One page per lesson, based on the teaching plan’s terms and concepts. Each page gives the vocabulary, the key rules, a homework helper, and common mistakes.
| Lesson |
Topic |
One-pager |
| 1 |
Geometry on a Sphere |
Lesson 1 |
| 2 |
The Saccheri Quadrilateral |
Lesson 2 |
| 3 |
The Geometries of Lobachevsky and Riemann |
Lesson 3 |
| 4 |
The Triangle Angle Sum Theorem Revisited |
Lesson 4 |
The three geometries at a glance
| Question |
Euclidean |
Lobachevskian |
Riemannian |
| Parallels through a point not on a line |
exactly one |
more than one |
none |
| Summit angles of a Saccheri quadrilateral |
right |
acute |
obtuse |
| Sum of a triangle’s angles |
180° |
less than 180° |
more than 180° |
The key theorems of the chapter
- Theorem 87: The summit angles of a Saccheri quadrilateral are equal.
- Theorem 88: The segment joining the midpoints of the base and summit is perpendicular to both.
- Theorem 89: Unequal legs give unequal summit angles in the same order.
- Theorem 90: Unequal summit angles give unequal legs in the same order.
- Lobachevskian Theorem 1: The summit of a Saccheri quadrilateral is longer than its base.
- Lobachevskian Theorem 2: A midsegment of a triangle is less than half the third side.
- Lobachevskian Theorem 3: The sum of a triangle’s angles is less than 180°.
- Lobachevskian Theorem 4: Similar triangles must be congruent.