Wentworth's Plane and Solid Geometry · Book II, Props. I–IV · Book 3: Book II: The Circle

23. Central angles, arcs and chords

Goal: Your student can prove that equal central angles intercept equal arcs, and that equal arcs are subtended by equal chords, with the converses.

⏱ About 40 minutes · 6 steps

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You'll need

  • The book, open to these pages
  • A ruler (straightedge) and a pair of compasses
  • Sharp pencil, eraser and plain paper (or a geometry notebook)

What's in the book

Propositions I–IV (§§ 235–244), including the Law of Converse Theorems (§ 240).

The lesson, step by step

  1. Warm up

    Ask: What is a central angle? A chord? What does "subtend" mean?

  2. The new idea

    A straight line cannot meet the circumference of a circle in more than two points.

    In the same circle or in equal circles, equal central angles intercept equal arcs; and of two unequal central angles the greater intercepts the greater arc.

    In the same circle or in equal circles, equal arcs are subtended by equal chords; and of two unequal arcs the greater is subtended by the greater chord.

    Prop. IV is the converse of Prop. III: equal chords subtend equal arcs.

  3. Draw the figure

    Draw two equal circles side by side (same compass setting). In each, draw a central angle of the same size, and the chords joining the ends of the arcs.

  4. Walk through the proof

    I: if a line met the circumference in three points, three equal radii would go from the centre to one line; but only two equal lines can be drawn from a point to a line (§ 102).

    II: place one circle on the other so the equal angles coincide; the arcs fall together. A bigger angle has a side falling outside, so a bigger arc.

    III: the triangles made by two radii and a chord are equal by § 143. For unequal arcs, use § 154 (bigger included angle, bigger third side).

    IV: reverse it, using three equal sides (§ 150) and § 155.

  5. Your student proves it back

    Close the book. Your student redraws the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("because the radii are equal", "by § 143"). Point to each line or angle as it is named.

    If a step is missing, ask "How do we know that?" rather than supplying it.

    Then read § 240 together: why the converse of a "group of three theorems" is automatically true.

  6. Wrap up

    Ask: In a circle, a chord of 5 cm cuts off an arc. Will a chord of 7 cm cut off a bigger or smaller arc (the minor arc)?

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