Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts

Duality (projective geometry) wikipedia , lookup

Euler angles wikipedia , lookup

Riemannian connection on a surface wikipedia , lookup

Rational trigonometry wikipedia , lookup

Perspective (graphical) wikipedia , lookup

Line (geometry) wikipedia , lookup

Euclidean geometry wikipedia , lookup

Transcript
```1
Lesson Plan #31
Class: Geometry
Date: Tuesday November 15th, 2012
Topic: Properties of parallel lines (continued)
Aim: How do we use the properties of parallel lines?
Objectives:
Students will be able to use the properties of parallel lines.
HW #31:
Page 81 #’s 1-10
Note:
Recall the previously stated postulate “Through a given point not on a given line, there exists one and only one line parallel to the
given line.”
Do Now:
1) Write the converse of the statement “If I live in New York City, then I live in New York State”
2) Write the converse of the statement “If two lines are cut by a transversal so that alternate interior angles are congruent, then
the two lines are parallel”
PROCEDURE:
Write the Aim and Do Now
Get students working!
Take attendance
Give Back HW
Collect HW
Go over the Do Now
Question:
If a statement is true is the converse of that statement always true?
Let’s see if we can prove the converse of the theorem “If two planar lines are cut by a transversal so that alternate interior angles
are congruent, then the two lines are parallel.
Let’s try a proof by contradiction:
Given: 𝑚 ∥ 𝑛
Prove: < 4 ≅< 6
Plan: Assume < 4 𝑛𝑜𝑡 𝑐𝑜𝑛𝑔𝑟𝑢𝑒𝑛𝑡 𝑡𝑜 < 6
Then draw a line through the vertex of angle 4 so it that it forms an angle congruent to < 6. This would lead to congruent alternate
interior angles, meaning this new line is parallel to line n. But we already had one line (m) parallel to the given line (n) through a
point not on the line
m
2
4
m
3
2
1
4
3
5
n
6
7
6
7
5
8
n
8
But this contradicts our previously stated postulate which states through a point not on a given line, there is one and only one line
parallel to the given line.
Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles formed are
congruent.
2
Theorems that are converses of other previously stated theorems (presented without proof)
 If two parallel lines are cut by a transversal, then the corresponding angles are congruent
 If two parallel lines are cut by a transversal, then the two interior angles on the same side of the transversal are
supplementary
 If a line is perpendicular to one of two parallel lines, it is perpendicular to the other.
3
4
```