Triangles and Geometric Diagrams

Standard Conventions for Sides and Angles

Imagine trying to deliver a parcel to a street where every house has the same number. Geometry without standard notation is just as confusing; examiners need to know exactly which part of a shape you are referring to.

• Vertex (corners) labelling: Always use capital letters (e.g., $A$, $B$, $C$). For polygons, letters must be written in a continuous loop around the perimeter (clockwise or anticlockwise).
• Line segment (sides) labelling: Sides can be named by their endpoints using two capital letters (e.g., side $AB$). Alternatively, standard practice uses the opposite side convention: side $a$ (lowercase) is always opposite vertex $A$, side $b$ is opposite vertex $B$, and so on.
• Interior angle notation: AQA prefers three-letter notation to avoid ambiguity. In $\angle ABC$ (or $\widehat{ABC}$), the middle letter ($B$) must be the vertex where the angle is located. Single letters (angle $A$) are only used if there is no possible confusion, while the Greek letter $\theta$ (theta) represents a general unknown angle.

Right-Angled Triangles and Geometry Shorthand

Mathematical diagrams use a visual shorthand that saves you from writing out long descriptive sentences.

• Right-angled triangle specifics: The hypotenuse is always the longest side, positioned opposite the $90^\circ$ angle. Relative to a chosen angle $\theta$, the opposite is the side facing the angle, and the adjacent is the side next to it (that is not the hypotenuse).
• Equidistant markers (Equality): Equal sides are indicated by matching "ticks" (short dashes) across the segments. Equal angles are indicated by matching arcs inside the corners. Multiple sets of equal parts use double or triple ticks/arcs.
• Line relationships: Parallel lines are shown by matching arrows ($>$ or $>>$) on the lines. Right angles are marked with a small square ($90^\circ$), and the symbol $\perp$ denotes that two lines are perpendicular.
• Whenever an AQA paper states "Diagram NOT accurately drawn", you must rely entirely on the written dimensions and geometric markings rather than measuring the image with a ruler or protractor.

Sketching Diagrams from Written Descriptions

Translating a wordy exam question into a simple visual representation is often the secret to unlocking the solution.

• A sketch is a freehand representation that identifies all important features (labels, lengths, angles). It does NOT need to be exactly to scale, but it must show relative properties (e.g., an obtuse angle should visually look wider than $90^\circ$).
• An accurate drawing requires precise use of a ruler, protractor, and/or compass to exact measurements. Never rub out your construction arcs, as examiners use these to award marks.
• When drawing bearings, always draw a three-figure angle (e.g., $045^\circ$) measured clockwise from a North line.

Worked Examples: Drawing from Text

Example 1: Sketching a Triangle Description: "Sketch triangle $ABC$ where $AB = 7\text{cm}$, $AC = 5\text{cm}$, and $\angle BAC = 40^\circ$."

• Step 1: Draw a horizontal line, label the vertices $A$ and $B$, and mark the length as $7\text{cm}$.

• Step 2: From vertex $A$, draw a line at an approximate $40^\circ$ angle (this should look like less than half of a right angle).

• Step 3: Label the end of this new line as vertex $C$ and mark the length $AC$ as $5\text{cm}$.

• Step 4: Join vertices $B$ and $C$ to complete the triangle.

Example 2: Bearings Sketch Description: "Ship $Q$ is on a bearing of $120^\circ$ from Ship $P$."

• Step 1: Mark a point $P$ and draw a vertical "North" line with an upward arrow and an '$N$'.

• Step 2: Draw a line from $P$ going clockwise, visually extending past the $90^\circ$ (East) position.

• Step 3: Draw a clockwise arc from the North line to your new line and label it $120^\circ$.

• Step 4: Mark the endpoint of the line as $Q$.