Why your triangle proof loses marks even when the answer is right

There is a particular kind of unfairness that students feel in the Triangles chapter. They get the right answer. The ratio is correct. The final number matches the answer key. And the marks are still gone.

It is not unfair, and once a student sees why, they usually stop losing those marks within a week. The board is not paying for the answer in this chapter. It is paying for the reasoning.

What is actually being marked

In a similarity question, the examiner has a marking scheme with a mark attached to each justification, not to the final ratio. Watch what happens when the justifications are removed but the answer is kept:

The order that cannot be reversed

Almost every lost mark in this chapter comes from doing these in the wrong order. The rule is simple and absolute:

1. State what is given, including the parallel line or the right angle.
2. Name the two triangles in matching order.
3. Justify the similarity, by AA, SAS, SSS, BPT or its converse.
4. Only now use the corresponding side ratios.

The correspondence trap

When a student writes $\triangle ABC \sim \triangle DEF$, they are making four claims at once: that $A$ matches $D$, $B$ matches $E$, $C$ matches $F$, and therefore that the sides pair up in that same order.

Write the letters in the wrong order and every ratio afterwards is wrong, even though the triangles genuinely are similar. This is the single most common source of a completely correct-looking answer that scores nothing.

The fix is a habit: before writing the similarity statement, mark the equal angles on the diagram and read the letters off the diagram in the order the angles match. Never write it from memory.

BPT and its converse are not the same tool

Students routinely use these two in place of each other, and examiners routinely notice.

• BPT: you are told a line is parallel, and you conclude the sides are divided in the same ratio.
• Converse of BPT: you are told the sides are divided in the same ratio, and you conclude the line is parallel.

Ask one question before choosing: is the parallel line something the question gave me, or something it wants me to prove? Given means BPT. To prove means the converse.

The same question applies to Pythagoras. If the right angle is given, use the theorem. If you are being asked to establish the right angle, you need the converse.

The pre-submission checklist

• Have I written down what was given, in words?
• Are my triangle letters in matching order, read off the diagram?
• Have I named the reason for similarity, not just asserted it?
• Did I prove the parallel or right angle before using it, if it was not given?
• Does every line have a reason I could say out loud?

This is a chapter where a student who writes slowly and justifies everything beats a faster student who writes only the answer. That is worth saying to your child directly, because it is the opposite of what they assume.