at 90 degrees
• Trapezium area = (1/2) x (sum of parallel sides) x height
• Sum of interior angles of n-sided polygon = (n-2) x 180 degrees

Coordinate Geometry

Coordinate geometry questions in CDS are mostly formula-based. You will be given coordinates and asked for distance, midpoint, slope, or equation of a line. The slope concept is crucial — parallel lines have equal slopes, perpendicular lines have slopes whose product is -1. Collinearity of three points is tested frequently.

• Distance between two points: sqrt((x2-x1)^2 + (y2-y1)^2)
• Midpoint: ((x1+x2)/2, (y1+y2)/2)
• Slope m = (y2-y1)/(x2-x1)
• Parallel lines: m1 = m2; Perpendicular lines: m1 x m2 = -1
• Area of triangle with vertices: (1/2)|x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|
• If area = 0, the three points are collinear

Lines, Angles, and Parallel Line Theorems

When a transversal cuts two parallel lines, several pairs of equal angles are formed. These appear in easy 1-mark questions in CDS. Alternate interior angles are equal. Corresponding angles are equal. Co-interior (same-side interior) angles are supplementary (add to 180 degrees).

• Alternate interior angles are equal when lines are parallel
• Corresponding angles are equal when lines are parallel
• Co-interior angles (same side of transversal) add up to 180 degrees
• Vertically opposite angles are always equal
• Linear pair (angles on a straight line) adds to 180 degrees
• Sum of all angles around a point = 360 degrees

• Confusing central angle and inscribed angle: many aspirants equate them. Remember central angle is always DOUBLE the inscribed angle on the same arc.
• Using Pythagoras without confirming the triangle is right-angled. Always check that the largest side squared equals sum of squares of other two before applying.
• In coordinate geometry, applying section formula with m and n swapped when dividing externally vs internally — read the question carefully.
• Forgetting that the diagonal of a rhombus bisects at 90 degrees but the diagonals are NOT equal — that is a rectangle property.
• For polygon angle sum, using n x 180 instead of (n-2) x 180. The correct formula subtracts 2, not uses n directly.

• Circle angle rule — HALF at the HAT (inscribed angle is HALF of central angle — the point on circle is like a hat far from the action).
• For parallel lines cut by transversal remember CAC: Corresponding equal, Alternate equal, Co-interior supplementary (adds to 180).
• Rhombus diagonals are PERPENDICULAR but NOT equal. Rectangle diagonals are EQUAL but NOT perpendicular. Square diagonals are BOTH. Use RPE: Rhombus Perpendicular, rEctangle Equal.
• Pythagorean triplets to memorise: 3-4-5, 5-12-13, 8-15-17, 7-24-25. Multiples of these also work (6-8-10, etc.). These appear directly in CDS options.

• CDS geometry questions are mostly direct theorem application — if you know 15 key theorems, you can solve 80 percent of questions in under 60 seconds each.
• Circle theorem questions frequently appear with a diagram described in text. Draw it on rough paper immediately — do not try to visualise without drawing.
• Coordinate geometry questions in CDS are straightforward distance/midpoint/slope problems. They are free marks if you know the 4-5 formulas cold.
• Pythagoras triplets (3-4-5, 5-12-13) appear as side lengths in 1-2 questions every paper. Spotting them saves 30 seconds of calculation.
• For angle questions with parallel lines, mark all equal angles in one colour mentally — most answers fall out in one step without any algebra.