Angle Measures 04Dc74

1. **State the problem:** We are given two angles, $\angle 1$ and $\angle 2$, with measures expressed in terms of $x$: $$\angle 1 = (x - 9)^\circ$$ $$\angle 2 = 6x^\circ$$ We need to find the values of these angles. 2. **Identify the relationship between the angles:** Usually, in geometry problems involving two angles labeled $\angle 1$ and $\angle 2$, they might be supplementary (sum to $180^\circ$) or complementary (sum to $90^\circ$). Since the problem does not specify, we assume they are supplementary angles (common in such problems). 3. **Write the equation for supplementary angles:** $$\angle 1 + \angle 2 = 180^\circ$$ Substitute the expressions: $$ (x - 9) + 6x = 180 $$ 4. **Simplify the equation:** $$ x - 9 + 6x = 180 $$ $$ 7x - 9 = 180 $$ 5. **Solve for $x$:** $$ 7x - 9 = 180 $$ $$ 7x = 180 + 9 $$ $$ 7x = 189 $$ $$ x = \frac{189}{7} $$ $$ x = 27 $$ 6. **Find the angle measures:** $$ \angle 1 = (x - 9)^\circ = (27 - 9)^\circ = 18^\circ $$ $$ \angle 2 = 6x^\circ = 6 \times 27^\circ = 162^\circ $$ 7. **Check the sum:** $$ 18^\circ + 162^\circ = 180^\circ $$ The angles are supplementary as expected. **Final answer:** $$ \angle 1 = 18^\circ, \quad \angle 2 = 162^\circ $$