(x + 2)° (9x-62)°
ChapterP: Prerequisites
SectionP.6: The Rectangular Coordinate System And Graphs
Problem 37E: An airplane flies from Naples, Italy, in a straight line to Rome, Italy, which is 120 kilometers...
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Question
Find the measure of each marked angle . Assume the lines are parallel.
![In the image, there is a diagram featuring a pair of parallel lines cut by a transversal. This creates two sets of alternate interior angles. The top left angle is labeled \((x + 2)^\circ\) and the top right angle is labeled \((9x - 62)^\circ\).
### Explanation:
1. **Parallel Lines:** The diagram shows two parallel lines intersected by a transversal line.
2. **Alternate Interior Angles:** The angles \((x + 2)^\circ\) and \((9x - 62)^\circ\) form a pair of alternate interior angles.
### Educational Notes:
- **Alternate Interior Angles:** When two parallel lines are cut by a transversal, the alternate interior angles are equal.
Therefore,
\[
(x + 2)^\circ = (9x - 62)^\circ
\]
To solve for \(x\), you set the expressions equal to each other:
\[
x + 2 = 9x - 62
\]
Re-arrange the equation to solve for \(x\):
\[
x - 9x = -62 - 2
\]
\[
-8x = -64
\]
\[
x = 8
\]
- **Substitute \(x\) back into one of the angle expressions to find the measure of the angles**:
If \(x = 8\), then:
\[
(x + 2)^\circ = (8 + 2)^\circ = 10^\circ
\]
\[
(9x - 62)^\circ = (9(8) - 62)^\circ = (72 - 62)^\circ = 10^\circ
\]
Therefore, both angles equal \(10^\circ\).
### Conclusion:
This example illustrates the property of alternate interior angles and how to solve for unknowns in algebraic expressions involving angle measures.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c3214b5-f2f5-4a89-904a-2906edf42c76%2F7595017a-5817-46d4-bf26-3e3c803f877b%2Fq1nbjnp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In the image, there is a diagram featuring a pair of parallel lines cut by a transversal. This creates two sets of alternate interior angles. The top left angle is labeled \((x + 2)^\circ\) and the top right angle is labeled \((9x - 62)^\circ\).
### Explanation:
1. **Parallel Lines:** The diagram shows two parallel lines intersected by a transversal line.
2. **Alternate Interior Angles:** The angles \((x + 2)^\circ\) and \((9x - 62)^\circ\) form a pair of alternate interior angles.
### Educational Notes:
- **Alternate Interior Angles:** When two parallel lines are cut by a transversal, the alternate interior angles are equal.
Therefore,
\[
(x + 2)^\circ = (9x - 62)^\circ
\]
To solve for \(x\), you set the expressions equal to each other:
\[
x + 2 = 9x - 62
\]
Re-arrange the equation to solve for \(x\):
\[
x - 9x = -62 - 2
\]
\[
-8x = -64
\]
\[
x = 8
\]
- **Substitute \(x\) back into one of the angle expressions to find the measure of the angles**:
If \(x = 8\), then:
\[
(x + 2)^\circ = (8 + 2)^\circ = 10^\circ
\]
\[
(9x - 62)^\circ = (9(8) - 62)^\circ = (72 - 62)^\circ = 10^\circ
\]
Therefore, both angles equal \(10^\circ\).
### Conclusion:
This example illustrates the property of alternate interior angles and how to solve for unknowns in algebraic expressions involving angle measures.
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