<p>In the figure above, what is the sum, in terms of n, of the degree measures of the four angles marked with arrows?</p>
<p>Thanks.</p>
<ol>
<li><p>“The measures of two angles on the same line add up to 180.”
Look at the angle just to the right of angle that has a measure of n. It is unmarked. This angle has a measure of 180-n.</p></li>
<li><p>“The sum of the measures of the angles in the inside of a triangle is 180.”
Therefore, we can find the sum of the other two angles by subtracting 180-n from 180. We get n.</p></li>
<li><p>“Vertical angles are congruent.”
Since both of the unmarked angles within the triangles are 180-n, we can do the same thing for the other side. The other angles also add up to n.</p></li>
</ol>
<p>The sum is 2n.</p>
<p>Other method: </p>
<p>Let the 2 marked angles on the left be x and y, the 2 angles on the right be a and b. And let the vertical angles be c. So for the triangle on the left, we know that 180=x+y+c -> c=180-x-y. For the triangle on the right, we know that 180=a+b+c -> c=180-a-b. Now that we solved for the vertical angles in terms of the marked angles, we set up the equation 360=n+n (vertical angle) +180-a-b +180-x-y => 360=2n+360-a-b-x-y -> 360+x+y+a+b-360 = 2n -> x+y+a+b = 2n. Therefore, the sum of the marked angles is 2n.</p>
<p>Here’s the quickest way:</p>
<p>The measure of an exterior angle of a triangle is equal to the sum of the 2 opposite interior angles. </p>
<p>Use this fact for EACH triangle. Do you see why the answer is 2n?</p>