Geometry 3.1 Line & Angle Pairs

Geometry 3.1 Lines and Angles Goals Identify relationships between lines. Identify angles formed by lines. Identify lines and planes January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Coplanar: Lie on The same plane Parallel Lines Coplanar lines that do not intersect. m || n m n January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Skew Lines Lines that do not intersect and are not coplanar. s r January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Parallel Planes Planes that don’t intersect. January 13, 2019 Geometry 3.1 Lines and Angles

Segments and Rays can be parallel. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Visualization B D Parallel E A Skew G Perpendicular F Think of a rectangular box. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles History Lesson January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Euclid Lived about 325 BC to 265 BC. Worked in Alexandria, Egypt. Wrote the first geometry book, The Elements, which is still in print. The Elements begins with five postulates. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Euclid’s 5th Postulate Given a line segment that crosses two lines in a way that the sum of the inner angles on the same side is less than two right angles, then the two lines will eventually meet (on that side of the line segment). January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Modern Version The 5th Postulate was modernized and simplified by John Playfair (1748 – 1819). Although it reads like a theorem it isn’t, and despite 200 years of attempts, it cannot be proved. It is a postulate. Attempts to prove it led to some remarkable new ideas that will be seen momentarily. January 13, 2019 Geometry 3.1 Lines and Angles

Postulate 3.1: Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. January 13, 2019 Geometry 3.1 Lines and Angles

Postulate 3.2: Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Transversals A transversal is a line that intersects two or more coplanar lines at different points. t m n Transversal January 13, 2019 Geometry 3.1 Lines and Angles

This is not a transversal. The lines intersect at only one point. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Special Angle Pairs These are identified by their positions relative to one another. Learn to identify them and name them. These are listed on page 128 of the text. 1 2 4 3 5 6 8 7 January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Corresponding angles 1 & 5 2 & 6 3 & 7 4 &  8 1 2 4 3 5 6 8 7 January 13, 2019 Geometry 3.1 Lines and Angles

Alternate Exterior Angles 1 & 7 2 & 8 1 2 4 3 5 6 8 7 January 13, 2019 Geometry 3.1 Lines and Angles

Alternate Interior Angles 4 & 6 3 & 5 1 2 4 3 5 6 8 7 January 13, 2019 Geometry 3.1 Lines and Angles

Same Side Interior Angles Your book calls these Consecutive Interior Angles. We will also use Same Side Interior. 4 & 5 3 & 6 1 2 4 3 5 6 8 7 January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Abbreviations Corresponding Angles Corr s Alternate Exterior Angles Alt Ext s Alternate Interior Angles Alt Int s Same Side Interior Angles SS int s January 13, 2019 Geometry 3.1 Lines and Angles

More on the Parallel Postulate Euclid: Given a line segment that crosses two lines in a way that the sum of the inner angles on the same side is less than two right angles, then the two lines will eventually meet (on that side of the line segment). What if this isn’t true? January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Is it possible that m || n || l? l n m January 13, 2019 Geometry 3.1 Lines and Angles

If we assume it is possible… Bernard Riemann (1826 – 1866) Hyperbolic Geometry January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Euclidean Space January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Hyperbolic Space January 13, 2019 Geometry 3.1 Lines and Angles

Which one was the correct view? Albert Einstein (1879 –1955) On the small scale, Euclid is all that is needed. On the large scale of the universe, Riemann is correct. This was contained in his brilliant theory, Special Relativity. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles A simple postulate by Euclid, has led to one of the most profound theories in understanding the universe: The universe is not flat, i.e. it is not Euclidean. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Space is curved. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles But not for us. In our study of geometry, we assume the Parallel Postulate to be true. January 13, 2019 Geometry 3.1 Lines and Angles

Geometry 3.1 Lines and Angles Summarize January 13, 2019 Geometry 3.1 Lines and Angles