Question

a line intersects two parallel lines, forming four acute angles and four obtuse angles. the measure of one of the acute angles is ((9x - 490)^circ). the sum of the measures of one of the acute angles and three of the obtuse angles is ((-18x + w)^circ). what is the value of (w)?

Explanation:

Step1: Relate acute and obtuse angles

Acute + obtuse = 180°
Obtuse = 180° - (9x - 490)°

Step2: Calculate sum of 1 acute + 3 obtuse

Sum = (9x - 490) + 3[180 - (9x - 490)]
= 9x - 490 + 540 - 27x + 1470
= -18x + 1520

Step3: Compare with given sum

-18x + 1520 = -18x + w
w = 1520 - 50 = 1470 (correction: arithmetic error fixed: -490+540=50; 50+1470=1520? Wait no: 9x-27x=-18x; -490+3180+3490= -490+540+1470= (540-490)+1470=50+1470=1520. Yes, so w=1520? Wait original problem: sum is (-18x + w). So sum expression after expansion is -18x +1520, so w=1520. But wait initial acute angle must be positive: 9x-490>0 →9x>490→x>490/9≈54.44. Obtuse angle:180-(9x-490)=670-9x>90→670-9x>90→9x<580→x<580/9≈64.44. So x is between ~54.44 and ~64.44. The sum expression is correct as -18x+1520, so w=1520. My earlier correction was wrong. So final w=1520.

Wait recheck Step2:
Sum = acute + 3 obtuse = A + 3(180 - A) = A + 540 -3A = 540 -2A.
Given A=9x-490, so sum=540 -2*(9x-490)=540-18x+980=1520-18x. Yes! That's simpler: 540+980=1520. So sum= -18x +1520. Therefore, w=1520. Correct. So answer is 1520.

Step1: Acute + Obtuse = 180°

Obtuse = 180° - Acute

Step2: Sum = Acute + 3×Obtuse

Sum = Acute + 3(180 - Acute) = 540 - 2×Acute

Step3: Substitute Acute = 9x - 490

Sum = 540 - 2(9x - 490) = 540 -18x +980 = -18x +1520

Step4: Equate to given sum

-18x +1520 = -18x +w → w=1520

Answer:

1470