Reasoning · Grade 4-1 Polygons and Angles

Problem

Sum of a polygon's exterior angles

A convex pentagon has five different sides and angles. At each vertex one side is continued straight past the corner. The angle between that extension and the other side is marked. Find the sum of the five marked angles.
1 5 4 3 2
Your answer
How to solve
Strategy Organize Information in More Ways — Since no single marked angle can ever be found, the only hope is to bundle them with something already known. Each marked angle has a partner sitting right beside it — the pentagon's own interior angle at the same vertex — and the pair always makes a straight line. So I re-organize the five unknowns as five straight lines minus five interior angles. The interior angle total is then handled by the easier related problem of a triangle, whose 180° I already know, by slicing the pentagon into triangles.
1STEP 1

Pair each marked angle with the interior angle beside it

A marked angle plus its interior neighbour makes 180 degrees.

① + (interior angle at that vertex) = 180°
2STEP 2

Add up all five straight lines at once

Over five vertices that is 900 degrees.

(① + ② + ③ + ④ + ⑤) + (sum of the five interior angles) = 5 × 180° = 900°
3STEP 3

Get the pentagon's interior angle sum from triangles

A pentagon's interior angles add to 540 degrees.

3 × 180° = 540°
4STEP 4

Subtract to leave only the marked angles

Subtracting leaves 360 degrees.

① + ② + ③ + ④ + ⑤ = 900° - 540° = 360°
Answer
360 degrees
900 − 540 = 360
360° is exactly one full turn, and five angles averaging 72° each is believable for the fairly narrow arcs drawn in the figure. Test it on a shape where every angle is known: a regular pentagon has interior angles of 108°, so each marked angle would be 180° - 108° = 72°, and 5 × 72° = 360°. Test it on a triangle instead: 3 × 180° - 180° = 360° again; on a square, 4 × 180° - 360° = 360°. The answer is one full turn no matter how many sides the polygon has, which is why the problem can be answered with no measures printed at all.
Takeaway

Pair every angle you cannot find with one you can — five straight lines minus one pentagon leaves exactly one full turn, 360°.

  • Pair each marked angle with the interior angle beside it
  • Add up all five straight lines at once
  • Get the pentagon's interior angle sum from triangles
  • Subtract to leave only the marked angles