CDS 2027 - Paper 3: Elementary Mathematics cheat sheet for Geometry! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Core Concepts & Formulas in Geometry

Mastering geometry for the CDS Elementary Mathematics paper requires a firm grasp of fundamental theorems, line properties, and polygon rules. This summary synthesizes essential geometric principles into a structured reference guide.

1. Lines and Angles

• Complementary Angles: Two angles whose sum is \(90^\circ\).
• Supplementary Angles: Two angles whose sum is \(180^\circ\).
• Parallel Lines Cut by a Transversal: Alternate interior angles are equal, corresponding angles are equal, and consecutive interior angles sum to \(180^\circ\).

2. Triangles & Congruence Criteria

• Angle Sum Property: The sum of interior angles in any triangle is \(180^\circ\). An exterior angle equals the sum of two opposite interior angles.
• Area Formulas: For a triangle with base \(b\) and height \(h\), \(\text{Area} = \frac{1}{2} b h\). By Heron’s formula, \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\), where semi-perimeter \(s = \frac{a+b+c}{2}\).
• Equilateral Triangle: Height \(h = \frac{\sqrt{3}}{2}a\) and \(\text{Area} = \frac{\sqrt{3}}{4}a^2\).
• Centroid & Medians: The centroid divides each median in a \(2:1\) ratio.

3. Circles and Tangents

• Circumference & Area: Perimeter \(C = 2\pi r\) and \(\text{Area} = \pi r^2\).
• Chord & Tangent Theorems: A line drawn from the center perpendicular to a chord bisects the chord. A tangent at any point on a circle is perpendicular to the radius through the point of contact.
• Cyclic Quadrilaterals: Opposite angles in a cyclic quadrilateral are supplementary (\(A + C = 180^\circ\)).

4. Polygons

• Sum of Interior Angles: For an \(n\)-sided polygon, \(\text{Sum} = (n-2) \times 180^\circ\).
• Number of Diagonals: Given by \(D = \frac{n(n-3)}{2}\).