22. The measure of \( \angle G \) is six more than twice the measure of \( \angle H \). If \( \angle G \) and \( \angle H \) are complementary angles, find \( m \angle H \).

Let the measure of \( \angle H \) be denoted as \( h \). According to the problem, the measure of \( \angle G \) can be expressed as: \[ g = 2h + 6 \] Since \( \angle G \) and \( \angle H \) are complementary angles, their measures add up to \( 90^\circ \). Therefore, we can write the equation: \[ g + h = 90 \] Substituting the expression for \( g \) into the equation gives: \[ (2h + 6) + h = 90 \] Now, combine like terms: \[ 3h + 6 = 90 \] Next, we will isolate \( h \) by first subtracting 6 from both sides: \[ 3h = 90 - 6 \] \[ 3h = 84 \] Now, divide both sides by 3 to solve for \( h \): \[ h = \frac{84}{3} = 28 \] Thus, the measure of \( \angle H \) is: \[ \boxed{28} \] To verify, we can calculate \( \angle G \): \[ g = 2h + 6 = 2(28) + 6 = 56 + 6 = 62 \] Now, check if \( \angle G \) and \( \angle H \) are indeed complementary: \[ g + h = 62 + 28 = 90 \] Since the sum is \( 90^\circ \), our solution is confirmed. The measure of \( \angle H \) is \( \boxed{28} \).