Question: User: \(\angle 1\) and \(\angle 2\) are complementary angles. If \(m\angle 1=(7x-13)\) \(^\circ\) and \(m\angle 2=(6x-1)\) \(^\circ\), then find the measure of \(\angle 1\).
1. **State the problem:** We are given two complementary angles \(\angle 1\) and \(\angle 2\) with measures \(m\angle 1 = (7x - 13)^\circ\) and \(m\angle 2 = (6x - 1)^\circ\). We need to find the measure of \(\angle 1\).
2. **Recall the definition of complementary angles:** Two angles are complementary if the sum of their measures is \(90^\circ\). So,
$$m\angle 1 + m\angle 2 = 90$$
3. **Set up the equation:** Substitute the given expressions:
$$(7x - 13) + (6x - 1) = 90$$
4. **Simplify the equation:**
$$7x - 13 + 6x - 1 = 90$$
$$13x - 14 = 90$$
5. **Solve for \(x\):** Add 14 to both sides:
$$13x - \cancel{14} + \cancel{14} = 90 + 14$$
$$13x = 104$$
Divide both sides by 13:
$$\frac{13x}{\cancel{13}} = \frac{104}{\cancel{13}}$$
$$x = 8$$
6. **Find the measure of \(\angle 1\):** Substitute \(x=8\) into \(m\angle 1 = 7x - 13\):
$$m\angle 1 = 7(8) - 13 = 56 - 13 = 43$$
7. **Final answer:** The measure of \(\angle 1\) is \(43^\circ\).