Question

1. find m∠abc.

diagram: lines ac and de intersect at point b. ∠abc is labeled (3x - 14)°, and the vertical angle (∠dbe) is labeled (2x + 10)°. below the diagram, multiple-choice options: 24°, 19°, 37°, 58° (each with a circle checkbox).

Explanation:

Step1: Identify vertical angles

Vertical angles are equal, so \(3x - 14 = 2x + 10\).

Step2: Solve for \(x\)

Subtract \(2x\) from both sides: \(3x - 2x - 14 = 2x - 2x + 10\) → \(x - 14 = 10\).
Add 14 to both sides: \(x - 14 + 14 = 10 + 14\) → \(x = 24\).

Step3: Calculate \(m\angle ABC\)

Substitute \(x = 24\) into \(3x - 14\): \(3(24) - 14 = 72 - 14 = 58\)? Wait, no—wait, wait, wait. Wait, actually, \(\angle ABC\) and \(\angle DBE\) are vertical angles? Wait, no, looking at the diagram, \(\angle ABC\) is \(3x - 14\), and the adjacent angle is \(2x + 10\), but actually, vertical angles: when two lines intersect, vertical angles are equal. So \(3x - 14 = 2x + 10\) (since they are vertical angles). Then solving \(x = 24\), then \(3x - 14 = 3*24 -14 = 72 -14 = 58\)? But wait, the options have 58? Wait, no, wait, maybe I mixed up. Wait, no, let's check again. Wait, the angle \(\angle ABC\) is \(3x -14\), and the other angle is \(2x +10\), which are vertical angles, so they are equal. So solving \(3x -14 = 2x +10\) gives \(x =24\), then \(3*24 -14 = 58\). So \(m\angle ABC = 58^\circ\)? Wait, but let's check the options. The options are 24, 19, 37, 58. So 58 is an option. Wait, but maybe I made a mistake. Wait, no, vertical angles are equal, so that's the key. So step by step:

1. Set vertical angles equal: \(3x - 14 = 2x + 10\)
2. Solve for \(x\): \(x = 24\)
3. Substitute \(x\) into \(3x -14\): \(3(24) -14 = 72 -14 = 58\)

Yes, that's correct.

Answer:

\(58^\circ\) (corresponding to the option with 58°)