Triangles and Geometric Diagrams

Why geometric labels matter

A geometric diagram turns a written description into a visible arrangement of points, sides and angles. Labels allow us to refer to each part precisely. Points and vertices are normally labelled with capital letters, such as $A$, $B$ and $C$.

The notation $\triangle ABC$ names the triangle whose vertices are $A$, $B$ and $C$. The line segments joining them are $AB$, $BC$ and $CA$. In this context, $AB$ may refer to the side itself or to its length; the surrounding statement normally makes the intended meaning clear. For example, $AB=5\text{ cm}$ concerns its length.

Labelling the sides of a triangle

A triangle may also use lower-case letters for its side lengths. Each lower-case side label is placed opposite the matching capital vertex:

• side $a$ is opposite vertex $A$, so it lies along $BC$;
• side $b$ is opposite vertex $B$, so it lies along $CA$;
• side is opposite vertex , so it lies along .

Thus, in the standard labelling of $\triangle ABC$,

$a=BC,\qquad b=CA,\qquad c=AB.$

“Opposite” is the key relationship: side $a$ does not touch vertex $A$. This convention later makes formulae such as the sine rule easier to read because each side is paired with its opposite angle.

Referring to angles accurately

An angle can be named with three capital letters. The middle letter is always the vertex of the angle. Therefore:

• $\angle ABC$ is the angle at $B$, between $BA$ and $BC$;
• $\angle BAC$ is the angle at , between and ;

The order of the outer two letters may be reversed without changing the angle, so $\angle ABC$ and $\angle CBA$ both name the angle at $B$. However, moving $B$ away from the middle would name a different angle. A single letter, such as angle $B$, is acceptable when only one angle at could be meant, but three-letter notation removes ambiguity.

Turning words into a diagram

A diagram should record the relationships given in the description rather than merely look attractive. First read the whole description, then identify the objects and constraints: which points are joined, which side or angle has a measurement, and whether anything is equal, parallel or perpendicular.

For example, consider:

Triangle $PQR$ has $PQ=PR$, $\angle QPR=40^\circ$, and is horizontal.

Start with a horizontal segment and label its endpoints $Q$ and $R$. Place $P$ away from that segment and join it to both endpoints. Because $P$ is the middle letter of $\angle QPR$, put $40^\circ$ at . Put identical single dashes across and to represent their equal lengths. Nothing in the description gives a side length, so none should be invented.

A sketch usually need not be accurately drawn unless the question asks for an accurate drawing or construction. It should nevertheless be large and clear enough for every label and mark to be seen. If a first sketch becomes crowded or gives a misleading special case, redraw it.

Reading names in the stated order

When a polygon is named, its vertices are normally listed consecutively around its boundary. For quadrilateral $ABCD$, the sides are $AB$, $BC$, $CD$ and $DA$; and join non-adjacent vertices and are the diagonals. Plotting the letters out of order can change the shape and make later statements appear to contradict one another.

Descriptions may also specify relative positions. Phrases such as “$D$ lies on $AB$”, “$C$ is inside the triangle” or “the lines meet at $X$” should determine where labels are placed. Draw the main shape first, add internal or external points next, and then add lines and measurements.

Using conventional marks

Geometry diagrams use marks as a compact language:

• matching dashes across sides show equal lengths;
• matching arcs show equal angles;
• a small square marks a right angle;
• matching arrow marks show parallel lines.

Marks must match the written information. Two sides with one dash each form one equal group; sides with two dashes form a different group. Do not add a right-angle square, equal-side dash or parallel arrow simply because the sketch appears to have that property.

Checking the finished representation

Return to the description and check it clause by clause. Confirm that every named point appears, required points are joined, measurements are placed at the correct side or angle, and each stated relationship has the appropriate mark. Also check that you have not introduced unsupported assumptions. A good diagram contains all the given information, but distinguishes that information from features that happen to result from a rough sketch.