Wentworth's Plane and Solid Geometry · Book I, Props. X–XIII · Book 2: Book I: Rectilinear Figures
8. Parallel lines and the transversal
Goal: Your student can define parallel lines, name the angles made by a transversal, and prove that the alternate-interior angles of parallels are equal (and the converse).
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)
- A protractor
What's in the book
The definition of parallels, the parallel axiom, Propositions X–XIII (§§ 103–111), the names of the angles a transversal makes, and Exercise 5.
The lesson, step by step
Warm up
Ask: State Prop. VI. Why can't there be two perpendiculars from one point to a line?
The new idea
Two straight lines in the same plane perpendicular to the same straight line are parallel.
If a straight line is perpendicular to one of two parallel lines, it is perpendicular to the other also.
If two parallel lines are cut by a transversal, the alternate-interior angles are equal.
Prop. XIII is the converse: if the alternate-interior angles are equal, the lines are parallel.
Also read the axiom in § 105: through a given point only one straight line can be drawn parallel to a given straight line.
Draw the figure
Draw two parallel lines (slide a set square along a ruler, or use the lines of ruled paper). Draw a slanting transversal across both.
Number the eight angles as in the book's figure for § 109, and have your student point out the interior, exterior, alternate-interior and exterior-interior angles.
Walk through the proof
Prop. X: if the two perpendiculars met, there would be two perpendiculars from one point to a line (impossible by Prop. VI).
Prop. XI: draw MN through C perpendicular to HK. MN is parallel to AB (Prop. X), and only one parallel goes through C, so EF is MN.
Prop. XII: through O, the midpoint of BC, draw a perpendicular to both lines. Turn the lower half-figure onto the upper one: the vertical angles at O match, C falls on B, and the perpendiculars coincide, so the two angles coincide.
Prop. XIII: draw MN through H parallel to CD. Its alternate angle equals angle HKD, which also equals angle AHK, so AB and MN coincide.
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.
For today, the most important one is Prop. XII.
Exercises
Ex. 5. Find the complement and the supplement of an angle that contains 37° 53' 49".
Remember: 90° = 89° 59' 60".
Wrap up
Point to pairs of angles on your figure and ask: "Equal or supplementary?"
Answers to the book's problems
Ex. 5: complement 52° 6' 11"; supplement 142° 6' 11".
Worked out for this site from the scan. If a number in your copy differs, trust the book.