Answer
$m \angle 1 = 62^{\circ}$
$m \angle 2 = 118^{\circ}$
$m \angle 3 = 118^{\circ}$
$x = \frac{5}{2}$
Work Step by Step
In an isosceles trapezoid, base angles are congruent to one another; therefore, $m \angle 1 = 62^{\circ}$.
In a trapezoid, interior angles add up to $360^{\circ}$. Let's set up an equation to find $m \angle 2$ and $m \angle 3$:
$62^{\circ} + 62^{\circ} + m \angle 2 + m \angle 3 = 360^{\circ}$
Add constants on the left side of the equation to simplify:
$124^{\circ} + m \angle 2 + m \angle 3 = 360^{\circ}$
Subtract $124^{\circ}$ from each side of the equation to move constants to the right side of the equation:
$m \angle 2 + m \angle 3 = 236^{\circ}$
Again, base angles of an isosceles trapezoid are congruent; therefore, $m \angle 2$ is half of $236^{\circ}$, and $m \angle 3$ is also half of $236^{\circ}$:
$m \angle 2 = m \angle 3 = 236^{\circ} \div 2$
Divide to solve:
$m \angle 2 = m \angle 3 = 118^{\circ}$
The legs of an isosceles trapezoid are congruent to one another, so let us set the two legs equal to one another:
$x + 2 = 5x - 8$
Subtract $2$ from each side of the equation to move constants to the right side of the equation:
$x = 5x - 10$
Subtract $5x$ from each side of the equation to move variable terms to the left side of the equation:
$-4x = -10$
Divide both sides by $-4$ to solve for $x$:
$x = \frac{10}{4}$
Divide both the numerator and denominator by their greatest common factor:
$x = \frac{5}{2}$
