Shape and Space - Classification of Quadrilaterals and Triangles

Classification of Triangles by Side Lengths: Triangles can be categorized based on how many sides are equal. An equilateral triangle has three equal sides and three equal angles of $60^{\circ}$, appearing perfectly balanced. An isosceles triangle has at least two equal sides and two equal base angles, looking like a symmetrical steeple. A scalene triangle has no equal sides and no equal angles, appearing irregular or leaning to one side.

Classification of Triangles by Internal Angles: Triangles are also named by their largest angle. A right-angled triangle contains one $90^{\circ}$ angle, which looks like an 'L' shape or the corner of a square. An acute triangle has all three angles smaller than $90^{\circ}$, giving it a 'sharp' appearance. An obtuse triangle has one angle greater than $90^{\circ}$, making it look wide and spread out.

The Parallelogram Family: A parallelogram is a quadrilateral with two pairs of parallel sides. This family includes the rectangle (four $90^{\circ}$ angles), the rhombus (four equal sides, resembling a tilted diamond), and the square. A square is unique because it is both a rectangle and a rhombus, featuring four equal sides and four $90^{\circ}$ angles, representing perfect four-way symmetry.

Non-Parallelogram Quadrilaterals: Some quadrilaterals do not have two pairs of parallel sides. A trapezium (or trapezoid) has exactly one pair of parallel sides, which visually looks like a triangle with its top cut off by a line parallel to the base. A kite has two pairs of equal-length sides that are adjacent to each other, forming a shape with one line of symmetry and no parallel sides.

Sum of Interior Angles: Every triangle, regardless of its shape, has interior angles that always add up to $180^{\circ}$. You can visualize this by cutting the corners of a paper triangle and placing them side-by-side to form a straight line. For quadrilaterals, the interior angles always sum to $360^{\circ}$, which is the equivalent of two triangles joined together ($180^{\circ} \times 2$).

Line Symmetry in Polygons: Symmetry is a visual property where a shape can be folded exactly in half. A regular pentagon has 5 lines of symmetry, while a rectangle has only 2 (horizontal and vertical through the center). A scalene triangle and a general trapezium have 0 lines of symmetry, meaning they cannot be folded onto themselves perfectly.

Regular vs. Irregular Polygons: A polygon is 'regular' if all its sides are the same length and all its angles are equal, such as an equilateral triangle or a square. These shapes look very uniform. 'Irregular' polygons have sides or angles of different sizes, making them look stretched or uneven.