Question

miguel solved the following problem below. is his work/answer correct? explain any mistakes he made. the diagram shows two parallel lines that are cut by a transversal.
miguel’s work:
3x + 15 = 4x + 25
-15 -15
3x = 4x + 10
-4x -4x
x = 10
his work and answer is correct.
he set up the problem correctly, but forgot the negative when solving for x.
he set up the problem wrong, these angles are supplementary, but his algebra is correct.
he set up the problem wrong, these angles are supplementary and he made algebra mistakes.

Explanation:

1. Angle Relationship: When two parallel lines are cut by a transversal, same - side interior angles are supplementary (their sum is \(180^{\circ}\)). The angles \(3x + 15\) and \(4x+25\) are same - side interior angles, so the correct equation should be \((3x + 15)+(4x + 25)=180\), not \(3x + 15=4x + 25\). So Miguel set up the problem wrong.
2. Algebra Mistakes: In his algebraic manipulation, when he subtracted \(15\) from both sides of \(3x + 15=4x + 25\), he should get \(3x=4x + 10\) (this step is correct). But when he subtracts \(4x\) from both sides, \(3x-4x=4x + 10-4x\) gives \(-x = 10\), so \(x=- 10\), not \(x = 10\). So he also made an algebra mistake.

Answer:

He set up the problem wrong, these angles are supplementary and he made algebra mistakes.