Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Angles




An angle is formed when two rays share an origin. There are many different types of angles formed when two lines/rays/line-segments intersect. The point of origin/intersection is called the vertex, and the two rays are called the sides. When we name an angle, we always put the vertex in the middle. For example, in the image on the right, $\angle ABC$ is an angle with vertex at point $B$.


Angles: All Different Kinds

  • A $90^{\circ}$ angle is a Right Angle. Lines, segments, or rays that form a right angle are said to be Perpendicular.
  • An angle smaller than $90^{\circ}$ is an Acute Angle.
  • An angle between $90^{\circ}$ and $180^{\circ}$ is an Obtuse Angle.
  • An angle that measures $180^{\circ}$ is a Straight Angle.
  • An angle of more than $180^{\circ}$ is a Reflex Angle.
  • Two angles that add up to $180^{\circ}$ are known as Supplementary Angles.
  • Two angles that add up to $90^{\circ}$ are known as Complementary Angles.
  • When two lines intersect, they form Vertical Angles that are opposite each other, and have equal measure. 
Angles: Formed by Parallel lines and Transversals
  • If two lines are extended in both directions forever and never intersect, they are called Parallel lines. 
  • A line that intersects two or more parallel lines is known as a Transversal.
We can form many angle relationships when we have a set of parallel lines and a transversal, as shown below.
      Corresponding Angles: $\angle 1 = \angle 5$
      Alternate Interior Angles: $\angle 3 = \angle 5$
      Same Side Interior Angles: $\angle 4 + \angle 5 = 180^{\circ}$
      Alternate Exterior Angles: $\angle 2 = \angle 8$

Angles: Formed in Triangles
  • When three points are connected by drawing line segments between them, we form the polygon known as a Triangle. The three points are known as the Sides of the triangle. The three angles formed on the inside of the triangle are known as Interior Angles. 
          Note: The sum of the angles of a triangle add up to $180^{\circ}$
  • If we were to extend one side past the vertex of a triangle, we would form an Exterior Angle of a triangle. For example, in the diagram below, $\angle BCD$ is an exterior angle. Further, $\angle CBA$ and $\angle CAB$ are your Remote Interior Angles.
          Note: $\angle BCD = \angle CBA + \angle CAB$

Let us look at some interesting facts about angles formed by broken lines within a set of parallel lines.




Note: Based on the diagrams above, we can also deduce that the sum of all the angles formed by the broken lines is equal to the number of broken lines multiplied by $180$.

Here are some additional relationships between angles formed by broken transversals between two parallel lines.


Signing off for now, be back in a couple of weeks. I'm off to Grand Teton and Yellowstone National Parks. Looking forward to coming back with some great experiences and spectacular pictures. 



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Equilateral and Equiangular Polygons

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