Question

if $m\angle ghj = m\angle ihj = 49^\circ$, $gj = a + 54$, and $ij = 3a$, what is the value of $a$?
$a = \square$

Explanation:

Step1: Identify congruent segments

Since $JH$ is an angle bisector of $\angle GHI$, and $GJ \perp GH$, $IJ \perp IH$, by the Angle Bisector Theorem, the distances from $J$ to the sides of the angle are equal: $GJ = IJ$.

Step2: Set up equation

Substitute the given expressions:
$a + 54 = 3a$

Step3: Solve for $a$

Subtract $a$ from both sides:
$54 = 2a$
Divide both sides by 2:
$a = \frac{54}{2} = 27$

Answer:

$27$