Lesson 2 — The Saccheri Quadrilateral
Big idea: A Saccheri quadrilateral has two equal legs perpendicular to its base, and its summit angles are equal — but it is not automatically a rectangle.
Words to know
| Word | Meaning |
|---|---|
| Birectangular quadrilateral | A quadrilateral that has two sides perpendicular to a third side. |
| Base | The side to which the two perpendicular sides are drawn. |
| Legs | The two sides perpendicular to the base. |
| Summit | The side opposite the base. |
| Base angles | The angles at the ends of the base; both are right angles. |
| Summit angles | The angles at the ends of the summit. |
| Saccheri quadrilateral | A birectangular quadrilateral whose legs are equal. |
Concepts to know
- In a birectangular quadrilateral, the legs are perpendicular to the base, so the two base angles are right angles.
- A Saccheri quadrilateral is the special case where the legs are equal.
- Girolamo Saccheri (1733) tried to prove Euclid’s Parallel Postulate by assuming the summit angles were acute or obtuse and looking for a contradiction.
- Theorem 87: The summit angles of a Saccheri quadrilateral are equal.
- Theorem 88: The line segment connecting the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both of them.
- Theorem 89: If the legs of a birectangular quadrilateral are unequal, the summit angles opposite them are unequal in the same order.
- Theorem 90: If the summit angles of a birectangular quadrilateral are unequal, the legs opposite them are unequal in the same order.
- Theorems 87–90 are proved without the Parallel Postulate, so they hold in Euclidean and non-Euclidean geometries.
- You may not assume the summit angles are right angles or that the figure is a rectangle: that conclusion uses the angle-sum theorems, which depend on the Parallel Postulate.
Formulas / rules
- No new formula.
- Base angles of a birectangular quadrilateral are right angles.
- In Euclidean geometry a Saccheri quadrilateral is a rectangle; in other geometries it need not be.
- Do not use “sum of quadrilateral angles = 360°” in these proofs.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Identify the parts | Base = side the legs are perpendicular to; legs = the two perpendicular sides; summit = side opposite the base. |
| Prove summit angles equal | Use Theorem 87; draw both diagonals and use congruent triangles. |
| Prove the midpoint segment is perpendicular | Use Theorem 88 and congruent triangles. |
| Compare legs and summit angles | Use Theorem 89 or 90; “same order” means the longer leg is opposite the larger summit angle. |
| Explain why it is not a rectangle yet | A rectangle needs right summit angles, which requires the Parallel Postulate. |
| Name the angles | Base angles are at the base; summit angles are at the summit. |
Watch out
- The legs are perpendicular to the base, not to the summit.
- Equal summit angles do not make them right angles.
- Do not use Theorem 32 (quadrilateral angle sum 360°) — it depends on the Parallel Postulate.
- “Same order” is about which angle is larger, not about equal measures.