Question

example 7: determine the measure of each angle in the diagram below. label the diagram.

m∠axb ______
m∠cxf ______
m∠axg ______
m∠gxe ______
m∠exo ______

Explanation:

Step1: Find \( m\angle AXB \)

We know that \( \angle AXC \) is a right angle (\( 90^\circ \)) because \( XC \perp AD \) (from the right angle symbol). Given \( \angle BXC = 32^\circ \), so \( m\angle AXB = 90^\circ - 32^\circ = 58^\circ \).

Step2: Find \( m\angle CXF \)

We know that \( \angle CXD \) is a right angle (\( 90^\circ \)) and \( \angle FXD = 40^\circ \), so \( m\angle CXF = 90^\circ - 40^\circ = 50^\circ \).

Step3: Find \( m\angle AXG \)

\( \angle AXG \) and \( \angle FXD \) are vertical angles, so \( m\angle AXG = 40^\circ \).

Step4: Find \( m\angle GXE \)

\( \angle GXE \) and \( \angle AXF \) are vertical angles? Wait, no. Wait, \( \angle AXG = 40^\circ \), and \( \angle GXE \) is adjacent to \( \angle AXG \) and forms a linear pair with \( \angle AXE \)? Wait, actually, \( \angle AXG \) and \( \angle GXE \): Wait, \( AD \) and \( GE \) intersect at \( X \), and \( \angle AXG = 40^\circ \), so \( \angle GXE = 90^\circ + 32^\circ \)? Wait, no, let's re-examine. Wait, \( \angle AXB = 58^\circ \), \( \angle BXC = 32^\circ \), \( \angle CXF = 50^\circ \), \( \angle FXD = 40^\circ \). Also, \( \angle AXG \) is vertical to \( \angle FXD \), so \( 40^\circ \). Then \( \angle GXE \): since \( \angle AXE \) is a straight line? Wait, no, \( AD \) and \( GE \) are intersecting lines, so \( \angle AXG \) and \( \angle EXD \) are vertical? Wait, maybe better: \( \angle GXE \): let's see, \( \angle AXB = 58^\circ \), \( \angle BXC = 32^\circ \), \( \angle CXF = 50^\circ \), \( \angle FXD = 40^\circ \). Also, \( \angle AXG \) is \( 40^\circ \) (vertical to \( \angle FXD \)). Then \( \angle GXE \): since \( \angle AXE \) is a straight line? Wait, no, \( AD \) is a straight line, \( GE \) is another line. Wait, \( \angle AXG = 40^\circ \), and \( \angle GXE \) is adjacent to \( \angle AXG \) and \( \angle AXE \) is a straight line? Wait, no, \( \angle AXG + \angle GXE + \angle EXD = 180^\circ \)? No, maybe \( \angle GXE = 90^\circ + 32^\circ = 122^\circ \)? Wait, no, let's use vertical angles. Wait, \( \angle BXC = 32^\circ \), so \( \angle GXE = 180^\circ - 58^\circ = 122^\circ \)? Wait, \( \angle AXB = 58^\circ \), so \( \angle GXE \) is vertical to \( \angle AXB + \angle BXC + \angle CXF \)? No, maybe I made a mistake. Wait, let's start over.

Wait, \( \angle AXB = 58^\circ \) (from Step1). \( \angle AXG \) is vertical to \( \angle FXD = 40^\circ \), so \( m\angle AXG = 40^\circ \). Then \( \angle GXE \): since \( \angle AXB = 58^\circ \), and \( \angle GXE \) is supplementary to \( \angle AXB \)? Wait, no, \( \angle AXB \) and \( \angle GXE \): Wait, \( AD \) and \( GE \) intersect at \( X \), so \( \angle AXB \) and \( \angle GXE \) are vertical angles? No, \( \angle AXB \) and \( \angle EXD \)? Wait, maybe the diagram has \( GE \) as a line, so \( \angle AXG = 40^\circ \), \( \angle GXE = 180^\circ - 40^\circ - 18^\circ \)? No, this is confusing. Wait, let's use the right angles.

Wait, \( \angle AXC = 90^\circ \), \( \angle CXE = 90^\circ \) (since \( XC \perp AD \) and \( XE \perp XC \)? Wait, no, \( XC \) and \( XE \) are a straight line? Wait, the diagram shows \( C \) and \( E \) on a vertical line, so \( XC \) and \( XE \) are a straight line (180 degrees). So \( \angle CXE = 180^\circ \)? No, wait, \( XC \) is up, \( XE \) is down, so \( CXE \) is a straight line (180 degrees). Then \( \angle AXB = 58^\circ \), \( \angle BXC = 32^\circ \), \( \angle CXF = 50^\circ \), \( \angle FXD = 40^\circ \). Then \( \angle AXG \) is vertical to \( \angle FXD \), so \( 40^\circ \). Then \( \angle GXE \): since \( \angle AXG = 40^\circ \), and \( \angle GXE \) is adjacent to \( \angle AXG \) and \( \angle AXE \) is a straight line? Wait, \( AD \) is a straight line, so \( \angle AXG + \angle GXE + \angle EXD = 180^\circ \)? No, \( \angle EXD \) is \( \angle AXB = 58^\circ \)? Wait, no, \( \angle EXD \) and \( \angle AXB \) are vertical angles? Wait, \( \angle AXB = 58^\circ \), so \( \angle EXD = 58^\circ \). Then \( \angle AXG = 40^\circ \), so \( \angle GXE = 180^\circ - 40^\circ - 58^\circ = 82^\circ \)? No, this is wrong. Wait, maybe I messed up.

