AMC 10 · 2021 · #6

Grade 8 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

angles-around-a-pointisosceles-triangleangle-sum-trianglesupplementary-angles identify-subproblemsconvert-to-algebra ↑ Prerequisites: angle-sum-triangle

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A square is drawn, and a point sits on the far side of one side's line. The angle it makes at a corner of the square is 110 degrees. On another side, a second point is marked at the same distance from that corner. Find the degree measure of the angle formed at the second point.

Pick an answer.

(A)

160

(B)

164

(C)

166

(D)

170

(E)

174

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) is the engine here. The whole problem is one picture read correctly: mark the three rays leaving D and notice they fill the complete turn around that point, which converts the given 110° into the angle ∠ ADE by plain subtraction. Tool #7 (Identify Subproblems) breaks the rest into two small, separate jobs — first find the apex angle of △ DEF, then find its base angle, then step outside the triangle to the linear pair at F. Tool #4 (Introduce a Variable) handles the middle job: name the unknown base angle x, and the triangle angle sum turns into a one-line equation. No coordinates, no trigonometry, no lengths are ever needed — only DE = DF matters, never how long they are.

1STEP 1

Three rays fill the turn at D

The angles around one point fill 360 degrees.

∠ ADC + ∠ CDE + ∠ EDA = 360°

2STEP 2

Subtract to get angle ADE

Subtract to get the remaining angle.

∠ EDA = 360° - 90° - 110° = 160°

3STEP 3

Carry that angle into triangle DEF

Carry that angle into the triangle.

DF = DA → ∠ FDE = ∠ ADE = 160°, DE = DF

4STEP 4

The two base angles match

Equal distances make the base angles match.

∠ DFE = ∠ DEF = x

5STEP 5

Solve for the base angle

Solve for the base angle.

x + x + 160° = 180° → x = (180° - 160°)/2 = 10°

6STEP 6

Finish with the straight angle at F

The straight angle finishes it at 170 degrees.

∠ AFE = 180° - ∠ DFE = 180° - 10° = 170°

Answer

170

First, an eyeball test against the figure: E is drawn just barely above the level of the top side AD, so segment FE leaves F almost horizontally, pointing back toward A's side of the picture only very slightly — an angle ∠ AFE just under a straight 180° is exactly what the drawing shows, and 170° fits. Second, a coordinate check. Put D at the origin with DA along the negative x-axis and DC along the negative y-axis, and take DE = DF = 1, so F = (-1, 0). Rotating DC (pointing at -90°) counterclockwise by 110° lands DE at 20°, on the far side of line CD from A as required, giving E = (cos 20°, sin 20°) ≈ (0.940, 0.342). Then FE ≈ (1.940, 0.342) and FA points along (-1, 0); the angle between them is 180° - arctan (0.342/1.940) ≈ 180° - 10° = 170°. Third, a sanity check on the answer choices: all five are between 160 and 174, so the intended answer is an obtuse angle very close to straight — consistent with a base angle of only 10°. Finally, a structural check: the answer never used the side length of the square, and it should not, since scaling the square changes no angle.

💡Key takeaway

This AMC 12 problem needs nothing past Grade 8 angle facts: the three angles around D add to 360°, so ∠ ADE = 360° - 90° - 110° = 160°; that 160° is the apex of an isosceles triangle because DE = DF, leaving base angles of (180° - 160°)/2 = 10°; and ∠ AFE is the rest of the straight line at F, so it is 180° - 10° = 170°.

Three rays fill the turn at D
Subtract to get angle ADE
Carry that angle into triangle DEF
The two base angles match
Solve for the base angle
Finish with the straight angle at F