Unit 2 - Postulates and Theorems(Pavan Govu)
The reason I chose the following postulates/theorems is because they all are appropriate representations of Unit 2 as a whole. They also, in some sort, tell a story. The first postulate/theorem starts off very basic and goes over some of what we learned in the last unit. Then the next, postulate/theorem introduces the basics of the new concept we are learning in Unit. The next 3 postulates/theorems get progressively harder as we go into depth. The last postulate/theorem, is advanced, but is the peak of what we'll learn. It is the conclusion. Overall, they contribute to a better understanding of the knowledge we gained in Unit 2.
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. In the diagram, Angle A is congruent to Angle D based on the Alternate Interior Angles Thrm. Parallel Lines Theorem In a coordinate plane, two non-vertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. In the diagram, the red line and the blue line are parallel based on the Parallel Lines Theorem. Parallel Postulate (In Spherical Geometry) There are no parallel lines on a sphere. In the diagram, the green great circle and the blue great circle can't be re-arranged to be parallel based on the Parallel Postulate (In Spherical Geometry.) Theorem 1 About Transformations (No name was given in the notes or lesson) Any translation or rotation is equivalent to a composition of two reflections. In the diagram, the reflection of the blue triangle over the y-axis to make the red triangle, and the red triangle is reflected over the x-axis to make the green triangle. The transformation from the blue triangle to the green triangle is also just a rotation according to Theorem 1 About Transformations. Theorem 2 About Transformations (No name was given in the Notes or lesson) A composition of two isometries is an isometry. In the diagram, the 2 transformations between the triangles are isometries. The composition of those 2 isometries is an isometry, based on Theorem 2 About Transformations. Theorem 3 About Transformations (No name was given in the Notes or lesson) The composition of two reflections across two parallel lines is equivalent to a translation. In the diagram, assume that line l is parallel to line m. The yellow figure was reflected over line m to make the blue figure and the blue figure was reflected over line l to make the red figure. The transformation between the yellow figure and the red figure is equivalent to a translation, based on Theorem 3 About Translations. |