Question

find the values of x and y.
(13x)°
38°
x =
y =

Explanation:

Step1: Find x using angle sum

The angle between \( BG \) and \( CG \) is \( 13x^\circ \), and between \( CG \) and \( GD \) is \( 38^\circ \). Since \( BG \perp GD \) (right angle), \( 13x + 38 = 90 \).

$$ 13x = 90 - 38 $$

$$ 13x = 52 $$

$$ x = \frac{52}{13} $$

$$ x = 4 $$

Step2: Find y using vertical angles

Angle \( y \) and the angle \( 38^\circ \) are vertical angles? Wait, no, \( y \) and the angle between \( CG \) and \( GD \)? Wait, actually, \( y \) is equal to \( 38^\circ \)? Wait, no, looking at the diagram, \( y \) and the angle \( 38^\circ \) are equal? Wait, no, the angle between \( FG \) and \( GE \) is \( y \), and the angle between \( CG \) and \( GD \) is \( 38^\circ \), which are vertical angles? Wait, no, \( y \) is equal to \( 38^\circ \)? Wait, no, let's recheck. Wait, the angle between \( BG \) and \( CG \) is \( 13x = 52^\circ \), and the angle between \( CG \) and \( GD \) is \( 38^\circ \), so \( 52 + 38 = 90 \), which matches the right angle. Then, \( y \) is equal to \( 38^\circ \) because they are vertical angles? Wait, no, \( y \) is the angle between \( FG \) and \( GE \), and \( 38^\circ \) is between \( CG \) and \( GD \), so they are vertical angles? Wait, yes, vertical angles are equal. So \( y = 38 \)? Wait, no, wait, \( y \) is adjacent to the right angle? Wait, no, the horizontal and vertical lines are perpendicular, so the angle between \( AG \) and \( BG \) is \( 90^\circ \), between \( BG \) and \( CG \) is \( 52^\circ \), between \( CG \) and \( GD \) is \( 38^\circ \), so \( 52 + 38 = 90 \), correct. Then, the angle \( y \) is equal to \( 38^\circ \) because they are vertical angles (since \( FG \) and \( CG \) are a straight line, and \( GE \) and \( BG \) are a straight line? Wait, no, \( y \) is the angle between \( FG \) and \( GE \), and \( 38^\circ \) is between \( CG \) and \( GD \), so they are vertical angles, so \( y = 38 \). Wait, but let's confirm. Alternatively, \( y \) and \( 38^\circ \) are equal because they are vertical angles. So \( y = 38 \).

Wait, no, wait, maybe I made a mistake. Wait, the angle between \( CG \) and \( GD \) is \( 38^\circ \), and \( y \) is the angle between \( FG \) and \( GE \). Since \( FG \) and \( CG \) are a straight line (so they are supplementary, but no, \( FG \) and \( CG \) are opposite rays, so \( \angle FGC \) is a straight angle, but \( y \) is between \( FG \) and \( GE \), and \( 38^\circ \) is between \( CG \) and \( GD \), so they are vertical angles, hence equal. So \( y = 38 \).

Answer:

\( x = 4 \), \( y = 38 \)