Question

directions: if l || m, find the value of each missing variable(s). 5. (5x + 2) 119 6. (12x - 8) 104 7. (5x + 7) (8x - 71) 8. (x - 7) (7x - 65) 9. (9x + 25) (13x - 19) (12x + 5) 10. (8y + 17) (6x - 7) (2x - 29) 11. (12x - 23) 49 (3x) (11x - 1) 12.

Explanation:

Step1: Identify angle - relationship for problem 5

Since \(l\parallel m\), the angle \((5x + 2)\) and the 119 - degree angle are corresponding angles, so \(5x+2 = 119\).
\[5x=119 - 2\]
\[5x=117\]
\[x=\frac{117}{5}=23.4\]

Step2: Identify angle - relationship for problem 6

Since \(l\parallel m\), the angle \((12x - 8)\) and the 104 - degree angle are corresponding angles, so \(12x-8 = 104\).
\[12x=104 + 8\]
\[12x=112\]
\[x=\frac{112}{12}=\frac{28}{3}\approx9.33\]

Step3: Identify angle - relationship for problem 7

Since \(l\parallel m\), the angle \((5x + 7)\) and \((8x-71)\) are alternate - interior angles, so \(5x + 7=8x-71\).
\[71 + 7=8x-5x\]
\[3x=78\]
\[x = 26\]

Step4: Identify angle - relationship for problem 8

Since \(l\parallel m\), the angle \((x + 75)\) and \((7x-65)\) are corresponding angles, so \(x + 75=7x-65\).
\[75 + 65=7x-x\]
\[6x=140\]
\[x=\frac{140}{6}=\frac{70}{3}\approx23.33\]

Step5: Identify angle - relationship for problem 9

Since \(l\parallel m\), the angle \((9x + 25)\) and \((12x-19)\) are corresponding angles, so \(9x + 25=12x-19\).
\[25 + 19=12x-9x\]
\[3x=44\]
\[x=\frac{44}{3}\approx14.67\]

Step6: Identify angle - relationship for problem 10

Since \(l\parallel m\), the angle \((8y + 17)\) and \((2x-29)\) and \((6x - 7)\) need more information about the angle - relationship. Assuming \((8y + 17)\) and \((6x - 7)\) are corresponding angles and \((2x-29)\) and \((6x - 7)\) are linear - pair related. First, if \((8y + 17)=(6x - 7)\) and \((2x-29)+(6x - 7)=180\) (linear pair). Solve \((2x-29)+(6x - 7)=180\):
\[8x-36 = 180\]
\[8x=180 + 36\]
\[8x=216\]
\[x = 27\]
Substitute \(x = 27\) into \(8y+17=6x - 7\), \(8y+17=6\times27-7\), \(8y+17=162 - 7\), \(8y+17=155\), \(8y=138\), \(y=\frac{69}{4}=17.25\)

Step7: Identify angle - relationship for problem 11

Since \(l\parallel m\), assume the angle \((3x)\) and \((11x - 1)\) are corresponding angles and \((12x-23)\) and \((3x)\) are vertical - angles related. First, if \(3x=12x-23\) (vertical angles), \(23=12x - 3x\), \(9x=23\), \(x=\frac{23}{9}\approx2.56\). But if we consider \(3x=11x - 1\) (corresponding angles), \(1=11x-3x\), \(8x=1\), \(x=\frac{1}{8}=0.125\). Let's assume the correct relationship is based on vertical - angles first. If \(3x=12x - 23\), \(x=\frac{23}{9}\).

Step8: Identify angle - relationship for problem 12

Since \(l\parallel m\), more information about the angle - relationship between \((7x-20)\), \((5x + 34)\) and other angles formed by the transversals is needed. Assuming they are corresponding angles, \(7x-20=5x + 34\).
\[7x-5x=34 + 20\]
\[2x=54\]
\[x = 27\]

Answer:

Problem 5: \(x = 23.4\)
Problem 6: \(x=\frac{28}{3}\)
Problem 7: \(x = 26\)
Problem 8: \(x=\frac{70}{3}\)
Problem 9: \(x=\frac{44}{3}\)
Problem 10: \(x = 27,y=\frac{69}{4}\)
Problem 11: \(x=\frac{23}{9}\)
Problem 12: \(x = 27\)