Competition · AMC preparation · step 4 of 4

AMC 8 · 2000 · #13

Grade 8 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

angle-sum-triangleisosceles-trianglesupplementary-angles identify-subproblems ↑ Prerequisites: angle-sum-triangleisosceles-triangle

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Triangle CAT is isosceles with ∠ ACT = ∠ ATC and apex angle ∠ CAT = 36°. Segment TR bisects ∠ ATC, with R on side CA. Find ∠ CRT.

Pick an answer.

(A)

$36^\circ$

(B)

$54^\circ$

(C)

$72^\circ$

(D)

$90^\circ$

(E)

$108^\circ$

AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The figure is given, so Tool #1 (Draw a Diagram) tells us to read every angle right off the picture rather than set up equations. The bisector TR cuts the big triangle △ CAT into a small triangle △ CRT that contains the angle we want. Tool #7 (Break Into Subproblems) splits the work into two short angle-sum steps: first use △ CAT to find the base angle ∠ ATC, then use △ CRT (whose two other angles we now know) to find ∠ CRT. No algebra needed beyond "angles in a triangle add to 180°."

1STEP 1

Find the base angles

Angle sum in △ CAT: the two equal base angles split 180° - 36°, so each base angle ∠ ATC = 72°.

∠ ACT = ∠ ATC = (180° - 36°)/2 = 144°/2 = 72°

2STEP 2

Halve the bisected angle

TR bisects ∠ ATC = 72°, so the half inside △ CRT is ∠ RTC = 36°.

∠ RTC = 1/2∠ ATC = 72°/2 = 36°

3STEP 3

Find the last angle

In △ CRT the angle at C stays ∠ ACT = 72° and at T is ∠ RTC = 36°, so ∠ CRT = 72°.

∠ CRT = 180° - ∠ RCT - ∠ RTC = 180° - 72° - 36° = 72° → (C)

Answer

72°

Two sanity checks. (i) Angle sum in △ CRT: 72° + 36° + 72° = 180°. (ii) The exterior-angle view: ∠ CRT is the exterior angle of △ ART at R, so it equals the sum of the two remote interior angles, ∠ RAT + ∠ ATR = 36° + 36° = 72° — same answer. The trap choices line up with common slips: (A) 36° stops at the apex angle or the bisected half; (B) 54° comes from halving 108° as if the apex were the base angle; (E) 108° is the supplement 180° - 72°, i.e. the other angle at R on segment CA.

💡Key takeaway

Label every angle you can on the picture, then the small triangle △ CRT has angles 72° and 36° already known — the third angle has to be 180° - 72° - 36° = 72°.

Find the base angles
Halve the bisected angle
Find the last angle

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Find the base angles

Halve the bisected angle

Find the last angle

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