1. x = 12
2. x = 7
3. x = 18
4. x = 5
5. x = 23
6. x = 9
7. x = 12
8. x = 15
9. x = 17; y = 8
10. x = 4; y = 10
To find the value of each missing variable(s), we would need to apply our knowledge of algebra. However, recall the following rules of angles in parallel lines:
- corresponding angles are congruent (i.e. equal to each other)
- vertically opposite angles are congruent
- alternate angles are congruent
- same-side interior angles are supplementary (i.e. sum up to give 180°
- alternate exterior angles are congruent.
Let's solve the given problems as shown below given that line l is parallel to line m:
1. [tex]58 = (5x - 2)[/tex] (corresponding angles)
solve for x
[tex]58 = 5x - 2\\58 + 2 = 5x - 2 +2[/tex] (addition property of equality)
[tex]60 = 5x \\\frac{60}{5} = \frac{5x}{5}[/tex](division property of equality)
[tex]12 = x\\x = 12[/tex]
2. [tex]134 = 16x+22[/tex] (alternate exterior angles)
[tex]\\134 - 22 = 16x +22-22[/tex] (subtraction property of equality)
[tex]134 - 22 = 16x +22-22\\112 = 16x\\\frac{112}{16} = \frac{16x}{16}[/tex](division property of equality)
[tex]7 = x\\x = 7[/tex]
3. [tex]7x - 1 = 125[/tex] (alternate angles)
[tex]7x - 1 + 1 = 125 + 1\\7x = 126\\\frac{7x}{7} = \frac{126}{7} \\x = 18[/tex]
4. [tex](9x+2) + 133 = 180[/tex] (same-side interior angles)
[tex]9x+2 + 133 = 180\\9x + 135 = 180\\9x + 135 - 135 = 180 - 135\\9x = 45\\\frac{9x}{9} = \frac{45}{9} \\x = 5[/tex]
5. [tex]3x+38 = 8x-77[/tex] (alternate exterior angles)
[tex]77 + 38 = 8x - 3x\\115 = 5x\\\frac{115}{5} = \frac{5x}{5} \\23 = x\\x = 23[/tex]
6. [tex]6x-2 = 11x-47[/tex] (corresponding angles)
[tex]6x-2 = 11x-47\\47 - 2 = 11x - 6x\\45 = 5x\\x = 9[/tex]
7. [tex]5x+75 = 13x - 21[/tex] (alternate angles)
[tex]5x+75 = 13x - 21\\21 + 75 = 13x - 5x\\96 = 8x\\12 = x\\x = 12[/tex]
8. [tex](5x+3)+(9x-33)= 180[/tex] (same-side interior angles)
[tex]5x+3+9x-33= 180\\14x - 30 = 180\\14x - 30 + 30 = 180 + 30\\14x = 210\\x = 15[/tex]
9. [tex]8x-31 = 6x+3[/tex] (alternate angles)
[tex]8x - 6x = 31 + 3\\2x = 34\\x = 17[/tex]
Solve for y
[tex](5y + 35) + (6x + 3) = 180[/tex] (same-side interior angles)
[tex]5y + 35 + 6x + 3 = 180\\[/tex]
Plug in the value of x
[tex]5y + 35 + 6(17) + 3 = 180\\5y + 35 + 102 + 3 = 180\\5y + 140 = 180\\5y = 180 - 140\\5y = 40\\y = 8[/tex]
10. [tex](29x-3)+(15x+7) = 180[/tex](supplementary angles)
Solve for y
[tex]13y - 17 = 29x - 3[/tex] (alternate exterior angles)
Plug in the value of x
[tex]13y - 17 = 29(4) - 3\\13y - 17 = 116 - 3\\13y - 17 = 113\\13y = 113 + 17\\13y = 130\\y = 10\\[/tex]
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