9. More angle pairs of parallels

The lesson, step by step

1. Warm up

   Ask: State Prop. XII and draw its figure. What are exterior-interior angles?

2. The new idea

   If two parallel lines are cut by a transversal, the exterior-interior angles are equal.

   If two parallel lines are cut by a transversal, the two interior angles on the same side of the transversal are supplementary.

   Props. XV and XVII are the converses: if those angles are equal (or supplementary), the lines are parallel.

3. Draw the figure

   Draw two parallels AB and CD, and a transversal EF cutting them at H and K. Mark angle EHB and angle HKD with arcs.

4. Walk through the proof

   Prop. XIV: angle EHB equals angle AHK (vertical angles), and angle AHK equals angle HKD (alternate-interior). So angle EHB = angle HKD.

   Prop. XVI: angle EHB + angle BHK is a straight angle, and angle EHB equals angle HKD. Substitute.

   Props. XV and XVII: the book gives no separate proofs. Have your student prove them the same way as Prop. XIII: draw a parallel through H and show it coincides with AB.

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.

6. Exercises

   Ex. 6. If the complement of an angle is one third of its supplement, how many degrees does the angle contain ?

7. Wrap up

   Your student sorts all the angle pairs at a transversal of parallels into "equal" and "supplementary".