Geometry and Measurement - Properties of 2D Shapes (Triangles and Quadrilaterals)

Sum of Interior Angles in a Triangle: The sum of the three interior angles of any triangle is always $180^\circ$. Visually, if you imagine cutting the three corners of a triangle and placing them side-by-side, they would form a perfect straight line.

Classification of Triangles by Sides: Triangles are categorized as Equilateral (three equal sides and three $60^\circ$ angles), Isosceles (two equal sides and two equal base angles), and Scalene (no equal sides or angles). Visually, an equilateral triangle looks perfectly symmetrical from all sides, while an isosceles triangle resembles a tall, balanced tent.

Classification of Triangles by Angles: Triangles are also defined by their largest angle. A Right-Angled triangle contains one $90^\circ$ angle (indicated by a small square in the corner), an Obtuse triangle has one angle greater than $90^\circ$ (looking 'wide'), and an Acute triangle has all angles less than $90^\circ$.

Sum of Interior Angles in a Quadrilateral: The interior angles of any four-sided shape (quadrilateral) always sum to $360^\circ$. This is visually logical because any quadrilateral can be split into two triangles by drawing a single diagonal line, and $2 \times 180^\circ = 360^\circ$.

Properties of Parallelograms and Rectangles: A parallelogram has two pairs of parallel and equal opposite sides. A rectangle is a special type of parallelogram where all four interior angles are exactly $90^\circ$. Visually, a parallelogram looks like a rectangle that has been pushed or tilted to one side.

Rhombus and Square Properties: A rhombus is a quadrilateral with four equal sides; its diagonals bisect each other at $90^\circ$. A square is a regular quadrilateral, meaning it is both a rectangle (four $90^\circ$ angles) and a rhombus (four equal sides). Visually, a rhombus looks like a 'diamond' shape.

Trapeziums and Kites: A trapezium (or trapezoid) has at least one pair of parallel sides. A kite has two pairs of equal-length sides that are adjacent to each other. Visually, a kite looks like a traditional flying kite with two shorter sides at the top and two longer sides at the bottom, while its diagonals cross at a right angle.

The Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of the two opposite interior angles. Visually, if you extend one side of a triangle outward, the angle formed outside is equal to the sum of the two angles 'inside' that aren't touching it.