Remote, Exterior and Interior Angles of A Triangle

Formula for remote, exterior and Interior Angles

The Exterior, Interior and Remote Interior Angles

An exterior angle of a triangle, or any polygon, is formed by extending one of the sides of the triangle (or polygon).

In a triangle, each exterior angle has two remote interior angles (see picture below). The remote interior angles are just the two angles that are inside the triangle and opposite from the exterior angle.

Picture of remote and interior angles of a triangle
Problem 1

If the exterior angle $$ \angle 1 = 110° $$ what is m $$ \angle 2 $$ ?

$$ \angle 1 + \angle 2 = 180° \\ \angle 2 = 180 - 110 = 70 $$


for Remote Interior Angles and Exterior Angles

As the picture below shows, an exterior angle (A) equals the sum of the remote interior angles.

To rephrase it, the angle 'outside the triangle' (exterior angle A) equals D + C (the sum of the remote interior angles).

Formula for Remote and Exterior Angles

Interactive Demonstration

of Remote and Exterior Angles of a Triangle

The interactive program below allows you to drag the points of the triangle around. Notice that the sum of the remote interior angles (C and D) equal the measure of exterior angle A.

animation of translation of math demo

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Problem 2

Use the formulas described on this page to calculate the values of x (a remote interior angle) and of Y.

Remote angles problem12

Since X and, $$ \angle J $$ are remote interior angles in relation to the 120° angle, you can use the formula.

120 = 45 + x
120- 45 = x
75° = x.

Now, since the sum of all interior angles of a triangle is 180°. You can solve for Y

75 + 45 + y = 180
120 + Y = 180
Y = 60

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