Question

problem 16: given ℓ || m, find the measure of ∠ptu. (first taught in lesson 24) after you enter your answer press go. m∠ptu =

Explanation:

Step1: Identify parallel lines and angles

Since \( \ell \parallel m \), the corresponding angles \( 3x^\circ \) and \( (5x - 32)^\circ \) are equal (corresponding angles postulate for parallel lines cut by a transversal). So we set up the equation:
\( 3x = 5x - 32 \)

Step2: Solve for \( x \)

Subtract \( 3x \) from both sides:
\( 0 = 2x - 32 \)
Add 32 to both sides:
\( 2x = 32 \)
Divide both sides by 2:
\( x = 16 \)

Step3: Find \( \angle PTU \)

First, find the measure of the angle related to \( \angle PTU \). The angle at \( Q \) is \( 89^\circ \), and since \( \ell \parallel m \), the angle adjacent to \( \angle PTU \) and the \( 89^\circ \) angle are supplementary? Wait, no, let's re - examine. Wait, actually, the angle \( 3x^\circ \) when \( x = 16 \) is \( 3\times16=48^\circ \)? Wait, no, wait. Wait, the angle \( \angle PTU \): let's see, the line \( QT \) is perpendicular? Wait, no, the angle at \( Q \) is \( 89^\circ \), but actually, since \( \ell \parallel m \), and the transversal \( NT \), the alternate interior angles: \( 3x \) and the angle at \( T \) (the one with \( 5x - 32 \)) are equal. Wait, we found \( x = 16 \), so \( 3x=48^\circ \). Now, what about \( \angle PTU \)? Wait, the line \( TU \) is parallel to \( PQ \) (since \( \ell \parallel m \)), and \( NT \) is a transversal. Wait, actually, \( \angle PTU \) and the angle \( 3x^\circ \): wait, no, let's look at the diagram again. The angle at \( Q \) is \( 89^\circ \), but maybe the vertical angle or something? Wait, no, let's correct. When we solved \( 3x=5x - 32 \), we get \( x = 16 \), so \( 3x = 48^\circ \). Now, \( \angle PTU \): since \( \ell \parallel m \), and the line \( QT \) is a transversal (perpendicular? Wait, no, the angle at \( Q \) is \( 89^\circ \), but maybe \( \angle PTU \) is equal to \( 3x^\circ \)? Wait, no, let's think again. Wait, the angle \( \angle PTU \): the lines \( \ell \) and \( m \) are parallel, \( PQ \) and \( TU \) are parts of \( \ell \) and \( m \) respectively, so \( PQ\parallel TU \). The transversal \( NT \) creates alternate interior angles \( \angle NPT \) (which is \( 3x^\circ \)) and \( \angle PTU \). Wait, no, \( \angle NPT \) and \( \angle PTU \): if \( PQ\parallel TU \), then alternate interior angles are equal. Wait, but also, the angle at \( Q \) is \( 89^\circ \), but maybe that's a distractor? Wait, no, let's check the calculation again. We had \( 3x=5x - 32 \), solving gives \( x = 16 \), so \( 3x = 48 \). Now, \( \angle PTU \): since \( \ell \parallel m \), and the transversal \( NT \), the angle \( \angle PTU \) is equal to \( 3x^\circ \), which is \( 48^\circ \). Also, let's check the other angle: \( 5x-32=5\times16 - 32=80 - 32 = 48^\circ \), which matches. Then, what about the \( 89^\circ \) angle? Maybe that's a right angle - adjacent angle? Wait, no, maybe the line \( QT \) is perpendicular to \( \ell \) and \( m \), but it's \( 89^\circ \), which is almost a right angle, but maybe that's a typo or a distractor. But according to the parallel lines and transversal, the alternate interior angles \( 3x \) and \( 5x - 32 \) are equal, so we solved for \( x \), and then \( \angle PTU \) is equal to \( 3x^\circ \) (or \( 5x - 32^\circ \)) which is \( 48^\circ \).

Answer:

48°