AMC 10 · 2007 · #11
Grade 8 geometry-2dPick an answer.
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
Naming the largest angle cascades all four into one letter.
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
The angle sum gives one equation.
A fixed total (360°) turns four unknown angles into one equation you can actually solve.
The four angles of a quadrilateral add to a full turn, which turns four unknowns into one equation.
▸ Why?
A diagonal cuts the quadrilateral into two triangles, and each contributes a straight angle.
▸ Why?
Two straight angles together make a full turn about a point, which is the fixed total.
Combine the fractions
Combining the fractions makes it a single coefficient.
A shared denominator lets the four pieces merge into one clean fraction of x.
5.NF.A.1Introduce A VariableSolve and round
Solving and rounding gives 173, choice (C).
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