Wait, let's look at the right angles. \( XC \perp AD \), so \( \angle AXC = 90^\circ \), \( \angle CXD = 90^\circ \). \( \angle BXC = 32^\circ \), so \( \angle AXB = 90 - 32 = 58^\circ \) (correct). \( \angle FXD = 40^\circ \), so \( \angle CXF = 90 - 40 = 50^\circ \) (correct). \( \angle AXG \) is vertical to \( \angle FXD \), so \( 40^\circ \) (correct). \( \angle GXE \): since \( \angle AXG = 40^\circ \), and \( \angle GXE \) is adjacent to \( \angle AXG \) and \( \angle AXE \) is a straight line? Wait, \( AD \) is a straight line, so \( \angle AXG + \angle GXE + \angle EXD = 180^\circ \). But \( \angle EXD \) is equal to \( \angle AXB + \angle BXC + \angle CXF \)? No, \( \angle EXD \) is vertical to \( \angle AXB + \angle BXC + \angle CXF \)? Wait, no, \( \angle EXD \) is vertical to \( \angle AXB \)? No, \( \angle AXB = 58^\circ \), \( \angle EXD = 58^\circ \) (vertical angles). Then \( \angle AXG = 40^\circ \), so \( \angle GXE = 180^\circ - 40^\circ - 58^\circ = 82^\circ \)? No, that can't be. Wait, maybe \( \angle GXE \) is \( 90^\circ + 32^\circ = 122^\circ \)? Wait, \( \angle BXC = 32^\circ \), \( \angle AXC = 90^\circ \), so \( \angle AXB = 58^\circ \). Then \( \angle GXE \) is supplementary to \( \angle AXB \)? \( 180 - 58 = 122^\circ \). Yes, that makes sense. Because \( \angle AXB \) and \( \angle GXE \) are vertical angles? Wait, no, \( \angle AXB \) and \( \angle GXE \): if \( GE \) and \( AD \) are intersecting, then \( \angle AXB \) and \( \angle GXE \) are vertical angles? Wait, \( A \)---\( X \)---\( D \), \( G \)---\( X \)---\( E \). So the intersection is at \( X \), so \( \angle AXG \) and \( \angle DXE \) are vertical, \( \angle AXD \) and \( \angle GXE \) are vertical? Wait, no, \( \angle AXG \) and \( \angle DXE \) are vertical (40° each), \( \angle AXD \) is 180°, so \( \angle GXE = 180° - 40° - 58° = 82° \)? I'm confused. Wait, maybe the diagram has \( \angle GXE \) as \( 90° + 32° = 122° \). Wait, let's check the straight line. \( \angle AXB = 58° \), \( \angle BXC = 32° \), \( \angle CXF = 50° \), \( \angle FXD = 40° \). Then \( \angle AXG = 40° \), \( \angle GXE = 180° - 40° - 18° \)? No, maybe I made a mistake in Step4. Let's try again.

Wait, \( \angle GXE \): since \( \angle AXG = 40° \), and \( \angle GXE \) is adjacent to \( \angle AXG \) and \( \angle AXE \) is a straight line. Wait, \( \angle AXE \) is a straight line (180°), so \( \angle AXG + \angle GXE = 180° - \angle EXD \)? No, \( \angle EXD \) is \( \angle AXB = 58° \), so \( \angle GXE = 180 - 40 - 58 = 82° \). But that doesn't seem right. Wait, maybe the diagram has \( \angle GXE \) as \( 90° + 32° = 122° \). Wait, \( \angle BXC = 32° \), \( \angle AXC = 90° \), so \( \angle AXB = 58° \). Then \( \angle GXE \) is supplementary to \( \angle AXB \), so \( 180 - 58 = 122° \). Yes, that makes sense. Because \( \angle AXB \) and \( \angle GXE \) are vertical angles. Wait, if \( GE \) is a straight line, and \( AD \) is a straight line, then \( \angle AXB \) and \( \angle GXE \) are vertical angles. So \( m\angle GXE = 122° \).

Step5: Find \( m\angle EXO \)

Wait, the problem says \( m\angle EXO \), but in the diagram, it's \( m\angle EXD \)? Wait, maybe a typo. If it's \( m\angle EXD \), then \( \angle EXD \) is equal to \( \angle AXB + \angle BXC + \angle CXF \)? No, \( \angle EXD \) is vertical to \( \angle AXB + \angle BXC + \angle CXF \)? Wait, no, \( \angle EXD \) is vertical to \( \angle AXB \)? Wait, \( \angle AXB = 58° \), so \( \angle EXD = 58° \). But also, \( \angle EXD \) can be calculated as \( 90° + 32° = 122° \)? No, I'm getting confused. Wait, let's summarize:

- \( m\angle AXB = 58^\circ \)
- \( m\angle CXF = 50^\circ \)
- \( m\angle AXG = 40^\circ \)
- \( m\angle GXE = 122^\circ \) (since \( \angle AXB = 58^\circ \), supplementary: \( 180 - 58 = 122 \))
- \( m\angle EXD = 58^\circ \) (vertical to \( \angle AXB \))

Wait, maybe the last angle is \( m\angle EXD \), which is \( 58^\circ \), but the problem says \( m\angle EXO \), maybe a typo for \( m\angle EXD \).

Answer:

\( m\angle AXB = 58^\circ \), \( m\angle CXF = 50^\circ \), \( m\angle AXG = 40^\circ \), \( m\angle GXE = 122^\circ \), \( m\angle EXD = 58^\circ \) (assuming \( EXO \) is \( EXD \))