Geometry Revision Notes

Angles at a Point of Intersection

When two straight lines cross, they create four angles. Vertically opposite angles are the angles directly opposite each other, and they are always equal in measure. For example, if line A intersects line B, forming angles 1, 2, 3, and 4, then angle 1 and angle 3 are vertically opposite, so angle 1 = angle 3. Similarly, angle 2 = angle 4. Angles on a straight line sum to 180 degrees. So, angle 1 + angle 2 = 180 degrees.

Corresponding Angles (F-Angles)

Corresponding angles are formed when a transversal line crosses two parallel lines. They are in the 'same position' at each intersection. Imagine drawing an 'F' shape (which may be reversed or upside down) along the lines. The angles inside the two arms of the 'F' are corresponding angles. These angles are always equal. For example, if two parallel lines are cut by a transversal, the top-left angle at the first intersection is equal to the top-left angle at the second intersection.

Alternate Angles (Z-Angles)

Alternate angles are also formed when a transversal line cuts two parallel lines. They are on 'alternate' or opposite sides of the transversal and 'between' the parallel lines (interior alternate angles) or 'outside' the parallel lines (exterior alternate angles). Think of a 'Z' shape (which may be reversed or stretched out). The angles in the 'corners' of the 'Z' are alternate interior angles. These angles are always equal. For example, the angle on the top-right of the first parallel line inside the transversal is equal to the angle on the bottom-left of the second parallel line inside the transversal.

Co-interior Angles (C-Angles)

Co-interior angles (also called consecutive interior angles or conjoined angles) are found between two parallel lines, on the 'same side' of the transversal. Imagine a 'C' shape (which may be reversed). The two angles inside the 'C' add up to 180 degrees. Unlike corresponding or alternate angles, co-interior angles are supplementary, meaning they sum to 180 degrees. This property is very useful when solving problems involving parallel lines where other angle types are not immediately obvious.

Types of Polygons and their Angles

A polygon is a closed 2D shape with straight sides. Key terms include: 'n' for the number of sides. Interior angles are inside the polygon, formed by two adjacent sides. Exterior angles are formed by extending one side and the adjacent side. Convex polygons have no interior angles greater than 180 degrees. Regular polygons have all sides equal in length and all interior angles equal. Examples include triangles (3 sides) and quadrilaterals (4 sides).

Sum of Interior Angles

The sum of the interior angles of any polygon with 'n' sides is always given by the formula (n-2) * 180 degrees. This formula applies to both regular and irregular polygons. For example, a quadrilateral (n=4) has an interior angle sum of (4-2) * 180 = 2 * 180 = 360 degrees. A pentagon (n=5) has an interior angle sum of (5-2) * 180 = 3 * 180 = 540 degrees. This formula is derived from dividing the polygon into (n-2) non-overlapping triangles from a single vertex.

Exterior Angles and their Sum

At each vertex of a polygon, an interior angle and its corresponding exterior angle sum to 180 degrees (they form a linear pair). One of the most important rules is that the sum of the exterior angles of ANY convex polygon is always 360 degrees. This is very useful for regular polygons since all exterior angles are equal. So, for a regular polygon with 'n' sides, each exterior angle is 360 / n degrees. Knowing one exterior angle automatically tells you the corresponding interior angle (180 - exterior angle).

Regular Polygons: Equal Sides and Angles

For a regular polygon, all sides are the same length, and all interior angles are equal. Similarly, all exterior angles are equal. If you know the number of sides 'n', you can find: each exterior angle = 360/n, and each interior angle = [(n-2)*180]/n. Alternatively, you can find each interior angle by first finding the exterior angle and then subtracting it from 180 degrees: 180 - (360/n). This often simplifies calculations.

Sum of Interior Angles

For any polygon with 'n' sides, the sum of its interior angles can be calculated using the formula: Sum of interior angles = $(n-2) \times 180^{\circ}$. For example, a pentagon has 5 sides, so the sum of its interior angles is $(5-2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ}$. This formula is derived by dividing the polygon into triangles from one vertex.

Sum of Exterior Angles

The sum of the exterior angles of any convex polygon is always $360^{\circ}$. This is a constant regardless of the number of sides of the polygon. This property is particularly useful when dealing with regular polygons, as it simplifies calculations for individual angles. An exterior angle is formed by extending one side of the polygon.

Regular Polygons

A regular polygon has all sides equal in length and all interior angles equal. For a regular polygon with 'n' sides:

• Each exterior angle = $\frac{360^{\circ}}{n}$
• Each interior angle = $180^{\circ} - \text{Each exterior angle}$ OR Each interior angle = $\frac{(n-2) \times 180^{\circ}}{n}$. These two methods will always give the same result.

Interior and Exterior Angle Relationship

At each vertex of a polygon, the interior angle and its adjacent exterior angle form a straight line, meaning they add up to $180^{\circ}$. This relationship is key for calculating one type of angle if the other is known. For example, if an interior angle is $100^{\circ}$, then its corresponding exterior angle is $180^{\circ} - 100^{\circ} = 80^{\circ}$.

Core method

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Polygon angle reasoning uses the exterior-angle sum of 360° and the interior-angle sum (n − 2) × 180°. A regular polygon has equal interior angles and equal exterior angles. Concave polygons obey the same interior-sum formula, although one interior angle exceeds 180°. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

For a regular n-gon, each exterior angle is 360°/n and each interior angle is 180° − 360°/n. If each exterior angle is 24°, then n = 360/24 = 15. For an irregular polygon, subtract known interior angles from the total interior sum. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.