Definition of one exterior angle
At each vertex the a triangle, one exterior edge of the triangle may be formed by extending ONE side of the triangle. See picture below.
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Calculating the Angles
We have the right to use equations to stand for the actions of the angles described above. One equation might tell united state the sum of the angle of the triangle. Because that example,
We know this is true, due to the fact that the amount of the angles inside a triangle is always 180 degrees. What is w? us don"t understand yet. But, we might observe that the measure of angle w plus the measure up of angle z = 180 degrees, due to the fact that they room a pair of supplementary angles. Notice how Z and also W together make a directly line? That"s 180 degrees. So, we have the right to make a new equation:
Then, if we combine the 2 equations above, we deserve to determine the the measure of angle w = x + y. Here"s exactly how to execute that:
x + y + z = 180 (this is the first equation) w + z = 180 (this is the 2nd equation)
Now, rewrite the second equation together z = 180 - w and substitute that for z in the first equation:
x + y + (180 - w) = 180 x + y - w = 0 x + y = w
Interesting. This tells us that the measure of the exterior angle equals the full of the other two interior angles. In fact, there is a theorem referred to as the Exterior edge Theorem which more explores this relationship:
Exterior edge Theorem
The measure up of an exterior edge (our w) that a triangle amounts to to the sum of the procedures of the two remote interior angles (our x and also y) that the triangle.
Let"s try two example problems.Example A:
If the measure of the exterior angle is (3x - 10) degrees, and also the measure up of the 2 remote internal angles are 25 degrees and also (x + 15) degrees, uncover x.
To solve, we usage the truth that W = X + Y. Note that below I"m introduce to the angles W, X, and also Y as displayed in the an initial image the this lesson. Their names room not important. What is crucial is that an exterior angle equals the amount of the remote interior angles.
We equate and also solve because that x.
exterior angle = interior angle + other interior angle$$ (3x - 10) = (25) + (x + 15) $$ $$ 3x - 10 = x + 40 $$ $$ 3x = x + 50 $$ $$ 2x = 50 $$ $$ x = 25 $$
Remember that "x" is no the answer here. We need the angle themselves, which space calculated as (3x-10), 25, and (x+15). The angles, then, room 65, 25, and 40 degrees.Example B
The exterior angle given is 110 degrees. 2 remote internal angles measure up 50 and (2x + 30). Discover x.
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Remember: exterior = amount of remote internal angles
We"re given the exterior angle (110). We equate 110 come (2x + 30) + 50 and also solve for x.$$ 110 = 2x + 30 + 50 $$ $$ 110 = 2x + 80 $$ $$ 30 = 2x $$ $$ 15 = x $$
A lesson listed by Mr. Feliz