Question

1. find the measure of angle a

Explanation:

Step1: Identify exterior angle property

The exterior angle equals sum of remote interior angles: $4x + 17 = (x + 23) + 80$

Step2: Solve for x

Simplify equation: $4x + 17 = x + 103$ → $3x = 86$ → $x = \frac{86}{3}$ (Wait, correction: $x + 23 + 80 = x + 103$, so $4x - x = 103 -17$ → $3x=86$? No, wait, maybe the exterior angle is supplementary? No, exterior angle property: exterior angle = sum of two non-adjacent interior angles. So yes, $4x+17 = (x+23)+80$. Then $4x -x = 103 -17$ → $3x=86$ → $x≈28.666$. Then angle A is $x+23≈51.666$? Wait, no, maybe I misread: is the angle labeled 80° adjacent to the exterior angle? If the exterior angle and the 80° angle are supplementary, then $4x+17 + 80 = 180$ → $4x=83$ → $x=20.75$. Then angle A is $x+23=43.75$. But original problem: the diagram shows exterior angle 4x+17, angle A is x+23, and the other interior angle is 80°. So exterior angle property is correct: exterior angle = sum of remote interiors (angle A and 80°). So $4x+17 = x+23+80$ → $3x=86$ → $x=86/3$. Then angle A is $86/3 +23 = 86/3 +69/3=155/3≈51.67°$. But maybe the problem has integer answer, so perhaps the exterior angle is the one supplementary to angle A? No, the standard exterior angle property applies here.

Wait, let's redo:

Step1: Apply exterior angle theorem

Exterior angle = sum of remote interior angles:
$4x + 17 = (x + 23) + 80$

Step2: Simplify the equation

$4x + 17 = x + 103$
$4x - x = 103 - 17$
$3x = 86$
$x = \frac{86}{3}$

Step3: Calculate angle A

Angle A = $x + 23 = \frac{86}{3} + \frac{69}{3} = \frac{155}{3} ≈ 51.67°$

But if the problem expects integer, maybe the exterior angle is adjacent to angle A: then $4x+17 + (x+23) =180$ → $5x+40=180$ → $5x=140$ → $x=28$. Then angle A=28+23=51°, and the other angle would be 180-51-80=49°, and exterior angle 4*28+17=129°, which is 51+78? No, 51+80=131≠129. So original exterior angle theorem is correct.

So angle A is $\frac{155}{3}°$ or approximately 51.67°.

Answer:

$\frac{155}{3}^\circ$ (or approximately $51.7^\circ$)