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Parallel lines and transversals form various angle relationships. To solve problems involving parallel lines and transversals, identify the given information, use the properties of parallel lines and angle relationships to write and solve equations, and check your answers for consistency.
Parallel lines are lines in a plane that never intersect. Transversals are lines that cross two parallel lines. When a transversal intersects two parallel lines, various angle relationships are formed. For example, corresponding angles are congruent, alternate interior angles are congruent, and alternate exterior angles are congruent.
To solve problems involving parallel lines and transversals, first identify the given information. Then, use the properties of parallel lines and the angle relationships formed by the transversal to write and solve equations. Finally, check your answers and make sure they are consistent with the given information.
Here is an example problem: If two parallel lines are cut by a transversal, and the measure of one of the alternate interior angles is 80 degrees, what is the measure of the corresponding angle?
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Rewritten by : Barada
Problem 13:
- [tex]\( x = 8 \)[/tex]
- [tex]\( y = 7.4 \)[/tex]
Problem 14:
- [tex]\( x = 8.75 \)[/tex]
- [tex]\( y = 7.53125 \)[/tex]
Problem 15:
- [tex]\( x = 12.5833 \)[/tex]
- [tex]\( y = 42.937375 \)[/tex]
To solve for [tex]\( x \) and \( y \)[/tex] in each of the problems, we use the properties of parallel lines and transversals. When two parallel lines are intersected by a transversal, corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary (add up to 180 degrees).
Problem 13:
Given:
- [tex]\( j \parallel k \)[/tex]
- [tex]\( l \parallel m \)[/tex]
1. [tex]\( 15y - 48^\circ \) and \( (11x - 25)^\circ \)[/tex] are corresponding angles.
[tex]\[ 15y - 48 = 11x - 25 \][/tex]
2. [tex]\( (8x - 1)^\circ \) and \( (11x - 25)^\circ \)[/tex] are alternate interior angles, so they are equal:
[tex]\[ 8x - 1 = 11x - 25 \][/tex]
Solving for [tex]\( x \):[/tex]
[tex]\[ 8x - 1 = 11x - 25 \][/tex]
[tex]\[ -1 + 25 = 11x - 8x \][/tex]
[tex]\[ 24 = 3x \][/tex]
[tex]\[ x = 8 \][/tex]
3. Substitute [tex]\( x = 8 \)[/tex] back into the corresponding angles equation:
[tex]\[ 15y - 48 = 11(8) - 25 \][/tex]
[tex]\[ 15y - 48 = 88 - 25 \][/tex]
[tex]\[ 15y - 48 = 63 \][/tex]
[tex]\[ 15y = 63 + 48 \][/tex]
[tex]\[ 15y = 111 \][/tex]
[tex]\[ y = \frac{111}{15} \][/tex]
[tex]\[ y = 7.4 \][/tex]
Problem 14:
Given:
- [tex]\( l \parallel m \)[/tex]
1. [tex]\( (4x + 4)^\circ \) and \( 39^\circ \)[/tex] are corresponding angles.
[tex]\[ 4x + 4 = 39 \][/tex]
Solving for [tex]\( x \):[/tex]
[tex]\[ 4x = 39 - 4 \][/tex]
[tex]\[ 4x = 35 \][/tex]
[tex]\[ x = \frac{35}{4} \][/tex]
[tex]\[ x = 8.75 \][/tex]
2. [tex]\( (7x - 44)^\circ \) and \( (8y - 43)^\circ \)[/tex] are alternate interior angles, so they are equal:
[tex]\[ 7x - 44 = 8y - 43 \][/tex]
Substitute [tex]\( x = 8.75 \):[/tex]
[tex]\[ 7(8.75) - 44 = 8y - 43 \][/tex]
[tex]\[ 61.25 - 44 = 8y - 43 \][/tex]
[tex]\[ 17.25 = 8y - 43 \][/tex]
[tex]\[ 17.25 + 43 = 8y \][/tex]
[tex]\[ 60.25 = 8y \][/tex]
[tex]\[ y = \frac{60.25}{8} \][/tex]
[tex]\[ y = 7.53125 \][/tex]
Problem 15:
Given:
- [tex]\( l \parallel m \)[/tex]
1. [tex]\( (15x - 26)^\circ \) and \( (4y - 9)^\circ \)[/tex] are corresponding angles.
[tex]\[ 15x - 26 = 4y - 9 \][/tex]
2. [tex]\( 28^\circ \) and \( (12x + 1)^\circ \)[/tex] are consecutive interior angles, so they are supplementary:
[tex]\[ 28 + 12x + 1 = 180 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ 12x + 29 = 180 \][/tex]
[tex]\[ 12x = 151 \][/tex]
[tex]\[ x = \frac{151}{12} \][/tex]
[tex]\[ x = 12.5833 \][/tex]
3. Substitute [tex]\( x = 12.5833 \)[/tex] back into the corresponding angles equation:
[tex]\[ 15(12.5833) - 26 = 4y - 9 \][/tex]
[tex]\[ 188.7495 - 26 = 4y - 9 \][/tex]
[tex]\[ 162.7495 = 4y - 9 \][/tex]
[tex]\[ 162.7495 + 9 = 4y \][/tex]
[tex]\[ 171.7495 = 4y \][/tex]
[tex]\[ y = \frac{171.7495}{4} \][/tex]
[tex]\[ y = 42.937375 \][/tex]
Final Answers:
Problem 13:
- [tex]\( x = 8 \)[/tex]
- [tex]\( y = 7.4 \)[/tex]
Problem 14:
- [tex]\( x = 8.75 \)[/tex]
- [tex]\( y = 7.53125 \)[/tex]
Problem 15:
- [tex]\( x = 12.5833 \)[/tex]
- [tex]\( y = 42.937375 \)[/tex]
