e then of a triangle is extended at one vertex, the angleit the other side at th

Question

e then of a triangle is extended at one vertex, the angleit the other side at that vertex is called an exterior angle of th at one vertex, the angle it creates with le of the triangle Construct an example of this in Sketchpad. Measure the c a. Compare this measurement with the measure of one of the interior angles at either of the to as the remote interior angles with respect to that ex other two vertices. These two angles are referred terior angle. Manipulate the triangle by moving the vertices so that the measure of the remote interior angle you chose comes as close as possible to the measure of the exterior angle. What do you observe? Express your observation as a conjecture by completing this sentence: "If an exterior angle is formed by extending one side of a triangle, then … b. Now compare the measure of the exterior angle to the measures of both of the remote interior angles. What do you observe? Express your observation as a conjecture by completing this sentence: "If an exterior angle is formed by extending one side of a triangle, then..." (This statement is intended to be different from the one you made in part (a).)

Explanation / Answer

Sum of Interior Angles of a polygon is 180(n2)180(n2) where nn is the number of sides (so is the number of angles);

We are told that the smallest angle is 136 degrees, the next will be 136+1 degrees, ..., and the largest one, nthnth angle, will be 136+(n1)136+(n1)degrees. The sum of the nn consecutive integers (the sum of nn angles) is given by

first+last/2*# of terms=(136+(136+n-1)/2 )*

So we have that 180(n2)=(271+n)/2n --> 360(n2)=(271+n)n

, now try the answer choices: in order RHS to end with zero (as LHS is because of 360) then nn, out of the options listed, could be either 10 or 9, n=9n=9 fits.

(b)

Exterior Angle Theorem

The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle.

Example:

If the measure of the exterior angle is (3x - 10) degrees, and the measure of the two remote interior angles are 25 degrees and (x + 15) degrees, find x

To solve, we use the fact that W = X + Y. Note that here I'm referring to the angles W, X, and Y as shown in the first image of this lesson. Their names are not important. What is important is that an exterior angle equals the sum of the remote interior angles.

We equate and solve for 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 not the answer here. We need the angles themselves, which are calculated as (3x-10), 25, and (x+15). The angles, then, are 65, 25, and 40 degrees

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