Question

categorize the following images with the correct equation setup. the lines are parallel. 2x + 3 = 5x; 2x + 3 + 5x = 180

Explanation:

Step1: Analyze the middle graph

The middle graph has two angles \(5x\) and \(2x + 3\) formed by a transversal intersecting two parallel lines. These angles appear to be vertical angles? Wait, no, looking at the position, if the lines are parallel and the angles are on a straight line (supplementary) or equal? Wait, the second equation is \(2x + 3+5x = 180\), which is for supplementary angles (sum to 180°). Wait, the middle graph: let's check the angle positions. If the two angles \(5x\) and \(2x + 3\) are adjacent and form a linear pair (sum to 180°), then the equation \(2x + 3+5x = 180\) applies. Wait, the right graph: the angles \(5x\) and \(2x + 3\) are alternate interior angles or corresponding angles? If the lines are parallel, alternate interior angles are equal, so \(2x + 3=5x\) would be for equal angles (like alternate interior or corresponding). Wait, the problem says "the lines are parallel". Let's re - examine the graphs:

• For the right graph (second from right, labeled 22?): The angles \(5x\) and \(2x + 3\) seem to be alternate interior angles (since the transversal crosses two parallel lines, alternate interior angles are congruent). So their measures are equal, so the equation \(2x + 3 = 5x\) is correct for the right graph.

• For the middle graph (labeled 23): The angles \(5x\) and \(2x + 3\) are adjacent and form a linear pair (they are on a straight line), so their sum is 180°, so the equation \(2x + 3+5x = 180\) is correct for the middle graph.

Assuming we need to match the middle graph (with angles \(5x\) and \(2x + 3\) on a straight line) with the equation \(2x + 3+5x = 180\) and the right graph with \(2x + 3 = 5x\). But the question is to categorize the images with the correct equation setup. Let's focus on the middle graph:

The middle graph has two angles \(5x\) and \(2x + 3\) that are supplementary (they form a linear pair, so their sum is 180°). So the equation for the middle graph is \(2x+3 + 5x=180\).

Step2: Analyze the right graph

The right graph has two angles \(5x\) and \(2x + 3\) that are equal (alternate interior angles or corresponding angles, since the lines are parallel). So the equation for the right graph is \(2x + 3=5x\).

Answer:

• Middle Graph (labeled 23): Equation \(2x + 3+5x = 180\)
• Right Graph (labeled 22): Equation \(2x + 3 = 5x\)