Question

1. find m∠abc.

(6x - 7)° (4x + 23)°
options: 90°, 15°, 83°, 41°

Explanation:

Step1: Identify Vertical Angles

Angles \( \angle ABC = (6x - 7)^\circ \) and \( \angle DBE \) are vertical angles? Wait, no, actually \( \angle ABC \) and \( \angle DBE \) are vertical? Wait, no, the two angles \( (6x - 7)^\circ \) and \( (4x + 23)^\circ \) are adjacent and form a linear pair? Wait, no, when two lines intersect, vertical angles are equal. Wait, actually, \( \angle ABC \) and the angle adjacent to it ( \( (4x + 23)^\circ \)) are supplementary? Wait, no, looking at the diagram, lines \( AE \) and \( CD \) intersect at \( B \), so \( \angle ABC \) and \( \angle DBE \) are vertical angles, but \( \angle ABC \) and \( \angle ABD \) are supplementary? Wait, no, the two angles given at point \( B \): \( (6x - 7)^\circ \) and \( (4x + 23)^\circ \) are actually vertical angles? Wait, no, when two lines intersect, vertical angles are equal. Wait, maybe I made a mistake. Wait, the angles \( (6x - 7)^\circ \) and \( (4x + 23)^\circ \) are vertical angles? Wait, no, if two lines intersect, the opposite angles are vertical and equal. Wait, maybe the two angles \( (6x - 7)^\circ \) and \( (4x + 23)^\circ \) are adjacent and form a linear pair? No, linear pair angles are supplementary (sum to \( 180^\circ \)), but vertical angles are equal. Wait, looking at the diagram, \( AE \) and \( CD \) intersect at \( B \), so \( \angle ABC \) and \( \angle DBE \) are vertical angles, and \( \angle ABC \) and \( \angle ABD \) are supplementary. Wait, but the angles given are \( (6x - 7)^\circ \) (which is \( \angle ABC \)) and \( (4x + 23)^\circ \) (which is \( \angle ABD \))? Wait, no, maybe \( (6x - 7)^\circ \) and \( (4x + 23)^\circ \) are vertical angles. Wait, let's check: if two lines intersect, vertical angles are equal, so \( 6x - 7 = 4x + 23 \). Let's solve that.

Step2: Solve for \( x \)

Set the two expressions equal (since they are vertical angles):
\( 6x - 7 = 4x + 23 \)
Subtract \( 4x \) from both sides:
\( 2x - 7 = 23 \)
Add 7 to both sides:
\( 2x = 30 \)
Divide by 2:
\( x = 15 \)

Step3: Find \( m\angle ABC \)

Substitute \( x = 15 \) into \( 6x - 7 \):
\( 6(15) - 7 = 90 - 7 = 83 \)
So \( m\angle ABC = 83^\circ \)

Answer:

\( 83^\circ \)