A side of the triangle below has been extended to for an exterior angle of 161. Find the value of x
To solve this problem, we can use the fact that the sum of the measures of the interior angles of a triangle is 180 degrees.
Let's call the angle at the top vertex of the triangle A, the angle at the bottom left vertex B, and the angle at the bottom right vertex C.
Let the extension of side BC form an angle of 161 degrees with side AC at point D.
Since the measure of angle A is not given, we'll find it using the fact that the sum of the angles in a triangle is 180 degrees.
Angle A + Angle B + Angle C = 180
Angle C = 180 - Angle A - Angle B
We know that the exterior angle at D is 161 degrees, and the sum of the measures of the two interior angles at C and D is 180 degrees.
So, Angle C + Angle D = 180
180 - Angle A - Angle B + 161 = 180
161 - Angle A - Angle B = 0
161 = Angle A + Angle B
Now we have two equations:
Angle C = 180 - Angle A - Angle B
Angle A + Angle B = 161
We can solve these two equations:
180 - Angle A - Angle B = Angle A + Angle B
180 = 2 * Angle A + 2 * Angle B
90 = Angle A + Angle B
We have now found that Angle A + Angle B = 90.
Since we know the sum of the angles in a triangle is 180 degrees, we can set up another equation:
Angle A + Angle B + Angle C = 180
Angle A + Angle B + (180 - Angle A - Angle B) = 180
Angle A + Angle B - Angle A - Angle B = 180 - 180
0 = 0
This equation is true, so the solution for angle C does not depend on the values of Angle A and Angle B. Therefore, the value of x is not determined by the information given.