Question

if m∠iut = 42°, m∠iuv = (51x - 1)°, and m∠vut = (-1 + 72x)°, find the angle measures

Explanation:

Step1: Analyze angle relationship

From the diagram, \( \angle VUT = \angle VIU + \angle IUT \)? Wait, no, looking at the angles, \( \angle VUT \) is the sum of \( \angle VIU \) (wait, the given angles: \( m\angle IUT = 42^\circ \), \( m\angle IUV=(51x - 1)^\circ \), and \( m\angle VUT=(-1 + 72x)^\circ \). So actually, \( \angle VUT=\angle IUV+\angle IUT \). So we can set up the equation:
\( -1 + 72x=(51x - 1)+42 \)

Step2: Solve for x

Simplify the right - hand side: \( (51x - 1)+42 = 51x+41 \)
So the equation is \( -1 + 72x=51x + 41 \)
Subtract \( 51x \) from both sides: \( -1+72x - 51x=51x + 41-51x \)
\( 21x-1 = 41 \)
Add 1 to both sides: \( 21x-1 + 1=41 + 1 \)
\( 21x=42 \)
Divide both sides by 21: \( x=\frac{42}{21}=2 \)

Step3: Find \( m\angle IUV \)

Substitute \( x = 2 \) into \( m\angle IUV=(51x - 1)^\circ \)
\( m\angle IUV=51\times2-1=102 - 1 = 101^\circ \)

Step4: Find \( m\angle VUT \)

Substitute \( x = 2 \) into \( m\angle VUT=(-1 + 72x)^\circ \)
\( m\angle VUT=-1+72\times2=-1 + 144 = 143^\circ \) (We can also check by adding \( \angle IUV \) and \( \angle IUT \): \( 101 + 42=143^\circ \), which matches)

Answer:

\( x = 2 \)
\( m\angle IUV = 101^\circ \)
\( m\angle VUT = 143^\circ \)