Geometry of Triangles and Quadrilaterals

Classification and Properties of Triangles

Triangles are the foundational building blocks of plane geometry. For the TMUA, you must be precise in your use of language and notation. Triangles are classified either by their side lengths or by their internal angles.

1. Equilateral Triangles: All three sides are equal in length, and all three internal angles are $60^\circ$. These shapes have three lines of reflectional symmetry and rotational symmetry of order 3.

2. Isosceles Triangles: At least two sides are equal in length. The angles opposite these sides (the base angles) are also equal. This symmetry implies that the altitude from the vertex between the equal sides bisects the base at a right angle.

3. Scalene Triangles: No sides or angles are equal.

Triangles are also defined by their largest angle. A right-angled triangle contains one $90^\circ$ angle, satisfying Pythagoras' Theorem. An acute triangle has all angles less than $90^\circ$, and an obtuse triangle has one angle greater than $90^\circ$. Two critical theorems always apply:

• The sum of the interior angles of any triangle is $180^\circ$.
• The Exterior Angle Theorem: The exterior angle at any vertex of a triangle is equal to the sum of the two opposite interior angles.

Special Types of Quadrilaterals

Quadrilaterals are four sided polygons. The sum of their interior angles is always $360^\circ$. The TMUA requires you to know the inclusive definitions of these shapes, where a more specific shape inherits the properties of a more general one.

The Parallelogram

A parallelogram is defined as a quadrilateral with two pairs of parallel sides. From this definition, we can derive several properties using the properties of parallel lines (alternate and corresponding angles):

• Opposite sides are equal in length.
• Opposite interior angles are equal.
• Consecutive angles are supplementary (sum to $180^\circ$).
• The diagonals bisect each other (the point of intersection is the midpoint of both diagonals).

The Rectangle and Rhombus

A rectangle is a special type of parallelogram that contains four right angles. Because it is a parallelogram, its opposite sides are equal and its diagonals bisect each other. Additionally, the diagonals of a rectangle are equal in length.

A rhombus is a special type of parallelogram with four equal sides. Its unique properties, derived from its symmetry, include:

• Diagonals bisect each other at right angles ($90^\circ$).
• Diagonals bisect the interior angles of the rhombus.

The Square

A square is the most regular quadrilateral. It is simultaneously a rectangle and a rhombus. It possesses four right angles, four equal sides, and diagonals that are equal, perpendicular, and bisect each other.

The Trapezium

A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides. An isosceles trapezium is a special case where the non-parallel sides are equal in length, resulting in equal base angles and equal diagonals.

The Kite

A kite is defined by two pairs of equal adjacent sides. Unlike a parallelogram, its opposite sides are not equal. Its properties include:

• One pair of opposite angles (the ones between the sides of different lengths) is equal.
• One diagonal (the main axis of symmetry) bisects the other diagonal at a right angle.
• One diagonal bisects the internal angles at its vertices.

Deriving Properties Using Congruence

Many geometric properties are derived by dividing a figure into triangles and proving they are congruent. For example, to prove that the diagonals of a parallelogram bisect each other, we can use the Angle-Side-Angle (ASA) rule on the two triangles formed by the diagonals and a pair of opposite sides. Since the sides are parallel, alternate interior angles are equal, and since opposite sides of a parallelogram are equal, the triangles must be congruent. This leads directly to the conclusion that the diagonal segments are of equal length.

General Polygons

You must be able to generalise these rules to any $n$ sided polygon:

• Sum of interior angles: $(n - 2) \times 180^\circ$.
• Sum of exterior angles: $360^\circ$ for any convex polygon.
• For a regular polygon, each interior angle is $\frac{(n - 2) \times 180^\circ}{n}$ and each exterior angle is $\frac{360^\circ}{n}$.