Critical Concept: Adjacent and Alternate Interior Angles

One of the most difficult concepts to understand in geometry is angles. It can be confusing to know which angle you are talking about when they have a different name depending on their location and relation to other angles. This article will discuss adjacent angles and alternate interior angles, going over what each one means in detail so that you can see how they relate to one another.

The adjacent angles: An adjacent angle is an angle that lies next to another on the same plane. The two angles share one side and do not touch each other, even though they lie near no gaps between them.

 Alternate interior angles: Alternate interior angles are congruent angles located across from one another in different parallel planes which have their sides intersecting perpendicularly (at a right-angle). It means that if you were looking at a line of symmetry or center point, these lines would cross perpendicularly through this intersection point. Since it touches only one side, there will be more than two but less than five total touching points along either plane’s edge because all four corners must meet together at the crossing point for both planes.

Difference between Adjacent and Alternate Interior Angles:

  • Adjacent angles are next to one another on the same side
  • Alternate interior angles share a common vertex, but only touch at one point because they lie in different planes.
  • The correct order of adjacent angles is clockwise. If an angle is greater than 90 degrees it becomes an obtuse angle and must be noted as such. For example, if you have two adjacent 30-degree angles then this would read “30/60” or simply 60 when no number goes before the slash mark for brevity.
  • Correct order of alternate interior angles has pointed up while angling down from left to right in descending size. Also, note that there’s no need for “/” between multiple sets beside each other since these angles are all the same size and would read “__/__” or simply ___ when no number goes before the slash mark for brevity.

Example: If you have three 30 degree angles then this would be written as “30/60 / 30”. This means that they alternate between 60 degrees and 30 degrees, but do not include 90 (or obtuse) even though it is bigger than both adjacent numbers at a total of 120 degrees because there’s only one occurrence of this angle across from each other in different parallel planes.

Formulas of Adjacent and Alternate Interior Angles:


adj.angle = (a + b) / 180 x sin(a)cos(b)

Alternate Interior Angles:

ai.ang = (a – b)/sin A cos B 

where a and b are opposites of each other, but not equal to 0 degrees or greater than 90 degrees as the case with obtuse angles which will always be noted accordingly because they cannot be adjacent nor alternate interior to one another.

Properties of Adjacent and Alternate Interior Angles:

  • Adjacent angles always add up to 180 degrees and Alternate Interior Angles must total 360.
  •  Adjacent angles are not equal, but alternate interior angles can be (and often are) congruent because they’re the same size to one another as a whole.
  • The sum of all adjacent angles is 720 degrees for a triangle, 900 for any three nonlinear lines or planes besides triangles which have two sets of 180-degree pairs across from each other so it would only be 540 instead, and 1080 for four parallel lines with an angle sum of 840 between them where there’s no overlap within the middle section unless you count 0/180 at every point since this doesn’t contribute anything towards totals compared to what should already be there.
  • If one has one adjacent angle and two alternate interior angles, then their sum is 180 degrees because they’re all the same size at 30 degrees each (a/b/c = a + b + c). So if you put them in order as “30 / 60” and “…90 / 90”, this would be written as “(60+90)/180”. And for four total points with three of those belonging to parallel lines or planes where we’d need to use sin A cos B from above along with basic trigonometry:

sin(A)cos(B) = ( Opposite Side ) / Hypotenuse

So, d/(hypotenuse) = (opposite side)*cos(A)*cos(B).

The common vertex of alternate interior angles can be seen as the “center” or focal point between all touching points on either plane’s edge which would break down into three separate triangles if one connected each intersection point. Since it touches only one side, there will be more than two but less than five total touching points because it’s a triangle with three points and two angles which both equal 90 degrees. This means that the other opposing side has to be parallel so there can only ever be one set of alternate interior angles, but depending on how many touching lines or planes exist between them would all contribute towards their total sum (if any exists).

So above was some primary information about angles which are very important concepts. One can always find more about different mathematics concepts on Cuemath website. It is an online platform to learn math and coding concepts by experts.



Helen is a multi-published, award-winning author of over 30 books, including the delightful Ivy & Bean series. She has written novels for young adults, including YA romantic comedies, and has written BBC drama.

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