A side of the triangle below has been extended to form an exterior angle of 76°. Find the value of x.
We can see that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. So, if one of the exterior angles of the triangle is 76 degrees, then the sum of the other two interior angles opposite the other two sides is also 76 degrees.
Let's label the three interior angles of the triangle as A, B, and C, where A is opposite the side with length x, B is opposite the side with length 12, and C is opposite the side with length 17.
We know that the sum of the angles in a triangle is 180 degrees, so we can set up the equation:
A + B + C = 180
We also know that B = 76, so we can substitute 76 for B in the equation:
A + 76 + C = 180
Let's simplify this equation:
A + C = 180 - 76
A + C = 104
Now we can see that A + C is the sum of the other two interior angles opposite the other two sides of the triangle, which is also equal to 76. So we can set up another equation:
A + C = 76
Since we already know that A + C = 104, we can substitute 104 for A + C in the equation:
104 = 76
This is not a true statement, so we have reached a contradiction. Therefore, there is no value of x that satisfies the conditions given in the problem.