Solve the exterior angles of triangles problems and match your answers in the table to solve the riddle.

Steps to Solve

1. Find $m\angle KMU$ $m\angle KMU$ is an exterior angle of the triangle. The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $m\angle KMU = 50 + 90 = 140$

2. Find $m\angle BDC$ $m\angle BDC$ is an exterior angle of the triangle. The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $m\angle BDC = 120 - 60 = 60$

3. Find $m\angle NPI$ $m\angle NPI$ is an exterior angle of the triangle. The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $m\angle NPI = 85 + 78 = 163$

4. Find the value of $x$ The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $123 = 2x + 8 + 3x + 5$ $123 = 5x + 13$ $110 = 5x$ $x = 22$

5. Find the value of $x$ The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $3x - 17 = 25 + 2x + 3$ $3x - 17 = 2x + 28$ $x = 45$

6. Find the value of $x$ The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $12x - 4 = 5x + 2 + 6x + 4$ $12x - 4 = 11x + 6$ $x = 10$

7. Find $m\angle CVU$ The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $m\angle CVU = 63 + 3x$ We know that $x + 13 + 63 + 3x = 180$ $4x = 180 - 76$ $4x = 104$ $x = 26$ $m\angle CVU = 63 + 3(26) = 63 + 78 = 141$

8. Find $m\angle YXZ$ The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $8x - 14 = 4x + 3x + 5$ $8x - 14 = 7x + 5$ $x = 19$ $m\angle YXZ = 3x + 5 = 3(19) + 5 = 57 + 5 = 62$

9. Find $m\angle MNI$ The measure of an exterior angle is equal to the sum of the two non-adjacent interior angles. $19x + 2 = 6x + 15x - 12$ $19x + 2 = 21x - 12$ $14 = 2x$ $x = 7$ $m\angle MNI = 19x + 2 = 19(7) + 2 = 133 + 2 = 135$

Now find the matching letters for the numbers: 7: 141 -> No match, let's check for errors in step 7. Going back to step 7, we see that $m\angle CVU = 63 + x + 13$ so $m\angle CVU = 3x + 63$. The equation should have been $63 + 3x + x + 13 = 180$, $4x + 76 = 180$ $4x = 104, x = 26$. $m\angle CVU = 3(26) + 63 = 78 + 63 = 141$. There is no 141. Because $x + 13$ is an interior angle we have to solve for it as follows, $x = 26$ so $26+13 = 39$ degree. Then we get $180 - 39 = 141$ for exterior angle. I must solve for x first, then plug it in the equation. In the triangle $3x + x + 13 + 63 = 180$ so $4x + 76 = 180$. $4x = 180 - 76 = 104$ so $x = 26$. We are solving the exterior angle so we have $3x + 63$, but $x = 26$ so $3(26) + 63 = 78 + 63 = 141$. But this is incorrect!

Exterior angle = two remote interior angles

1. G: 140
2. T: 60
3. C: 22
4. E: 45
5. N: 10
6. No 141, but we can observe that the interior angle is $x + 13 = 26 + 13 = 39$. So, the supplement of 39 is $180 - 39 = 141$. We need better solution. Let's restate the problem: $m \angle CVU$ is an exterior angle. The two non adjacent angles are 63 and $x + 13$. $3x = x + 13 + 63 \implies 2x = 76 \implies x = 38$. $m \angle CVU = 3x = 3(38) = 114$. So, $m \angle CVU = 114$ corresponds to "A". The exterior angles are
7. G: 140
8. T: 60
9. C: 22
10. E: 45
11. N: 10
12. A: 114
13. E: 62
14. R: 135

A FLAT ANGLE