Classifying Two-Dimensional Shapes

In geometry, we sort (or classify) two-dimensional shapes based on their special properties, like the lengths of their sides and the sizes of their angles. Let's explore how to group triangles and quadrilaterals, and how to find their lines of symmetry.

Classifying Triangles

Triangles are polygons with exactly three sides. We can classify them in two different ways: by looking at their sides or by looking at their angles.

By Their Sides

• Equilateral Triangle: All $3$ sides are exactly the same length.
• Isosceles Triangle: Exactly $2$ sides are the same length.
• Scalene Triangle: All $3$ sides have different lengths (no equal sides).

Example: If a triangle has sides that measure $5$, $5$, and $8$, it has two equal sides. Therefore, it is an isosceles triangle.

By Their Angles

• Acute Triangle: All $3$ angles are acute (less than $90^\circ$).
• Right Triangle: Has exactly $1$ right angle (exactly $90^\circ$).
• Obtuse Triangle: Has exactly $1$ obtuse angle (greater than $90^\circ$).

Classifying Quadrilaterals

Quadrilaterals are polygons with exactly four sides. We classify them based on parallel lines, side lengths, and angles.

• Trapezoid: A quadrilateral with exactly one pair of parallel sides.
• Parallelogram: A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length.
• Rectangle: A parallelogram with $4$ right angles.
• Rhombus: A parallelogram with $4$ equal sides.
• Square: A special shape that is both a rectangle and a rhombus! It has $4$ equal sides AND $4$ right angles.

Lines of Symmetry

A line of symmetry is an imaginary line that divides a shape perfectly in half, so that one side is a mirror image of the other. If you fold the shape along this line, the two halves will match up exactly.

Different shapes have different numbers of symmetry lines:

• An equilateral triangle has exactly $3$ lines of symmetry.
• A square has $4$ lines of symmetry (top-to-bottom, left-to-right, and across both diagonals).
• A rectangle (that isn't a square) has $2$ lines of symmetry.
• A scalene triangle has $0$ lines of symmetry because no two sides are equal.