AMC 10 · 2007 · #15
Grade 8 geometry-2dPick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The chain ∠ A=2∠ B=3∠ C=4∠ D ties every angle to ∠ A, so naming ∠ A=x (Tool #4) writes all four angles in terms of one unknown. The fact that a quadrilateral's angles sum to 360° then turns the picture into a single equation (Tool #13). Solving gives a decimal, and rounding lets us match one of the five listed choices (Tool #3).
Name ∠ A and cascade the others
Let ∠ A=x. The chain then forces ∠ B=x/2, ∠ C=x/3, ∠ D=x/4 — four angles, one unknown.
One equal-value chain lets you pin every angle to a single unknown, so there is really only one number to find.
7.EE.B.4Introduce A VariableUse the quadrilateral angle sum
Any quadrilateral splits along a diagonal into two triangles, so its angles total 360°: x+x/2+x/3+x/4=360.
A fixed total (360°) turns four unknown angles into one equation you can actually solve.
A four-sided figure's angles add to a fixed total, so four unknowns become one solvable equation.
▸ Why?
A diagonal cuts the figure into two triangles, so its total is twice a triangle's fixed total.
▸ Why?
Going once around inside the figure sweeps that same fixed total, whatever the shape.
Combine the fractions
Over the common denominator 12 the numerators add to 12+6+4+3=25, so 25x/12=360.
A shared denominator lets the four pieces merge into one clean fraction of x.
5.NF.A.1Introduce A VariableSolve and round
Multiply by 12 and divide by 25: x=4320/25=172.8, which rounds to 173° — choice (D).
Undoing the 25/12 coefficient isolates x, and 0.8 rounds up to the next whole degree.
5.NBT.A.4Eliminate PossibilitiesWhen angles are chained by an equal-value string, name the biggest one, write the rest as fractions of it, and let the fixed 360° total do the solving.
- Name ∠ A and cascade the others
- Use the quadrilateral angle sum
- Combine the fractions
- Solve and round