Wentworth's Plane and Solid Geometry · Book I, Props. XXXV–XXXVIII · Book 2: Book I: Rectilinear Figures
16. More on parallelograms, and parallels cutting equal parts
Goal: Your student can prove the remaining parallelogram theorems and that parallels cutting equal parts on one transversal cut equal parts on every transversal.
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 XXXV–XXXVIII (§§ 183–190) with the important corollaries on the midpoints of the sides of a triangle and the median of a trapezoid, and Exercises 13–16.
The lesson, step by step
Warm up
Ask: Prove that the opposite sides of a parallelogram are equal.
The new idea
If two sides of a quadrilateral are equal and parallel, then the other two sides are equal and parallel, and the figure is a parallelogram.
The diagonals of a parallelogram bisect each other.
If three or more parallels intercept equal parts on one transversal, they intercept equal parts on every transversal.
Prop. XXXVII (§ 185) gives a fourth test: two parallelograms are equal if two sides and the included angle are equal.
Draw the figure
Draw a parallelogram and both diagonals, crossing at O. Separately, draw a triangle ABC, mark D and E, the midpoints of AB and AC, and join DE.
Walk through the proof
XXXV: the diagonal makes two triangles equal by two sides and the included angle (alternate-interior angles), so the other sides are equal, and their alternate angles show they are parallel.
XXXVI: triangles AOE and COB are equal (a side and two angles), so AO = OC and BO = OE.
XXXVIII: draw lines through A, B, C parallel to the first transversal; the small triangles are equal (a side and two angles), so the parts are equal.
Corollaries § 188–190: the line through the midpoint of one side parallel to the base bisects the other side; the line joining midpoints is parallel to the third side and half as long.
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.
Exercises
Try Exercises 13–16 on page 52. For example:
Ex. 13. The median from the vertex to the base of an isosceles triangle is perpendicular to the base, and bisects the vertical angle.
Wrap up
Measure DE and BC in your triangle drawing. Is DE half of BC?
Answers to the book's problems
Ex. 13: the two triangles made by the median are equal by three sides (Prop. XXIV). Ex. 14: alternate-exterior angles equal means the vertical angles, which are alternate-interior, are equal; use Prop. XIII. Ex. 15: each exterior angle equals an interior angle on the same side (vertical angles); use Prop. XVI. Ex. 16: reverse the reasoning of Ex. 15 and use Prop. XVII.
Worked out for this site from the scan. If a number in your copy differs, trust the book.