Wentworth's Plane and Solid Geometry · Book I, Props. I–IV · Book 2: Book I: Rectilinear Figures
5. Straight angles, supplements and vertical angles
Goal: Your student can prove that all straight angles are equal, that adjacent angles with exterior sides in a line are supplementary (and the converse), and that vertical angles are equal.
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
Propositions I to IV (§§ 81–94) with their corollaries, the first proofs in the book, then Exercises 1–3 on complements and supplements.
The lesson, step by step
Warm up
Ask: What is a straight angle? What are supplementary angles? How many degrees in a right angle?
The new idea
These are the first real proofs. Each one is short. The statements:
All straight angles are equal.
If two adjacent angles have their exterior sides in a straight line, these angles are supplementary.
Conversely: If two adjacent angles are supplementary, their exterior sides are in the same straight line.
If one straight line intersects another straight line, the vertical angles are equal.
Draw the figure
For Prop. II: draw a long line AB, mark O on it, and draw a slanting line OD up from O. That makes the adjacent angles AOD and BOD.
For Prop. IV: draw two lines crossing at C. Label one line OP and the other AB, so the vertical angles are OCB and ACP.
Walk through the proof
Prop. I: lay one straight angle on the other with vertex on vertex and one side on a side; the other sides must fall together, because two straight lines with two common points coincide.
Prop. II: the two angles together make the straight angle AOB, so their sum is a straight angle: they are supplementary.
Prop. III (converse): suppose another line CF were in line with AC; then two angles would have the same supplement, so they are equal and the lines coincide.
Prop. IV: angles OCB and ACP each have angle OCA as a supplement (Prop. II), and supplements of the same angle are equal (§ 85).
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 to say back is Prop. IV (vertical angles).
Exercises
Try Exercises 1–3 on page 18. For example:
Ex. 1. Find the complement and the supplement of an angle of 49°.
Ex. 3. Find the number of degrees in an angle if it is double its supplement; if it is one third of its supplement.
For Ex. 2 and 3, call the angle x and write an equation.
Wrap up
Ask: Why are vertical angles equal? Your student should answer with the reason, not just "they look equal."
Tip
The author suggests writing proofs at first "in his own language", step under step, with a neat figure. Keep that format from the start.
Answers to the book's problems
Ex. 1: complement 41°, supplement 131°. Ex. 2: 60° (x = 2(90 − x)); 18° (x = (90 − x)/4). Ex. 3: 120° (x = 2(180 − x)); 45° (x = (180 − x)/3).
Worked out for this site from the scan. If a number in your copy differs, trust the book.