1. **State the problem:** We are given three angles with expressions in terms of $x$ and $y$: - $m\angle 5 = 2x + 3y$ - $m\angle 6 = 5x + y$ - $m\angle 7 = -2x + 15y$ We know $m\angle 5$ is a right angle, so $m\angle 5 = 90^\circ$. 2. **Use angle relationships:** Since angles 5, 6, and 7 are formed by intersecting lines, and angle 5 is a right angle, angles 6 and 7 are adjacent to it. The sum of angles 6 and 7 equals angle 5 because they form a straight line (linear pair). So, $$m\angle 6 + m\angle 7 = m\angle 5 = 90$$ 3. **Write the equation:** Substitute the expressions: $$5x + y + (-2x + 15y) = 90$$ Simplify: $$5x - 2x + y + 15y = 90$$ $$3x + 16y = 90$$ 4. **Use the expression for $m\angle 5$:** $$2x + 3y = 90$$ 5. **Solve the system:** From the two equations: $$\begin{cases} 2x + 3y = 90 \\ 3x + 16y = 90 \end{cases}$$ Multiply the first equation by 3: $$6x + 9y = 270$$ Multiply the second equation by 2: $$6x + 32y = 180$$ 6. **Subtract the second from the first:** $$\cancel{6x} + 9y - (\cancel{6x} + 32y) = 270 - 180$$ $$9y - 32y = 90$$ $$-23y = 90$$ 7. **Solve for $y$:** $$y = \frac{90}{-23} = -\frac{90}{23}$$ 8. **Substitute $y$ back into $2x + 3y = 90$:** $$2x + 3\left(-\frac{90}{23}\right) = 90$$ $$2x - \frac{270}{23} = 90$$ Add $\frac{270}{23}$ to both sides: $$2x = 90 + \frac{270}{23} = \frac{90 \times 23}{23} + \frac{270}{23} = \frac{2070 + 270}{23} = \frac{2340}{23}$$ 9. **Solve for $x$:** $$x = \frac{2340}{23 \times 2} = \frac{2340}{46} = 50.8695652174$$ 10. **Find $m\angle 5$:** Recall: $$m\angle 5 = 2x + 3y$$ Substitute values: $$2(50.8695652174) + 3\left(-\frac{90}{23}\right) = 101.7391304348 - 11.7391304348 = 90$$ **Final answer:** $$m\angle 5 = 90^\circ$$ This confirms the given right angle measure.