Tuesday October 21

Assume all lines and planes that appear parallel are parallel. 1. What segments are skew to segment WX? 2. What segments are parallel to segment UW? 3. What segments intersect segment QR?

October 31 is known across the United States and much of the world as Halloween. What is October 30 known as in parts of the United States?

In parts of the East Coast, the night before Halloween is known as Mischief Night. It was a night to gently prank neighbors by soaping windows, toilet-papering trees and playing Ding-Dong Ditch, or ringing a door bell and running away.

In Detroit, the night before Halloween is known as Devil’s Night. It was a night marked by arson to unoccupied buildings and widespread damage.

CONGRUENCE Experiment with transformations in the plane 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Prove geometric theorems 9. Prove theorems about lines and angles.

We have looked at transversals and pairs of lines and talked about the angles. Almost every pair of angles in the picture above have a special name. Some you already know. What name do we have for angles 1 and 4?

Angles 1 and 4 are vertical angles. Any time we have a transversal crossing two lines, there are 4 sets of vertical angles. In the above diagram, 1 and 4, 2 and 3, 5 and 8, and 6 and 7 are all vertical pairs. What name do we have for angles 7 and 8?

Angles 7 and 8 are a linear pair, or are supplementary. Any time we have a transversal intersect two lines, we have 8 linear pairs that are supplementary. But that leaves many other pairs that we don’t know the names of. Let’s give some definitions first.

We know about interior and exterior angles already. These are defined in relation to the two lines. Angles 1, 2, 7 and 8 are exterior angles, and angles 3, 4, 5 and 6 are interior angles.

Now lets define angles based on their relation to the transversal. Since angles 1, 3, 5 and 7 are on the same side of the transversal, they are called same side angles. So are angles 2, 4, 6 and 8.

We can get even more precise. Since angles 3 and 5 are interior angles and on the same side of the transversal, they are called same side interior angles. What other angle pair are same side interior angles?

Likewise, since angles 2 and 8 are exterior angles and on the same side of the transversal, they are called same side exterior angles. What other angle pair are same side exterior angles?

Angles 3 and 6 are on opposite sides of the transversal. And because they are both interior angles, the pair are called alternate interior angles. What is the other pair of alternate interior angles?

And angles 2 and 7 are on opposite sides of the transversal. And because they are both exterior angles, the pair are called alternate exterior angles. What is the other pair of alternate exterior angles?

The last special angle pair is defined in relation to both the transversal and the lines. Angles 1 and 5 are both on the same side of the transversal, and on the same side of their respective line. This angle pair is called corresponding angles. What are the other 3 pairs of corresponding angles?

Now you know the special names for all the pairs of angles when a transversal intersects two lines. What is the pair name for angles 6 and 7? What is the pair name for angles 2 and 4? What is the pair name for angles 2 and 6?

I have a cheat sheet ready with all of the angle pairs written down so that you will not have to memorize them. But you should be aware of how they all get their name so that you can apply the theorems about them in the next section.

Let’s talk about the first Postulate from Chapter Three. In the diagram above the two lines are parallel. This brings up some very interesting relationships among the pairs of angles

Postulate 3.1 says that if two parallel lines are intersected by a transversal, then each of the pairs of corresponding angles are congruent. What are the pairs of corresponding angles in this diagram?

I will have a second cheat sheet of all of the Theorems from Chapter 3 ready tomorrow so that we can start going over some of the special relationships that exist with these angle pairs. There are 11 Theorems and 4 Postulates…

Do Now: Start on homework, problems , 37-42, page