A side of the triangle below has been extended to form an exterior angle of 121°. Find the value of x.
Since an exterior angle of a triangle is equal to the sum of the two remote interior angles, the two remote interior angles must be $180^\circ - 121^\circ = 59^\circ$. Since the three angles in a triangle sum to $180^\circ$, $2x = 59^\circ$ so $x = \boxed{29.5^\circ}$. [asy]
unitsize(5mm);
pair A,B,C;
A=(0,0);
B=(15,0);
C=(5,8);
draw(A--B--C--cycle);
draw(B--(-28,0));
draw(rightanglemark(B,C,A,18));
label("$121$",(-26,0),W);
label("$2x$",(1,.7),E);
[/asy]
what..?
I apologize for the confusion in my previous response. Let me explain it correctly this time.
We have an exterior angle of $121^\circ$. We know that the exterior angle of a triangle is equal to the sum of the two remote interior angles.
Let the three interior angles of the triangle be $a$, $b$, and $c$, with $c$ being the exterior angle of $121^\circ$. We can set up the equation $a + b = c$.
However, we want to find the value of $x$, which is the measurement of one of the interior angles. The exterior angle formed has been extended from side $x$. This means that $a = x$.
We can now rewrite the equation as $x + b = 121^\circ$.
Since the sum of the interior angles of a triangle is $180^\circ$, we also have the equation $a + b + c = 180^\circ$. Substituting $x$ for $a$ and $121^\circ$ for $c$, we get $x + b + 121^\circ = 180^\circ$.
Subtracting $121^\circ$ from both sides, we have $x + b = 59^\circ$.
Combining this equation with our previous equation $x + b = 121^\circ$, we can solve for $x$.
Subtracting $x + b$ from both sides of $x + b = 121^\circ$, we get $0 = 121^\circ - 59^\circ$.
Simplifying, we find $62^\circ = x$.
Therefore, the value of $x$ is $\boxed{62^\circ}$.
To find the value of x, we can use the fact that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
Let's call the angle adjacent to the exterior angle as angle A, and the angle opposite to it as angle B. According to the given information, the measure of the exterior angle is 121°. Therefore, the sum of angles A and B is equal to 121°.
Now, let's look at the triangle and identify the angles.
A
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x | \ y
| \
________________
B C
Since the exterior angle is formed by extending side BC, angle A is the angle between sides AC and BC, and angle B is the angle between sides AB and BC.
Using the triangle angle sum property, we know that the sum of angles in a triangle is 180°. Therefore, we can write the equation:
Angle A + Angle B + Angle C = 180°
Since we know that Angle A + Angle B = 121°, we can substitute it into the equation:
121° + Angle C = 180°
By isolating Angle C, we can calculate its value:
Angle C = 180° - 121°
Angle C = 59°
Therefore, the value of angle C is 59°.
Now, let's focus on finding the value of x. Angle A is the angle between sides AC and BC, so it is equal to Angle C. Therefore, x = 59°.
Hence, the value of x is 59°.