UNIT I
ANGLE
RELATIONSHIPS


Introduction to geometry: Through the work of this lesson students have the opportunity to learn about points, lines, rays, angles, and planes in the coordinate plane and as normally it is covered in plane geometry. For the coordinate geometry the distance formula, and midpoint formulas are introduced for one and two dimensions. The Pythagorean Theorem is briefly introduced since it is used early in some lessons. For it there is a more in depth coverage in more advanced sections of the geometry section in this website.


Finding
complementary, supplementary, and congruent angles:
Students need to understand that in geometry you start with a point, then lines, then rays, and finally you get to angles. These in turn have some important relationships when they are in pairs. These are complementary, supplementary, and vertical angle relationships. This lesson introduces those concepts and definitions, including the case of supplementary angles forming a linear pair.


Word
Problems:
In geometry as in any other discipline is of paramount importance that students learn to “read” word problems, and how to distinguish the relevant from the nonrelevant information in the solution of a problem. This lessons highlights that process by underlining in different colors the information that corresponds to the right and left side of the equations that needs to be setup for the solution of the problems.


Concrete to abstract activity: Modeling complementary, supplementary,
linear pairs and
vertical angles using GeoLegs:
Mathematics in general implies to deal with abstraction. For some students the concepts behind angle pairs: Complementary, supplementary and vertical. Turns to be somewhat “too abstract.” They are not to be blamed for that. Opportunities to develop this are different from individual to individual. The advantage of the lesson for students is to get a visual approach to deal with these concepts including addressing the issue of not knowing how to read angular scales.
Geolegs are used to present the concepts. These a available online and chances are students themselves have them already at their homes.


Introduction: Determining the difference with parallel lines and perpendicular lines in the coordinate plane includes to deal with the concept of the inclination of a line or slope. This in turn may be defined in terms of the coordinates of two points, or in terms of the change in “x” and the change in “y”.
This lesson shows students the two ways of approaching the slope.


Finding
angles involving parallel lines. Once students were exposed to the concept of parallel lines, and the concepts behind angle pair relationships. They are ready to apply it to parallel lines cut by a transversal and the angle relationships that take place in both intersections and how they are related in both of them. Further in more advanced sections this knowledge will enable them to solve problems that involve triangles, quadrilaterals, circles, and, so on. In this lesson, those skills are fully developed.


From
conditionals: If...Then; to logical reasoning: deductive & inductive A big big in geometry is to develop the ability to use inductive and deductive reasoning in the solution of problems in geometry, and to extend it to the real life of the students. They need to know what is a logical “if …then” statement, its converse, its inverse, and its contrapositive. They need to learn to identify and use the Law of Syllogism and The Law of Detachment. These are vastly used in the geometric proofs: Both formal and informal, and are part of our natural thinking tools to interpret the world around us and our relationship with it.
Therefore the working the lesson, you will develop this way of thought.


Proofs: Involving
segments and angle relationships. These proofs
apply segment addition postulate, angle addition postulate and angle
relationships like complementary, supplementary and vertical angles.
Includingangle relationships in parallel lines. You will be able to setup the proof and see step by step how it is developed to reach the final conclusion.


Writing across the curriculum: PARAGRAPH PROOFS. Proofs may be developed using a “T” table format, a flow chart, or in a paragraph. We do proofs everyday. When you want to put in order your room, you look around and check what is out of order, and then you determine the right place for each item and then carry out the necessary actions to put it in order.
If you put these steps on a “T” table, and then use this to put it in a paragraph, then you would be doing exactly what is done here in the lesson, but with geometric proofs involving segments, angles, and angle relationships.
