Angle Abc 6C8752

1. **State the problem:** We need to find the measure of $\angle ABC$ given the angles around point $B$ formed by intersecting lines. 2. **Analyze the figure and given angles:** The lines intersect at $B$, creating vertical and supplementary angles. We have angles labeled as $(x + y)^\circ$, $96^\circ$, $y^\circ$, and $2x^\circ$ around $B$. 3. **Use angle relationships:** Vertical angles are equal, and angles on a straight line sum to $180^\circ$. 4. **Set up equations:** - Since $A-B-E$ is a straight line, angles on it sum to $180^\circ$. - The angle adjacent to $96^\circ$ on the straight line is $x + y$. So, $$ (x + y) + 96 = 180 $$ 5. **Simplify:** $$ x + y = 180 - 96 = 84 $$ 6. **Use vertical angles:** - $y^\circ$ and $2x^\circ$ are vertical angles, so they are equal: $$ y = 2x $$ 7. **Substitute $y = 2x$ into $x + y = 84$:** $$ x + 2x = 84 $$ $$ 3x = 84 $$ 8. **Solve for $x$:** $$ x = \frac{84}{3} = 28 $$ 9. **Find $y$:** $$ y = 2x = 2 \times 28 = 56 $$ 10. **Find $\angle ABC$:** - $\angle ABC = (x + y)^\circ = 28 + 56 = 84^\circ$ **Final answer:** $$ m\angle ABC = 84^\circ$$