Question

find the measure of \\( \angle g j k \\).
\\( m \angle g j k= \\) \\( { }^{circ} \\)

Explanation:

Step1: Identify parallel lines and alternate interior angles

Since \( FH \parallel FG \) (assuming \( FH \) and \( FG \) are parallel, and \( EK \) is a transversal), the alternate interior angles \( \angle GJK \) and \( \angle IKH \) are equal? Wait, no, actually, looking at the diagram, \( FJ \parallel HK \) (the two vertical parallel lines), and \( ED \) is a transversal. So \( \angle GJK \) and \( \angle IKH \) are alternate interior angles? Wait, no, the angles given are \( (2x + 92)^\circ \) at \( J \) ( \( \angle GJK \)?) Wait, no, the angle at \( J \) is \( (2x + 92)^\circ \) and at \( K \) is \( (3x + 63)^\circ \). Since \( FJ \parallel HK \), these are alternate interior angles, so they should be equal. So set \( 2x + 92 = 3x + 63 \).

Step2: Solve for \( x \)

\( 2x + 92 = 3x + 63 \)
Subtract \( 2x \) from both sides: \( 92 = x + 63 \)
Subtract \( 63 \) from both sides: \( x = 92 - 63 = 29 \)

Step3: Find \( m\angle GJK \)

Now substitute \( x = 29 \) into \( 2x + 92 \):
\( 2(29) + 92 = 58 + 92 = 150 \)? Wait, no, wait, maybe I mixed up the angles. Wait, maybe they are same - side interior angles? Wait, no, if the lines are parallel, and the transversal cuts them, alternate interior angles are equal. Wait, maybe I made a mistake. Wait, let's re - examine. The two parallel lines are \( FJ \) (with points \( F, J, G \)) and \( HK \) (with points \( H, K, I \)). The transversal is \( ED \) (with points \( E, K, J, D \)). So \( \angle GJK \) and \( \angle IKH \) are alternate interior angles, so they should be equal. So \( 2x + 92 = 3x + 63 \). Solving for \( x \):

\( 2x+92 = 3x + 63\)

\(92 - 63=3x - 2x\)

\(x = 29\)

Now, substitute \( x = 29 \) into the expression for \( \angle GJK \), which is \( 2x + 92 \):

\(2\times29+92=58 + 92=150\)? Wait, that can't be right. Wait, maybe they are supplementary? Wait, maybe I got the angle labels wrong. Wait, if the lines are parallel, and the angles are same - side interior angles, they should be supplementary. Let's check: \( (2x + 92)+(3x + 63)=180 \)

So \( 5x+155 = 180\)

\(5x=180 - 155 = 25\)

\(x = 5\)

Then \( 2x + 92=2\times5 + 92 = 10 + 92 = 102\), \( 3x + 63=15 + 63 = 78\), and \( 102+78 = 180\), which is supplementary. Ah, I see, I misidentified the angle relationship. The two angles \( \angle GJK=(2x + 92)^\circ \) and \( \angle IKH=(3x + 63)^\circ \) are same - side interior angles, so they are supplementary (since the lines are parallel). So the correct equation is \( (2x + 92)+(3x + 63)=180 \)

Step2 (corrected): Solve for \( x \) with supplementary angles

\(2x + 92+3x + 63 = 180\)

Combine like terms: \(5x+155 = 180\)

Subtract 155 from both sides: \(5x=180 - 155 = 25\)

Divide by 5: \(x = 5\)

Step3 (corrected): Find \( m\angle GJK \)

Substitute \( x = 5 \) into \( 2x + 92 \):

\(2\times5+92=10 + 92 = 102\)

Wait, but let's check the diagram again. The lines \( FJ \) and \( HK \) are parallel (both vertical with arrows), and \( ED \) is a transversal. So \( \angle GJK \) and \( \angle HKE \) (or \( \angle IKH \)): if \( FJ \parallel HK \), then same - side interior angles are supplementary. So the sum of \( \angle GJK \) and \( \angle IKH \) is \( 180^\circ \). So the equation \( (2x + 92)+(3x + 63)=180 \) is correct.

So \( x = 5 \), then \( m\angle GJK=2x + 92=2\times5+92 = 102^\circ \)

Wait, but let's re - do the equation:

\(2x+92 + 3x+63=180\)

\(5x+155 = 180\)

\(5x=25\)

\(x = 5\)

Then \( 2x + 92=10 + 92 = 102\), and \( 3x + 63=15 + 63 = 78\), and \( 102 + 78 = 180\), which works. So the measure of \( \angle GJK \) is \( 102^\circ \)

Answer:

\(102\)