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Remote Interior Angles - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

If one side of a triangle is extended beyond the vertex, an exterior angle is formed. This exterior angle is supplementary with its adjacent, linear angle. Since the angle sum in a triangle is also 180 degrees, the exterior angle must have a measure equal to the sum of the remaining angles, called the remote interior angles.

When we talk about exterior angles and the remote interior angles it helps that we think about what in English what do they mean do they mean? Exterior means outside, so this is angle 1 outside of our triangle. The angles that are inside the triangles are the interior but there's only two that are remote. Remote means far away which is why when you're trying to use your TV you use a remote because your far away, so the remote interior angles are 3 and 4 so again 1 is your exterior angle because it's outside and the two angles that are not adjacent to angle 1 are your remote interior angles.
There's a special relationship that exists here and that is angle 1 is equal to angle 3 plus angle 4 but you're not just going to take my word for it you're going say "Mr. McCall you need to prove that," so what I'm going to do is I'm going to say angle 1 and angle 2 must sum to 180 degrees because if I add those two angles up we get a straight line. The second thing I'm going to say is that these 3 angles 2, 3, 3. 2, 3 and 4 must sum to 180 degrees because they make a triangle.
If I solve this equation for 2, I'm sorry that 2 is a little messy, then I can substitute in to my first equation, so I'm going to subtract angle 3 and I'm going to subtract angle 4 so what I'm doing is just moving everything to the other side of that equation so subtract angle 3 subtract angle 4 and I find that angle 2 must equal 180 minus those two angles, so 180 degrees minus angle 3 minus angle 4 so I know angle 2 in terms of angle 3 and 4 and I'm going to substitute that in right over there, so we're going to shift and I'm going to say angle 1 plus angle 2 which we said was 180 minus angle 3 minus angle 4 if we go back to our original equation here that has to equal 180 degrees. I see I have 180 degrees on both sides so I'm just going to minus 180 and then that will make them disappear and if I move negatives angle 3 and negative angle 4 to the other side by adding angle 3 and angle 4 then all I have left is angle 1 is equal to 180 and negative 180 is 0 so we have angle 3 plus angle 4 which has proven that the remote exterior angle excuse me the exterior angle is equal to the sum of the remote interior angles.

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