A side of the triangle below has been extended to form an exterior angle of 133°. Find the value of x.
We can see that the exterior angle $133^\circ$ is equal to the sum of the two remote interior angles of the triangle, which are $x$ and $x+5^\circ$. So we have $$x + (x+5^\circ) = 133^\circ.$$ Simplifying the left side gives $2x+5^\circ=133^\circ$, so $2x = 128^\circ$ and $x=\boxed{64^\circ}$.
To find the value of x, we need to understand the relationship between exterior and interior angles formed by the extension of a side of a triangle.
The exterior angle formed by extending a side of a triangle is equal to the sum of the two opposite interior angles.
Let's denote the three interior angles as A, B, and C. The exterior angle formed by extending the side opposite angle A is given as 133°.
Therefore, we can set up an equation:
A + B = 133° (1)
Now, let's analyze the given triangle:
A
*------* B
\ /
\ /
\/
C
Since we have a triangle, the sum of all interior angles is always 180°.
So, we can write another equation:
A + B + C = 180° (2)
To find the value of x, we need to determine the value of angle A. We can use the fact that angles A and C are vertically opposite angles, and hence they are equal.
Now, let's solve the two equations simultaneously to find the value of x:
Substitute A = C into equation (2):
C + B + C = 180°
2C + B = 180° (3)
Now, substitute the value of B from equation (1) into equation (3):
2C + (133° - A) = 180°
2C + 133° - A = 180°
Rearrange the equation:
2C - A = 180° - 133°
2C - A = 47° (4)
Since A = C:
2C - C = 47° (5)
Simplify equation (5):
C = 47°
Since A = C:
A = 47°
Now, we can substitute the value of A into equation (1) to find the value of B:
A + B = 133°
47° + B = 133°
B = 133° - 47°
B = 86°
Therefore, the values of the three angles in the triangle are A = 47°, B = 86°, and C = 47°.
To find the value of x, we need to look at the side that has been extended. Let's denote the extended side as a.
We know that the sum of the measures of any two sides of a triangle is always greater than the measure of the third side. Therefore, we can set up an inequality:
a + x > a
x > 0
Hence, the value of x can be any positive real number.
To find the value of x, we need to use the Exterior Angle Theorem.
The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
In this case, since the side of the triangle has been extended to form an exterior angle of 133°, we can say that the sum of the two non-adjacent angles is equal to 133°.
Let's call the two non-adjacent angles A and B.
So, A + B = 133°.
Now, let's look at the triangle and identify the angles.
Since the exterior angle is formed by extending a side of the triangle, the two non-adjacent interior angles will be the other two angles of the triangle.
Let's call the third angle in the triangle C.
Now, we can write the equation: A + B + C = 180°, because the sum of the three angles in a triangle is always 180°.
Since we have A + B = 133°, we can substitute this value in the equation to get:
133° + C = 180°.
Now, let's solve for C:
C = 180° - 133°.
C = 47°.
So, we have the value of angle C as 47°.
Now, let's find the value of x. Since angle A and angle C are adjacent angles, they share a side. We can see that x and 47° are opposite angles of parallel lines, so they are congruent.
Therefore, x = 47°.
Hence, the value of x is 47°.