IM Geo.1.21 Lesson: One Hundred and Eighty

Tyler thinks angle  is congruent to angle  because they are corresponding angles and a translation along the directed line segment from  to  would take one angle onto the other.  Here are his reasons.

• The translation takes  onto , so the image of  is .
• The translation takes  somewhere on ray  because it would need to be translated by a distance greater than  to land on the other side of .
• The image of  has to land somewhere on line  because translations take lines to parallel lines and line  is the only line parallel to  that goes through .
• The image of , call it , has to land on the right side of line  or else line  wouldn’t be parallel to the directed line segment from  to .

In what circumstances are corresponding angles congruent? Be prepared to share your reasoning.

• Use the Text tool to label the 3 interior angles as  and .
• Mark the midpoints of 2 of the sides.
• Extend the side of the triangle without the midpoint in both directions to make a line.
• Use what you know about rotations to create a line parallel to the line you made that goes through the opposite vertex.

• Translate triangle  along the directed line segment from  to  to make triangle . Label the measures of the angles in triangle .
• Translate triangle  along the directed line segment from  to  to make triangle . Label the measures of the angles in triangle .
• Label the measures of the angles that meet at point . Explain your reasoning.

What is the value of ? Explain your reasoning.

This can open up new areas to explore if we change those assumptions. For example, both of our proofs that the measures of the angles of a triangle sum to 180 degree were based on rigid transformations that take lines to parallel lines. If our assumptions about parallel lines changed, so would the consequences about triangle angle sums. Any study of geometry where these assumptions change is called non-Euclidean geometry.  In some non-Euclidean geometries, lines in the same direction may intersect while in others they do not. In spherical geometry, which studies curved surfaces like the surface of Earth, lines in the same direction always intersect. This has amazing consequences for triangles. Imagine a triangle connecting the north pole, a point on the equator, and a second point on the equator one quarter of the way around Earth from the first. What is the sum of the angles in this triangle?