Geometry Revision Notes

Angle Sum Property of Triangles

A fundamental property of all triangles is that the sum of their interior angles always equals 180 degrees. This is crucial for finding missing angles. For example, if two angles are 70° and 50°, the third angle is 180° - (70° + 50°) = 180° - 120° = 60°.

Types of Triangles Based on Sides and Angles

Triangles are classified by their side lengths and angle measures. An equilateral triangle has three equal sides and three equal angles (each 60°). An isosceles triangle has two equal sides, with the angles opposite these sides (base angles) also being equal. A scalene triangle has all sides of different lengths and all angles of different measures.

Right-angled Triangles

A right-angled triangle is defined by having one angle exactly equal to 90 degrees. The side opposite the right angle is known as the hypotenuse, and it is always the longest side of the triangle. The other two angles must be acute (less than 90 degrees) and sum to 90 degrees.

Exterior Angles of a Triangle

The exterior angle of a triangle is equal to the sum of the two opposite interior angles. For any vertex, the interior angle and its corresponding exterior angle sum to 180 degrees because they form a straight line. This property is useful for a variety of angle problems.

Types of Triangles by Sides

Triangles can be categorised based on the lengths of their sides:

1. Equilateral Triangle: All three sides are equal in length. Consequently, all three interior angles are also equal, each measuring 60 degrees. If you know one side, you know all three.
2. Isosceles Triangle: At least two sides are equal in length. The angles opposite the equal sides (called base angles) are also equal. This property is very useful for finding unknown angles.
3. Scalene Triangle: All three sides have different lengths. As a result, all three interior angles are also different. You cannot assume any angle or side equality based on this type alone.

Types of Triangles by Angles

Triangles can also be grouped by the types of angles they contain:

1. Acute-angled Triangle: All three interior angles are acute (less than 90 degrees).
2. Right-angled Triangle: One of its interior angles is a right angle (exactly 90 degrees). The side opposite the right angle is called the hypotenuse, which is always the longest side.
3. Obtuse-angled Triangle: One of its interior angles is obtuse (greater than 90 degrees). Only one obtuse angle is possible as the sum of angles must be 180 degrees.

Key Angle Property

A fundamental property of all triangles is that the sum of their interior angles is always 180 degrees. This is a crucial rule for solving problems involving unknown angles within any triangle. For example, if you know two angles, you can find the third by subtracting the sum of the known angles from 180 degrees. This holds true for all types of triangles, regardless of their side lengths or other angle properties.

The Sine Rule

The Sine Rule connects the lengths of sides of a triangle to the sines of their opposite angles. For a triangle with sides a, b, c and angles A, B, C opposite those sides, the rule is: a/sin(A) = b/sin(B) = c/sin(C). This rule is used to find an unknown side when you know two angles and one side (AAS or ASA), or to find an unknown angle when you know two sides and one non-included angle (SSA). Be aware that SSA can sometimes lead to an 'ambiguous case', where two different triangles can be formed with the given information. Always check for a second possible angle (180° - calculated angle) if finding an angle.

The Cosine Rule

The Cosine Rule is a generalisation of Pythagoras' theorem for non-right-angled triangles. It relates all three sides of a triangle to one of its angles. The formulas are: c² = a² + b² - 2ab cos(C), a² = b² + c² - 2bc cos(A), and b² = a² + c² - 2ac cos(B). This rule is used to find an unknown side when you know two sides and the included angle (SAS), or to find an unknown angle when you know all three sides (SSS). When rearranging to find an angle, ensure you isolate cos(A) before taking the inverse cosine: cos(A) = (b² + c² - a²) / (2bc).

Area of a Triangle Using Sine

The formula for the area of a triangle, given two sides and the included angle, is Area = ½ab sin(C). Similarly, Area = ½bc sin(A) or Area = ½ac sin(B). This is an alternative to the standard ½ base × height formula when the perpendicular height is not known or easily found. It is particularly useful in problems involving the Sine or Cosine Rule, as it integrates the concept of angles into area calculations.

Problem-Solving Strategies

When tackling problems involving general triangles, first identify what information is given (sides, angles). Then determine what needs to be found. If you have a pair (side and its opposite angle), consider the Sine Rule. If you have two sides and the included angle (SAS) or all three sides (SSS), consider the Cosine Rule. For area, if you have two sides and the included angle, use the ½ab sin(C) formula. Always draw a clear diagram and label it. Sometimes, you may need to use a combination of these rules or even apply angle properties (e.g., angles in a triangle sum to 180°) first.

Types of Quadrilaterals

A quadrilateral is a four-sided polygon. The sum of its interior angles is always 360°. Common types include:

• Square: All four sides are equal, and all four angles are 90°. Opposite sides are parallel.
• Rectangle: Opposite sides are equal and parallel, and all four angles are 90°.
• Rhombus: All four sides are equal. Opposite angles are equal. Opposite sides are parallel. Diagonals bisect each other at 90°.
• Parallelogram: Opposite sides are equal and parallel. Opposite angles are equal. Consecutive angles sum to 180°.
• Trapezium (or Trapezoid): Has at least one pair of parallel sides.
• Kite: Has two pairs of equal-length sides that are adjacent to each other. One pair of opposite angles are equal. Diagonals intersect at 90°.

Sides of Quadrilaterals

The properties of a quadrilateral's sides are key to its classification:

• Parallel sides: Indicated by arrows on the sides. For example, a parallelogram has two pairs of parallel sides, while a trapezium has at least one pair.
• Equal sides: Often marked with tick marks. A square and a rhombus have all four sides equal. A rectangle and a parallelogram have opposite sides equal. A kite has two pairs of adjacent equal sides.
• Lengths: The actual measurements of sides help differentiate between a square and a rectangle, or a rhombus and a parallelogram.

Angles of Quadrilaterals

Angles provide another crucial set of properties:

• Sum of interior angles: Always 360° for any quadrilateral.
• Right angles (90°): Present in squares and rectangles.
• Opposite angles: Equal in parallelograms, rhombuses, and squares/rectangles. In a kite, one pair of opposite angles are equal.
• Consecutive angles: In parallelograms, consecutive angles (angles next to each other) sum to 180° due to parallel lines.

Diagonals of Quadrilaterals

The lines connecting opposite vertices (diagonals) also have specific properties:

• Bisect each other: This means they cut each other in half. True for parallelograms, rectangles, squares, and rhombuses.
• Perpendicular: Terminating at a 90° angle. True for squares, rhombuses, and kites.
• Equal length: True for squares and rectangles.