TEXT: Elementary College Geometry by Africk
Geometry Labs by Henri Picciotto. Videos from Khan Academy.
Two lines which never intersect are parallel, e.g., the 2 rails in straight, infinite railroad tracks. A line which passes through 2 other lines is a transversal. The angles inside the letter Z or some reflection or rotation of it are alternate interior angles. The angles inside and below the parallel lines in the letter F or some reflection or rotation of it are corresponding angles. The angles inside the letter C or some reflection or rotation of it are consecutive or same side interior angles. The following are equivalent:
- two lines are parallel;
- alternate interior angles are equal;
- corresponding angles are equal;
- same side angles are supplementary (sum to 180°).
Here is a site with interactive diagrams that will help with the terminology and ideas.
Here is a video from Khan Academy with additional exercises worked out for angles and parallel lines.
The area of a figure is how many squares, 1 unit by 1 unit (such as inches or meters), can fit inside the figure, counting pieces of the squares as well.
Many tiles are one foot by one foot. By counting the number of such tiles on a floor, one can determine the area of a floor in square feet.
As did circumference, the area of a circle also uses π: A=π r 2.
To prove the formula for the area of circle, take a filled circle and slice it up like a pizza pie, perhaps into 8 slices. Arrange the slices on on side of the pie so that the crusts all line up and do the same for the opposite side of the pie so that they look like teeth of a monster’s open mouth. Then slide the 2 sides together (close the mouth). Here is a good animation. Another proof uses the formula for the area of a triangle A=½ b h, where b is the base and h is the height. Here is the animation for this latter proof.
We are doing things somewhat backwards. The best way to prove A=½ b h is to note that a triangle and its image rotated 180° around the midpoint of any edge forms a parallelogram and then to use the formula for the area of parallelogram A= b h (which we will see later in the course). This site uses this approach and has many examples and exercises.
A polygon is a closed planar figure made from a finite number of line segments, called its sides. (A regular polygon has all its sides equal.) A vertex is where 2 line segments are joined. A triangle is a 3 sided polygon. The most important theorem regarding a triangle is the fact that its interior angles sum to 180°. Here is a proof. A more elaborate/formal proof can be found here and has an excellent embedded video. The bottom half of this web page has a treatment of the exterior angle theorem. An exterior angle is the angle between any side of a shape and a line extended from the next side. The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior anglesThe standard proof for the exterior angle theorem uses the sum of the interior angles theorem together with supplementary angles. It can be found in your text, as well as on the 2nd site mentioned above as a corollary (a statement that can be easily proved by applying a theorem).
In LAB 1.5, you will explore the classification of triangles, which corresponds to section 1.6 in your textbook. You can classify triangles according to their largest angle as well as by how many sides (and angles!) are equal.
Additional questions for the laboratory:
- Has this laboratory made you look at triangles with a more critical/inquisitive eye? Have you seen acute triangles in buildings and bridges? obtuse triangles?
- Since so much of our world is based on right angles, it is no surprise that the world has many examples of such triangles. Can you think of some?
- Symmetry is something that is part of many designs, for aesthetic and practical reasons. Give some examples of isosceles triangles in construction or art.