The missing-letter rule

Lesson 1 · ordering a triangle's sides and angles · about 5 minutes · fixes Version 3, questions 2 and 8

The bigger the side, the bigger the angle across from it. The side across from an angle is the one whose name does not contain that angle's letter.

That's the whole rule (Math Open Reference: "the shortest side is always opposite the smallest interior angle; the longest side is always opposite the largest"). Your student's mistakes almost always come from the matching, not the rule. Kids tend to pair an angle with a side that touches it. The missing-letter trick removes that guesswork.

The method

  1. Match. Next to each side, write the letter its name is missing. \(YZ\) is missing \(X\), so \(YZ\) is across from \(\angle X\).
  2. Rank. Order whichever you were given, sides or angles. (Given only two angles? The three angles add to \(180°\), so find the third first.)
  3. Carry over. The ranking transfers straight across: shortest side ↔ smallest angle, and so on.

Question 2, worked

Version 3 · Q2

In \(\triangle XYZ\), \(XY=12\), \(YZ=8\), \(XZ=17\). List the angles from least to greatest.

X Y Z 17 12 8
Each side is coloured the same as the angle across from it. Not to scale.
SideMissing letterRank
\(YZ=8\)\(\angle X\)least
\(XY=12\)\(\angle Z\)middle
\(XZ=17\)\(\angle Y\)greatest

Answer: \(\angle X,\ \angle Z,\ \angle Y\).

Question 8, worked (the other direction)

Version 3 · Q8

In \(\triangle DEF\), \(m\angle D=48°\) and \(m\angle E=75°\). List the sides from shortest to longest.

Rank first, and don't skip the hidden third angle: \(\angle F=180°-48°-75°=57°\). So \(D\ (48°) < F\ (57°) < E\ (75°)\).

AngleSide without that letterRank
\(\angle D=48°\)\(EF\)shortest
\(\angle F=57°\)\(DE\)middle
\(\angle E=75°\)\(DF\)longest

Answer: \(EF,\ DE,\ DF\).

What to watch for with your student. Ask him "which side is across from \(\angle Y\)?" If he points at a side that touches \(Y\), that's the bug. Have him cover the letter \(Y\) with a finger and read the two letters that are left.

Your turn

Do each one on paper before clicking. The effort of recalling is what makes it stick.