Wentworth's Plane and Solid Geometry · Book II, Props. XVII–XX · Book 3: Book II: The Circle
29. Inscribed angles and angles made by chords, tangents and secants
Goal: Your student can prove that an inscribed angle is measured by half its arc, and the related theorems for chords, tangents and secants.
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 XVII–XX (§§ 289–296) with corollaries, the notes on positive and negative quantities and the Principle of Continuity (§§ 297–299), and Exercises 97–100.
The lesson, step by step
Warm up
Ask: What is a central angle measured by?
The new idea
An inscribed angle is measured by half the arc intercepted between its sides.
An angle formed by two chords intersecting within the circumference is measured by half the sum of the intercepted arcs.
An angle included by a tangent and a chord drawn from the point of contact is measured by half the intercepted arc.
Prop. XX (§ 296): an angle formed outside the circle by two secants, two tangents, or a tangent and a secant is measured by half the difference of the intercepted arcs. The book leaves its proof to the student.
Draw the figure
Draw a circle, centre C. Draw an inscribed angle PBA with one side BP passing through C. Join CA. Then draw two more figures: one with C inside the angle and one with C outside it.
Walk through the proof
XVII, case 1: triangle ACB is isosceles, so angle A = angle B. The exterior angle PCA = A + B = 2B. PCA is measured by arc PA, so B is measured by half of it.
Cases 2 and 3: draw the diameter from B and add (or subtract) two angles of case 1.
XVIII: draw AE parallel to BD; the arcs AB and DE are equal (§ 257), and angle COD equals inscribed angle CAE.
XIX: draw a chord parallel to the tangent and use alternate-interior angles and Prop. XVII.
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 let your student try to prove Prop. XX alone, using an exterior angle of a triangle.
Exercises
Try Ex. 97–100 on pages 104–105. For example:
Ex. 97. The opposite angles of an inscribed quadrilateral are supplementary.
Wrap up
Read §§ 290–293: an angle in a semicircle is a right angle. Have your student test it with a set square in a drawn semicircle.
Answers to the book's problems
Ex. 97: the two opposite angles are measured by halves of two arcs that together make the whole circumference, so together by a semicircumference: two right angles. Ex. 100: draw the common internal tangent at A, meeting BC at M. Tangents from M are equal (Prop. XII), so MB = MA = MC; a median equal to half the side it bisects makes a right angle (Ex. 48).
Worked out for this site from the scan. If a number in your copy differs, trust the book.