Question

find the value of x.
m\angle fgj=(2x + 23)^circ

Explanation:

Step1: Identify right angles and angle sum

From the diagram, $\angle F$ and $\angle H$ are right angles ($90^\circ$). Also, $FG$ and $JH$ seem to be equal (marked congruent), so the triangles or angles might be related. Notice that $\angle FGJ + \angle JGH + 49^\circ = 90^\circ + 90^\circ$? Wait, no, actually, since $FG \perp GF$ (wait, $\angle F$ is right, so $FG$ is perpendicular to the vertical line, and $GH$ is horizontal with $\angle H$ right. So $FG$ and $JH$ are both perpendicular to their respective lines, so $FG \parallel JH$? Wait, maybe the angles at $G$: $\angle FGJ + 49^\circ = 90^\circ$? Wait, no, let's look at the right angles. $\angle F = 90^\circ$, $\angle H = 90^\circ$, and $FG = JH$ (marked), $GJ$ is common? Wait, maybe $\angle FGJ$ and the angle with $49^\circ$ add up to $90^\circ$? Wait, no, the diagram shows $\angle FGJ = (2x + 23)^\circ$ and there's a $49^\circ$ angle, and since $\angle F$ and $\angle H$ are right angles, the figure might be a rectangle or have some congruent triangles. Wait, actually, since $\angle F = 90^\circ$ and $\angle H = 90^\circ$, and $FG = JH$ (marked), $GJ$ is a transversal, so maybe $\angle FGJ + 49^\circ = 90^\circ$? Wait, no, let's check the angle at $G$. The total angle at $G$ between the two right angles: since $\angle F$ and $\angle H$ are right, the sum of $\angle FGJ$ and $49^\circ$ should be $90^\circ$? Wait, no, maybe $\angle FGJ + 49^\circ = 90^\circ$? Wait, let's set up the equation. If $\angle FGJ + 49^\circ = 90^\circ$, then:

Step2: Solve for x

Set up the equation: $2x + 23 + 49 = 90$? Wait, no, wait. Wait, $\angle FGJ$ and the $49^\circ$ angle are complementary? Wait, no, the right angle at $F$: $\angle F = 90^\circ$, so the angle $\angle FGJ$ and the angle adjacent to $49^\circ$? Wait, maybe the correct equation is $2x + 23 + 49 = 90$? Wait, no, let's re-examine. The diagram has $\angle FGJ = (2x + 23)^\circ$ and a $49^\circ$ angle, and since $\angle F$ and $\angle H$ are right angles, the sum of $\angle FGJ$ and $49^\circ$ should be $90^\circ$? Wait, no, maybe the total angle at $G$ between the two perpendicular lines is $90^\circ$, so $\angle FGJ + 49^\circ = 90^\circ$. So:
$2x + 23 + 49 = 90$? Wait, no, $2x + 23 + 49 = 90$? Wait, $2x + 72 = 90$? Then $2x = 18$, $x = 9$? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, the right angle at $F$: $\angle F = 90^\circ$, so the angle $\angle FGJ$ and the angle between $GJ$ and $GH$? Wait, no, the other right angle is at $H$, so $GH \perp JH$, so $JH$ is vertical, $GH$ is horizontal. $FG$ is horizontal (since $\angle F$ is right), $JH$ is vertical? Wait, no, the diagram: $F$ has a right angle, so $FG$ is horizontal, and the line from $F$ is vertical. $H$ has a right angle, so $GH$ is horizontal, $JH$ is vertical. So $FG$ and $GH$ are both horizontal, $F$ and $H$ are vertical? Wait, maybe $FG$ and $JH$ are both vertical, $GH$ and $FJ$ are horizontal. So $FGHJ$ is a rectangle? Then $FG = JH$ (marked), $FJ = GH$. Then the angle at $G$: $\angle FGJ$ and $\angle JGH$: since $FG$ and $JH$ are vertical, $FJ$ and $GH$ are horizontal, so $GJ$ is a diagonal. Then the angle $\angle FGJ$ and $49^\circ$: since $\angle H = 90^\circ$, $\angle JGH = 49^\circ$, so $\angle FGJ + 49^\circ = 90^\circ$? Wait, no, $\angle FGJ$ is at $G$ between $FG$ (vertical) and $GJ$, and $\angle JGH$ is between $GJ$ and $GH$ (horizontal). So together, they make a right angle? Wait, $FG$ is vertical, $GH$ is horizontal, so the angle between $FG$ and $GH$ is $90^\circ$, so $\angle FGJ + 49^\circ = 90^\circ$. So:
$2x + 23 + 49 = 90$? Wait, no, $2x + 23 + 49 = 90$? Wait, $2x + 72 = 90$? Then $2x = 18$, $x = 9$? Wait, that seems low. Wait, maybe the equation is $2x + 23 = 90 - 49$? Wait, $90 - 49 = 41$, so $2x + 23 = 41$? Then $2x = 18$, $x = 9$. Wait, that's the same. Wait, let's check again. The angle $\angle FGJ$ and the $49^\circ$ angle are complementary because they form a right angle (since $FG$ is vertical and $GH$ is horizontal, the angle between them is $90^\circ$). So:
$m\angle FGJ + 49^\circ = 90^\circ$
Substitute $m\angle FGJ = 2x + 23$:
$2x + 23 + 49 = 90$
Simplify:
$2x + 72 = 90$
Subtract 72 from both sides:
$2x = 90 - 72$
$2x = 18$
Divide by 2:
$x = 9$
Wait, but let's confirm. If $x = 9$, then $2x + 23 = 18 + 23 = 41$, and $41 + 49 = 90$, which is a right angle, so that makes sense. So the equation is correct.

Answer:

$x = 9$