Geometry at the IGCSE level focuses on the properties of shapes, angles, and the mathematical reasoning used to solve spatial problems. 
1. Angle Properties


-Angles on a straight line: Always sum to 180°.


-Angles at a point: Always sum to 360°.



-Vertically opposite angles: Are equal.
w=y, x=z



Parallel Lines:
Corresponding angles (F-shape) are equal. ($\alpha = \gamma$)
Alternate angles (Z-shape) are equal. ($\beta = \theta$)
Co-interior angles (C/U-shape) sum to 180°. ($\beta + \gamma = 180 ^\circ$)




2. Polygons
A polygon is any closed 2D shape with straight sides.

Triangles
Angle Sum: The interior angles of any triangle sum to 180°.
Exterior Angle: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
In Right-angled Triangle,
    by Pythaoras Theoram,
            $a^2+b^2=c^2$ 

General Polygons (n = number of sides)

Sum of Interior Angles: $(n - 2) \times 180^\circ$
Sum of Exterior Angles: Always 360° for any convex polygon.
Regular Polygons: All sides and angles are equal.
Individual Interior Angle: $\frac{(n-2) \times 180^\circ}{n}$
Individual Exterior Angle: $\frac{360^\circ}{n}$

3. Circle Theorems

Angle at the center: The angle subtended at the center is twice the angle at the circumference.
Angles in the same segment: Angles subtended by the same arc are equal.
In cycle A,  $\angle D= \angle E$
                    $\angle A= 2\times \angle D$


Angle in a semicircle: The angle subtended by a diameter is always 90°. 
(GH is diameter, $\angle I= \angle J = 90 ^ \circ$
Cyclic Quadrilaterals: Opposite angles sum to 180°.($\angle H+ \angle G = \angle  + \angle I = 180^ \circ$





Tangents:
A tangent is perpendicular $(90^\circ) $ to the radius at the point of contact.
Tangents from an external point to a circle are equal in length.
Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.
CD is tangent. AB is radius.
$AB \perp CD$
$\alpha = \theta$




4. Symmetry
Line Symmetry: The number of ways a shape can be folded onto itself (reflection).
Rotational Symmetry: The number of times a shape looks the same during a full $360^\circ$ turn.

5. Similarity and Congruency

Congruent - identical in size and shape
Similar - same shape, same or different size proportionally ( Zoom in / Zoom out of a photo) 
    If lenght scale factor is k
    - Area Ratio = $k^2$
    - Volume ratio - $k^3$