Understanding angles (age 12+)

Activities to try

1. 1

   Construct the perpendicular bisector with compasses and straight edge

   Why it works: Every point on the perpendicular bisector is equidistant from P and Q. Equal-radius arcs locate two such points, and the straight line through them is the perpendicular bisector; visible arcs make the method inspectable, though arcs alone do not prove they were placed correctly.

   1. Take a line segment PQ drawn diagonally (not horizontal or vertical), a pair of compasses and a straight edge — no ruler measuring and no protractor.
   2. Open the compasses to MORE than half of PQ; discuss why an opening of half or less means the arcs only touch or never meet, and why an enormous opening just wastes space.
   3. Place the compass point on P and sweep an arc above and below the segment; WITHOUT changing the radius, place the point on Q and sweep again so the arcs cross at two points.
   4. Rule the straight line through the two crossing points, leaving all construction arcs visible as evidence of method.
   5. Explain why the line passes through the midpoint at 90°: the four equal radii form a rhombus, whose diagonals perpendicularly bisect each other.
2. 2

   Bisect an angle using the same equal-arc idea

   Why it works: The same equidistance principle that bisects a segment also bisects an angle — one property, reused, rather than a second memorised routine.

   1. Take a marked angle (try one at a non-standard orientation), compasses and a straight edge.
   2. Place the compass point at the vertex and sweep an arc that crosses both arms of the angle at two points.
   3. WITHOUT changing the radius, place the point on each of those two crossing points in turn and sweep equal arcs that meet inside the angle.
   4. Rule the line from the vertex through that meeting point, leaving the arcs visible.
   5. Explain how this reuses the equal-radius property: equal lengths from the vertex and equal arcs from the arm-crossings force the two halves to be congruent, so the line bisects the angle.
3. 3

   See the rhombus break into a kite with a dynamic compass (concept-before-dexterity scaffold)

   Why it works: When two circles cross, their common chord is perpendicular to the line through their centres. With equal radii, that chord also bisects PQ; with unequal radii it does not. Keep both circles crossing during the comparison, and explain the equal-radius rule rather than memorising it.

   1. Use an on-screen compass whose radius locks once set: plant it on P and sweep an arc, then on Q and sweep an equal arc so the two equal circles cross at two points.
   2. Rule the line through the crossings and observe the highlighted locus showing every point on it is equidistant from P and Q.
   3. Now drag a slider that shrinks the second circle's radius (keeping it large enough that the two circles still cross); watch the crossing points drift, the line slide sideways while staying perpendicular to PQ, and the label flip from RHOMBUS to KITE — the line visibly misses the midpoint.
   4. Articulate in your own words why 'keep the compass fixed' is a consequence of the rhombus property, not an arbitrary rule.
   5. Transfer to real compasses and paper once the concept is secure.