Question

find the value of x in the triangle below:
(there is a triangle with angles 2x - 17°, x + 1°, and an exterior angle x + 89°)
x = \boxed{}

Explanation:

Step1: Use the exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, we have the equation: $x + 89=(2x - 17)+(x + 1)$

Step2: Simplify the right - hand side

Simplify $(2x - 17)+(x + 1)$: $2x-17+x + 1=3x-16$
So our equation becomes $x + 89=3x-16$

Step3: Solve for x

Subtract $x$ from both sides: $89 = 2x-16$
Add 16 to both sides: $89 + 16=2x$, so $105 = 2x$
Divide both sides by 2: Wait, no, wait. Wait, let's re - do the equation solving.
Wait, from $x + 89=3x-16$, subtract $x$ from both sides: $89=2x - 16$
Then add 16 to both sides: $89+16 = 2x$, $105=2x$? Wait, that can't be. Wait, maybe I made a mistake in the exterior angle. Wait, the exterior angle is equal to the sum of the two remote interior angles. Let's check the angles again. The two remote interior angles are $2x-17$ and $x + 1$, and the exterior angle is $x + 89$. So the correct equation is $x + 89=(2x-17)+(x + 1)$
Let's expand the right - hand side: $2x-17+x + 1=3x-16$
So $x + 89=3x-16$
Subtract $x$ from both sides: $89=2x-16$
Add 16 to both sides: $89 + 16=2x$, $105=2x$? Wait, that gives $x = 52.5$, but maybe I messed up the exterior angle. Wait, maybe the exterior angle is supplementary to the adjacent interior angle? Wait, no, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.
Wait, let's re - express the angles. Let's denote the adjacent interior angle to the exterior angle $x + 89$ as $y$. Then $y+(x + 89)=180$ (linear pair). And also, by triangle angle sum, $(2x-17)+(x + 1)+y = 180$. So substituting $y = 180-(x + 89)$ into the triangle angle sum equation:
$(2x-17)+(x + 1)+180-(x + 89)=180$
Simplify the left - hand side:
$2x-17+x + 1+180-x - 89=180$
Combine like terms:
$(2x+x - x)+(-17 + 1+180 - 89)=180$
$2x+( - 16+91)=180$
$2x + 75=180$
Subtract 75 from both sides: $2x=180 - 75=105$
Wait, no, $180-75 = 105$, so $2x=105$? No, that's the same as before. Wait, maybe the exterior angle is equal to the sum of the two remote interior angles. So the two remote interior angles are $2x-17$ and $x + 1$, so $x + 89=(2x-17)+(x + 1)$
$x + 89=3x-16$
$89 + 16=3x-x$
$105 = 2x$? Wait, that would mean $x = 52.5$, but maybe the problem has integer values. Wait, maybe I made a mistake in identifying the exterior angle. Wait, let's look at the diagram again. The exterior angle is $x + 89$, and the two non - adjacent interior angles are $2x-17$ and $x + 1$. So the equation is correct.
Wait, let's solve $x + 89=3x-16$
$89+16=3x - x$
$105 = 2x$
$x=\frac{105}{2}=52.5$? But maybe the problem has a typo, or maybe I misread the angles. Wait, maybe the exterior angle is $x + 89$, and the two non - adjacent angles are $2x-17$ and $x + 1$. Let's check with $x = 29$. Wait, no, let's do the equation again.
Wait, maybe the exterior angle is equal to the sum of the two remote interior angles. So:
$x + 89=(2x-17)+(x + 1)$
$x + 89=3x-16$
$89 + 16=3x - x$
$105 = 2x$
$x = 52.5$
But maybe the problem expects an integer. Wait, maybe I made a mistake in the angle labels. Let's assume that the exterior angle is equal to the sum of the two non - adjacent interior angles. So the calculation is as above.

Wait, maybe I messed up the sign. Let's re - write the equation:

$x + 89=2x-17+x + 1$

$x + 89=3x-16$

$89 + 16=3x - x$

$105 = 2x$

$x=\frac{105}{2}=52.5$

But if we consider that maybe the exterior angle is supplementary to the adjacent interior angle, and the adjacent interior angle plus the other two interior angles equals 180. Let's denote the adjacent interior angle as $z$. Then $z+(x + 89)=180$, so $z = 180-(x + 89)=91 - x$

And by triangle angle sum: $(2x-17)+(x + 1)+z = 180$

Substitute $z = 91 - x$:

$(2x-17)+(x + 1)+91 - x=180$

$2x-17+x + 1+91 - x=180$

$2x+75=180$

$2x=180 - 75=105$

$x = 52.5$

So the value of $x$ is $52.5$ or $\frac{105}{2}$. But maybe the problem has a mistake, or maybe I misread the angles. If we assume that the exterior angle is equal to the sum of the two non - adjacent interior angles, then $x = 52.5$.

Answer:

$x = 52.5$ (or $\frac{105}{2}$)