Geometry (Lines, Angles, Triangles, Polygons)

Geometry deals with the properties of shapes, sizes, and positions of figures.

Lines & Angles: Understand concepts like parallel lines, transversals, vertically opposite angles, corresponding angles, alternate interior angles, and linear pairs. The sum of angles on a straight line is 180°.
Triangles: A polygon with 3 sides. The sum of angles is always 180°. Key types are Equilateral (all sides/angles equal), Isosceles (two sides/angles equal), and Right-angled (one angle is 90°).
Polygons: A closed figure made of straight line segments. A 'regular' polygon has all sides and all angles equal.

Pythagorean Theorem: In a right-angled triangle, \( (Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2 \).
Sum of Interior Angles of an n-sided Polygon: \( (n-2) \times 180° \).
Each Interior Angle of a Regular n-sided Polygon: \( \frac{(n-2) \times 180°}{n} \).
Sum of Exterior Angles of any Polygon: Always 360°.
Number of Diagonals in an n-sided Polygon: \( \frac{n(n-3)}{2} \).
Area of a Triangle: \( \frac{1}{2} \times base \times height \). Also, Heron's formula for a triangle with sides a, b, c is \( \sqrt{s(s-a)(s-b)(s-c)} \) where s is the semi-perimeter \( \frac{a+b+c}{2} \).

Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of its two opposite interior angles. This is a very frequently used shortcut.
Similarity & Congruence: Understand the conditions (AAA, SAS, SSS etc.). In similar triangles, the ratio of sides is equal to the ratio of heights, medians, and perimeters. The ratio of areas is the square of the ratio of corresponding sides.
Look for Pythagorean Triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25) and their multiples often appear in right-angled triangles. Recognizing them saves calculation time.

If a diagram is not provided, draw one yourself. A neat, reasonably scaled diagram can provide clues and prevent silly mistakes.
Do not assume information that is not given. For example, don't assume a triangle is right-angled unless specified or provable.
Memorize all the basic formulas and theorems, especially those related to triangles and regular polygons, as they form the basis for most questions.