Question

find the value of each variable.

1. (8y - 102)° 31° (2y + 6)°
2. (2y + 50)° (7x - 248)° (5y - 17)° (x + 44)°
3. (5x + 4)° 74° (2x - 24)° 23°

Explanation:

Step1: Identify vertical - angle relationship for problem 2

Vertical angles are equal. So, \(2y + 6=31\).
\[2y+6 = 31\]
\[2y=31 - 6\]
\[2y = 25\]
\[y=\frac{25}{2}=12.5\]
Also, \(8y-102\) and \(2y + 6\) are supplementary (linear - pair). Substitute \(y = 12.5\) into \(8y-102\): \(8\times12.5-102=100 - 102=-2\) (This is wrong. Let's use the linear - pair relationship for the correct approach).
Since \(31+(8y - 102)=180\) (linear - pair of angles).
\[8y-102=180 - 31\]
\[8y-102 = 149\]
\[8y=149 + 102\]
\[8y=251\]
\[y=\frac{251}{8}=31.375\]
And \(2y+6\) and \(31\) are vertical angles.

Step2: Identify vertical - angle and linear - pair relationships for problem 3

Vertical angles: \(2y + 50=5y-17\)
\[5y-2y=50 + 17\]
\[3y=67\]
\[y=\frac{67}{3}\approx22.33\]
Also, \((7x-248)+(x + 44)=180\) (linear - pair)
\[7x-248+x + 44=180\]
\[8x-204 = 180\]
\[8x=180 + 204\]
\[8x=384\]
\[x = 48\]

Step3: Identify vertical - angle relationships for problem 4

Vertical angles: \(5x+4=74\)
\[5x=74 - 4\]
\[5x=70\]
\[x = 14\]
Also, \(2x-28\) and \(23\) are related to vertical - angle and linear - pair relationships. Let's use the fact that \(5x+4\) and \(74\) are vertical angles to find \(x\) first.

Answer:

Problem 2: \(y = 31.375\)
Problem 3: \(x = 48,y=\frac{67}{3}\approx22.33\)
Problem 4: \(x = 14\)