AMC 10 · 2011 · #10
Grade 8 geometry-2dPick an answer.
No picture is given, so the first move is to draw the rectangle and mark the two equal angles at M (Tool #1). The drawing exposes two facts you would otherwise miss: AB ∥ DC turns ∠ AMD into an alternate interior angle equal to ∠ MDC, and that makes triangle CMD isosceles. From there Tool #7 (Identify Subproblems) splits the work into a short isosceles-triangle step and a right-triangle step, and Tool #4 (Introduce a Variable) lets the straight-line angle sum finish it in one equation.
Draw and label the rectangle
Labelling the corners makes every length explicit.
A labeled picture turns hidden geometry conditions into lines and angles you can actually chase.
4.G.A.1Draw A DiagramSpot the alternate interior angles
The parallel sides give a pair of alternate angles.
When two parallel lines are cut by a slanted line, the two inside angles on opposite sides are twins.
When two parallel lines are cut by a slanted line, the two inside angles on opposite sides are twins.
▸ Why?
A line crossing two parallels makes matching angles at both crossings.
▸ Why?
Two equal base angles then face two equal sides, so the triangle they sit in is isosceles.
Find the isosceles triangle
That makes one triangle isosceles.
Equal base angles force equal sides, so a triangle with two matching angles is isosceles.
8.G.A.5Identify SubproblemsRead the right triangle at B
The familiar right triangle gives 30 degrees.
A right triangle with a leg half the hypotenuse is always the 30-60-90 triangle.
8.G.B.7Identify SubproblemsAdd the angles along the line
Angles along the line give 75 degrees, choice (E).
Angles resting on one straight line always total a straight angle, 180°.
4.MD.C.7Introduce A VariableParallel sides made the angle bounce back equal, turning a triangle isosceles; then a 30-60-90 corner and the straight-line total of 180° pinned the angle at 75°.
- Draw and label the rectangle
- Spot the alternate interior angles
- Find the isosceles triangle
- Read the right triangle at B
- Add the angles along the line