Question

ray uw is the angle bisector of ∠vut. if m∠vuw = (4x + 6)° and m∠wut = (6x - 10)°, what is the measure of ∠wut?
options: 32°, 38°, 48°, 76°

Explanation:

Step1: Recall angle bisector property

Since \( UW \) is the angle bisector of \( \angle VUT \), we have \( m\angle VUW = m\angle WUT \).
Given \( m\angle VUW=(4x + 6)^\circ \) and \( m\angle WUT=(6x - 10)^\circ \), so we set up the equation:
\( 4x + 6 = 6x - 10 \)

Step2: Solve for \( x \)

Subtract \( 4x \) from both sides:
\( 6 = 2x - 10 \)
Add 10 to both sides:
\( 16 = 2x \)
Divide both sides by 2:
\( x = 8 \)

Step3: Find \( m\angle WUT \)

Substitute \( x = 8 \) into \( m\angle WUT=(6x - 10)^\circ \):
\( m\angle WUT = 6(8)-10 = 48 - 10 = 38^\circ \)? Wait, no, wait. Wait, angle bisector means \( \angle VUT = 2\angle VUW = 2\angle WUT \). Wait, maybe I made a mistake. Wait, let's re - check.

Wait, the angle bisector divides \( \angle VUT \) into two equal angles, so \( \angle VUW=\angle WUT \). So \( 4x + 6=6x - 10 \). Solving:

\( 6 + 10=6x - 4x \)

\( 16 = 2x \)

\( x = 8 \)

Then \( m\angle VUW=4(8)+6 = 32 + 6=38^\circ \), and \( m\angle WUT=6(8)-10 = 48 - 10 = 38^\circ \). Wait, but maybe the question is asking for \( \angle VUT \)? Wait, no, the options are 32, 38, 48, 76. Wait, maybe I misread the problem. Wait, maybe \( \angle VUW=(4x + 6) \) and \( \angle WUT=(6x - 10) \), and since \( UW \) is the bisector, \( \angle VUW=\angle WUT \), so \( 4x + 6=6x - 10 \), \( x = 8 \). Then \( \angle VUT=\angle VUW+\angle WUT = 2\angle VUW=2\times(4\times8 + 6)=2\times(32 + 6)=2\times38 = 76^\circ \)? Wait, the options have 76. Let's check again.

Wait, if \( \angle VUW=(4x + 6) \) and \( \angle WUT=(6x - 10) \), and \( UW \) bisects \( \angle VUT \), so \( \angle VUW=\angle WUT \). So:

\( 4x+6 = 6x - 10 \)

\( 6 + 10=6x - 4x \)

\( 16 = 2x \)

\( x = 8 \)

Then \( \angle VUW=4\times8 + 6=38^\circ \), \( \angle WUT=6\times8 - 10 = 38^\circ \), so \( \angle VUT=38^\circ+38^\circ = 76^\circ \). So the measure of \( \angle WUT \) is \( 38^\circ \)? Wait, no, if the question is asking for \( \angle WUT \), then it's \( 38^\circ \), but if it's asking for \( \angle VUT \), it's \( 76^\circ \). Wait, the problem says "what is the measure of \( \angle WUT \)"? Wait, let's check the options. The options are 32, 38, 48, 76.

Wait, when \( x = 8 \), \( m\angle WUT=6x - 10=6\times8 - 10 = 48 - 10 = 38^\circ \). So the answer is \( 38^\circ \)? Wait, but let's re - check the equation.

Angle bisector: \( m\angle VUW=m\angle WUT \)

\( 4x + 6=6x - 10 \)

\( 6 + 10=6x - 4x \)

\( 16 = 2x \)

\( x = 8 \)

Then \( m\angle WUT=6(8)-10 = 38^\circ \). So the measure of \( \angle WUT \) is \( 38^\circ \).

Answer:

\( 38^\circ \) (corresponding to the option with \( 38^\circ \))