Neutral geometry
And then we add Euclid’s parallel postulate Saccheri’s dilemma Options are:
Summit angles are right Wants Summit angles are obtuse
Was able to rule out, and we’ll see how Summit angles are acute
The hypothesis of the acute angle is absolutely false, because it is repugnant to the nature of the straight line! Rule out obtuse angles:
If we knew that a quadrilateral can’t have the sum of interior angles bigger than 360, we’d be fine.
We’d know that if we knew that a triangle can’t have the sum of interior angles bigger than 180.
Hold on! Isn’t the sum of the interior angles of a triangle EXACTLY180? Theorem: Angle sum of any triangle is less than or equal to 180º Suppose there is a triangle with angle sum greater than 180º, say anglfle sum of ABC i s 180º + p, w here p> 0. Goal: Construct a triangle that has the same angle sum, but one of its angles is smaller than p. Why is that enough? We would have that the remaining two angles add up to more than 180º: can that happen? Show that any two angles in a triangle add up to less than 180º. What do we know if we don’t have Para lle l Pos tul at e??? Alternate Interior Angle Theorem: If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parallel. Converse of AIA
Converse of AIA theorem: If two lines are parallel then the aliilblternate interior angles cut by a transversa l are congruent.
Converse of AIA Parallel Postulate If the converse of AIA holds then the sum of the interior angles of a triangle is 180
So if the parallel postulate holds then we know that the sum of the interior angles of a triangle is EXACTLY180. But what if we don’t? Exterior angle theorem: An exterior angle of a triangle is greater than either remote interior angle.
Proof: Suppose contrary. Then either: 1. DCB ABC, or 2. DCB < ABC. Supply the arguments in each case: 1. We have
B
A C 2. Here D B
A C D Show that any two angles in a triangle add up to less th an 180º
B
A
D C Consequences:
Theorem (longer side): Given two non-congruent sides in a triangle, the angle opposite the longer side is greater than the angle opposite the shorter side. Theorem (larger angle): Given two non-congruent sides in a triangggppggle, the angle opposite the longer side is greater than the angle opposite the shorter side. Theorem (triangle inequality): The sum of the lengths of any two s ides o f a tr iang le is greater t han t he lengt h o f the third side. Consequences:
Show SAA: If AC DF, A D, and B E, then ABC DEF. In neutral geometry: Angle sum of any tri ang le is less than or equal to 180º Suppose there is a triangle with angle sum greater than 180º, say angle sum of ABC is 180º + p, where p>0.
GlCGoal: Construct a tr iang lhhle that has t he same ang le sum, but one of its angles is smaller than p. Why is that enough? We would have that the remaining two angles add up to more than 180º: can that happen? Show that any two angles in a triangle add up to less than 180º. Construct a triangle with angle sum as that of ABC (180º + p), but one of its angles is at most half of m( A) Saccheri’s dilemma Options are:
Summit angles are right
Summit angles are obtuse
Summit angles are acute: