Lesson 4.5 Number 4 Use the following diagram to solve the problem, given that \( \overline{\mathbf{A B}} \| \overline{\mathbf{C D}} \) and lin the transversal. Justify your answers using angle relationships in parallel lin intersected by a transversal. What is \( \mathrm{m} \angle 4 \) if \( \mathrm{m} \angle 4=2(5 \mathrm{x}-9) \) and \( \mathrm{m} \angle 5=3(\mathrm{x}+8) \) ? Answer format: 150 degrees

To solve for \( \mathrm{m} \angle 4 \), we will use the properties of parallel lines intersected by a transversal. Given that \( \overline{\mathbf{A B}} \| \overline{\mathbf{C D}} \) and line \( \ell \) is the transversal, we can identify the relationships between the angles. 1. **Identify the angle relationships**: - Since \( \overline{\mathbf{A B}} \| \overline{\mathbf{C D}} \), angles \( \angle 4 \) and \( \angle 5 \) are corresponding angles. Therefore, they are equal: \[ \mathrm{m} \angle 4 = \mathrm{m} \angle 5 \] 2. **Set up the equation**: - We have the expressions for the angles: \[ \mathrm{m} \angle 4 = 2(5x - 9) \] \[ \mathrm{m} \angle 5 = 3(x + 8) \] - Setting these equal gives us: \[ 2(5x - 9) = 3(x + 8) \] 3. **Solve for \( x \)**: - Expand both sides: \[ 10x - 18 = 3x + 24 \] - Rearranging the equation: \[ 10x - 3x = 24 + 18 \] \[ 7x = 42 \] - Dividing by 7: \[ x = 6 \] 4. **Substitute \( x \) back to find \( \mathrm{m} \angle 4 \)**: - Substitute \( x = 6 \) into the expression for \( \mathrm{m} \angle 4 \): \[ \mathrm{m} \angle 4 = 2(5(6) - 9) \] \[ = 2(30 - 9) \] \[ = 2(21) \] \[ = 42 \] 5. **Substitute \( x \) back to find \( \mathrm{m} \angle 5 \)**: - Substitute \( x = 6 \) into the expression for \( \mathrm{m} \angle 5 \): \[ \mathrm{m} \angle 5 = 3(6 + 8) \] \[ = 3(14) \] \[ = 42 \] Since both angles are equal, we confirm that \( \mathrm{m} \angle 4 = 42 \) degrees. Thus, the final answer is: \[ \mathrm{m} \angle 4 = 42 \text{ degrees} \]