Answer
$a = 6$
$m \angle G = 150^{\circ}$
$m \angle K = 150^{\circ}$
$m \angle H = 30^{\circ}$
$m \angle J = 30^{\circ}$
Work Step by Step
The diagram is that of a parallelogram.
Opposite angles of a parallelogram are congruent. We see that $\angle G$ is opposite $\angle K$ and $\angle H$ is opposite $\angle J$, so $m \angle G$ is equal to $m \angle K$ and $m\angle H$ is equal to $m \angle J$.
Let's set $m \angle G$ and $m \angle K$ equal to one another so we can solve for $a$:
$m \angle G = m \angle K$
Let's plug in what we know:
$20a + 30 = 17a + 48$
Subtract $30$ from each side of the equation to isolate constants on one side:
$20a = 17a + 18$
Subtract $17a$ from each side of the equation to isolate the variable on one side of the equation:
$3a = 18$
Divide both sides by $3$ to solve for $a$:
$a = 6$
Let's substitute $6$ for $a$ into the expression for the value of $\angle G$:
$m \angle G = 20(6) + 30$
Multiply first, according to order of operations:
$m \angle G = 120 + 30$
Add to solve:
$m \angle G = 150$
If $m \angle G$ is $150^{\circ}$, then $m \angle K$ is also $150^{\circ}$ because they are congruent.
Now let's find $m \angle H$ and $m \angle J$. Set up the expression for $m \angle H$ first:
$m \angle H = 5(6)$
Multiply to solve:
$m \angle H = 30$
If $m \angle H$ is $30^{\circ}$, then $m \angle J$ is also $30^{\circ}$ because they are congruent.
