Geometry Cheat Sheet - CDS 2027 - Paper 3: Elementary Mathematics

1. CDS Exam Pattern & PYQ Analysis

In the Combined Defence Services (CDS) examination conducted by UPSC, Paper 3 tests candidates on Elementary Mathematics. A rigorous analysis of Previous Years' Questions (PYQs) reveals that Geometry and Mensuration combined yield between 20 to 25 questions out of 100. Understanding fundamental angle relations, parallel lines, transversals, and triangle laws is essential for quick problem-solving under tight time constraints.

2. Lines, Angles & Parallel Lines

Understanding basic classifications of angles and their relationships forms the bedrock of Euclidean geometry.

Angle Types

• Acute Angle: \(0^\circ < \theta < 90^\circ\)
• Right Angle: \(\theta = 90^\circ\)
• Obtuse Angle: \(90^\circ < \theta < 180^\circ\)
• Straight Angle: \(\theta = 180^\circ\)

Angle Relationships

• Complementary Angles: Two angles whose sum is \(90^\circ\) (\(\alpha + \beta = 90^\circ\)).
• Supplementary Angles: Two angles whose sum is \(180^\circ\) (\(\alpha + \beta = 180^\circ\)).
• Vertical Angles: Opposite angles formed by intersecting lines are equal (\(a = b\)).

Parallel Lines Cut by a Transversal

When two parallel lines \(l_1\) and \(l_2\) are intersected by a transversal line \(t\), key geometric angle pairs are established:

• Corresponding Angles: \(\angle 1 = \angle 5\), \(\angle 2 = \angle 6\)
• Alternate Interior Angles: \(\angle 3 = \angle 6\), \(\angle 4 = \angle 5\)
• Consecutive Interior Angles: Sum up to \(180^\circ\) (\(\angle 3 + \angle 5 = 180^\circ\))

3. Triangles & 2D Geometry

Triangles form the core of 2D geometry in the CDS Elementary Mathematics syllabus. Below are the fundamental formulas every aspirant must memorize.

Fundamental Formulas

• Sum of Interior Angles: \(\alpha + \beta + \gamma = 180^\circ\)
• Area of a Triangle: \[A = \frac{1}{2} bh = \frac{1}{2} ab \sin(\theta)\]
• Heron's Formula: For a triangle with sides \(a\), \(b\), and \(c\), \[A = \sqrt{s(s-a)(s-b)(s-c)}\] where semi-perimeter \(s = \frac{a+b+c}{2}\).

4. Right Triangle Theorems & Laws of Triangles

Properties of right-angled triangles frequently appear in direct numericals and height-and-distance problems.

Right Triangle Theorems

• Pythagorean Theorem: In a right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \[a^2 + b^2 = c^2\]
• Trigonometric Ratios: \[\sin\theta = \frac{\text{Opp}}{\text{Hyp}}, \quad \cos\theta = \frac{\text{Adj}}{\text{Hyp}}, \quad \tan\theta = \frac{\text{Opp}}{\text{Adj}}\]

5. Frequently Asked Questions (People Also Ask)