Download Congruence and stuff

Constructions
And some old homework
Euclid’s Bridge of Fools
How does Euclid’s proof work?
In Fano’s geometry every point lies on exactly 3
lines - Melissa
In incidence geometry, for every point there is a
line not passing through that point. - Emma
What is this thing?
Definition of a circle
How would you make it?
How would you construct a midpoint of a segment?
How would you construct an angle congruent to a given
angle?
How would you bisect an angle?
How would you construct an equilateral triangle?
How would you construct a square?
In the world of Neutral geometry (one
without parallel postulate)
 If in □ABCD, AD≅BC, ∠A and ∠ B are right then
∠C≅∠D
Saccheri’s dilemma
 Options are:
 Summit angles are right
Wants
 Summit angles are obtuse
 Summit angles are acute
Was able to rule out, and we’ll see how
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.
 Then 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 angle sum of ∆ABC is 180º + p, where 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
Parallel Postulate???
 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
alternate interior angles cut by a transversal 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?
Can we at least say something about the
sum of two interior angles?
B
A
C
D
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
2.
Here
C
B
D
A
C
D
Consequences:
 Show that any two angles in a triangle add up to less
than 180º
 Show SAA: If AC ≅ DF, ∠ A ≅ ∠ D, and ∠ B ≅ ∠ E,
then ∆ABC ≅ ∆DEF.
In neutral geometry: Angle sum of any
triangle 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.
 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.
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