AMC 10 · 2023 · #11
Grade 11 geometry-2dPick an answer.
The problem hands over slopes and asks for degrees, so the real work is a translation between two languages: the algebraic language of rise over run and the geometric language of angles. The dictionary entry that connects them is that a line's slope equals the tangent of the angle the line makes with the positive x-axis. So the first move is to give those two tilt angles names, α and β. Once they have names, the angle between the lines is nothing more than their difference, because each line's direction is measured from the same reference, the x-axis. Angles subtract cleanly; slopes do not. The tangent subtraction formula is exactly the tool that turns tan(α - β) back into arithmetic performed on the two given slopes. The absolute value in the standard statement of that rule is not decoration, so it gets a step of its own instead of being copied silently: it is the part that selects the acute angle out of the two angles present at any crossing. After that the work is ordinary fraction arithmetic plus one special-angle fact. A sketch runs alongside the whole way to keep track of which line is steeper, and the five listed measures give a cheap final filter, since the tangent takes each value only once between 0° and 90°.
Four angles, one acute
The four angles take only two values.
Two crossing lines make only one shape of corner and its leftover partner, so asking for "the acute angle" is not ambiguous.
7.G.B.5Draw A DiagramTurn slopes into angles
A slope is a tangent.
A slope is a tilt angle in disguise: the number reports the tangent, the angle reports the same fact in degrees.
10.G-SRT.C.6Introduce A VariableSubtract angles, not slopes
You must subtract the angles.
Angles are the things that actually subtract, and the subtraction formula is the dictionary that turns their difference back into slope arithmetic.
Angles are the things that actually subtract, and the subtraction formula turns their difference back into a tangent.
▸ Why?
The crossing angle is the difference of the two tilts, since angles along one line add and subtract.
▸ Why?
A slope is the tangent of a tilt, the far leg over the near one, so slopes and tilts are one thing.
The absolute value picks acute
The absolute value picks the acute one.
The sign of the tangent is the only thing separating the two angles at a crossing, so throwing the sign away is the same as demanding the acute one.
11.F-TF.A.2Change Focus Count The ComplementEvaluate the fraction
The fraction collapses to one.
Split a messy fraction into a top job and a bottom job; when both land on 5/3, the division is over before it starts.
7.NS.A.3Identify SubproblemsRead the angle
A tangent of one means 45 degrees.
Between 0° and 90° the tangent never repeats itself, so a single tangent value pins down a single acute angle.
10.G-SRT.C.8Eliminate PossibilitiesA slope is a tilt angle in disguise, and angles subtract where slopes do not — so trade the slopes for tilt angles, subtract those with the tangent subtraction formula, and let the absolute value keep the acute one.
- Four angles, one acute
- Turn each slope into an angle
- Subtract the angles, not the slopes
- Why the absolute value picks acute
- The fraction collapses to 1
- Read the angle off the tangent