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.
Kiana Gardiner Mr.Reitz
I learned that in the area of a circle, you can never have a negative radius. Since the radius is half the diameter, the diameter must be positive. The area of a circle is just how much space the circle takes up.
Mr. Reitz
Name: XiaoJing Ao
I learned that Parallel Lines and transversals line they must be supplementry and complementry, also the internal angles and the external angles are equal. however, some some angle are congress from each other.
Horacio Moller
Mr.Reitz
I learned that no matter if the triangles are Equilateral, Obtuse, Acute, Isosceles or Scalene its interior angles will always amount to a total of 180 degrees.
Rebecca Kogan
Professor Ezra Halleck
Section 5147
Going over triangles, I remembered that the interior angles of a triangle MUST always sum up to 180 degrees. In the lab, we went over the classifications of triangles and figured out different angle combinations for triangles such as equilaterals and obtuse scalenes.
Diego Gutierrez
threw out my highschool years i never understood about shapes and geometry and trig but this lesson really made me open my eyes. As i was taught about parallel lines i just remember thinking how i couldnt understand something so simple and easy, i learned alot about interior and exterior angles , i also learned the tricks to figuring out the size of the angles inside and out.
Tamika Cadet
Mr. Reitz
I remembered almost all the types of triangles and angles, until we went over scalene triangles. I knew there was one that I was missing but I just couldn’t put my finger on it. I learned that there are three types of triangles, equilateral, isosceles, and scalene. An equilateral triangle has three equal sides and three equal angles that always add up to 60°. An isosceles triangle has two equal sides and two equal angles. Lastly, a scalene triangle has no equal sides and no equal angles.
Jesse Mohammed
Mr. Reitz
i learned that pi derived from the diameter going around the circle 3.14.. times. I thought it was just a random calculation that they came up with.
Victor Chen
Mr. Reitz
In class you gave us a helpful tip to remember complementary and supplementary, and i would like to share it here. If you imagine or draw a line through the first letter of (c)omplementary and (s)upplementary the “C” makes a nine which makes you remember its means 90 degree and for “S” you would get an eight, which is 180 degree.
Candice Wright
Professor Halleck
After watching the video titled “Area of a circle”, I mastered the formula in which to do so. I learned that in order to find the area of a circle you have to undergo a variety of different steps all depending on the information your given in an equation. I also learned that the diameter of a circle is euqal to two times the radius and the formula to find the area of a circle is equal to pi (r) squared.
Diana Guaman
Professor Halleck
I learn that the area of a circle can be found by multiplying pi by the square of the radius . Also, if you know the diameter, the radius is 1/2 as larger.
Wilmer Baez
Hello Prof. Reitz
I am not familiar with geometry, but I am getting into it, i don’t have everything clear yet but I am trying to. I learned a couple of things like the amount of a triangle is always 180 degrees.
Professor Reitz
Before this lesson I was sharp on the basic angles, such as right, and straight angles, now i have more knowledge. Now I don’t just know of supplementary, complementary,interior, exterior angle but I can find them as well. I needed to brush up on my area skills especially on the area of a triangle. I love math so i can’t wait to continue learning more of it.
Shaquana Oriol – Professor Reitz
Before this lesson i didn’t remember anything about angles, areas, and lines. I was used to the basic straight angle, right angles, parallel lines, intersecting lines, and the area of a square. The other were all refreshing my memory as we spoke of them all I started doing things on my own it was like all my knowledge started coming back to me.
Pr. Halleck
okay so I posted my comments on the discussion form on the main page but I guess I’ll repost, it was nice to refresh my memories on the different types of triangles, scalene, isosceles and equilateral. Also that angles add up to 180 degrees and that alternate angles are the same as well as corresponding angles
i learned that transversal is the line that cuts through parallel lines, and creates supplementary, exterior, interior, alternating and corresponding angles.
Prof. Reitz
I like how you helped us how to remember complementary and supplementary.
All the work we were doing help me complete my work.
Professor Reitz
Going over these lessons made me remember about the different types of triangles their are scalene, isosceles, and equilateral. Also it reminded me that triangles have 180 degrees. the video refreshed my memory on areas of a circle.
Cindy soto – Prof. Reitz It was good going over parallel lines, area and triangles. It has been a while I haven’t taken a math class so going over some of these topics is good. And working in groups is also very helpful.
John Perez
Professor Reitz
The lesson reminded me of triangles and the different types of them. The lesson also refreshed my memory on the area of circles.
I honestly don’t know how to comment on this due to the fact that I know triangles very well. I did have a problem with transversals in the past. Apparantly, i just needed a straight to the point explanation. Transversals,interior, and exterior angles are now easy for me to understand.
Shakirah Greenidge
Professor Halleck
I know triangles very well, so this was really a refresher type lesson. Even though we breezed through it, I think that the review was much needed.