Question

find ( mangle t ) in parallelogram ( stuv ).( s ) angle is ( 2z + 9^circ ), ( v ) angle is ( z ). ( mangle t = square^circ )

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary, and opposite angles are equal. Also, angle \( S \) and angle \( V \) are not consecutive, wait, actually in parallelogram \( STUV \), angle \( S \) and angle \( T \) are consecutive? Wait no, let's see the vertices: \( S, T, U, V \) in order. So \( ST \parallel UV \) and \( SV \parallel TU \). So angle \( S \) and angle \( T \) are consecutive? Wait no, angle at \( S \) and angle at \( V \): wait, the given angle at \( S \) is \( 2z + 9 \) degrees, and angle at \( V \) is \( z \) degrees. Wait, in a parallelogram, consecutive angles are supplementary, but opposite angles are equal? Wait no, actually, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, angle \( S \) and angle \( U \) are opposite? No, let's list the angles: angle at \( S \), angle at \( T \), angle at \( U \), angle at \( V \). So \( \angle S = \angle U \), \( \angle T = \angle V \)? Wait no, that's not right. Wait, in a parallelogram, opposite angles are equal. So \( \angle S = \angle U \), \( \angle T = \angle V \)? Wait no, let's think again. If \( STUV \) is a parallelogram, then \( ST \parallel UV \) and \( SV \parallel TU \). So the consecutive angles: \( \angle S \) and \( \angle T \) are consecutive (since \( S \) to \( T \) to \( U \) to \( V \) to \( S \)), so \( \angle S + \angle T = 180^\circ \)? Wait no, no: in a parallelogram, consecutive angles (adjacent angles) are supplementary. Wait, but also, \( \angle S \) and \( \angle V \): are they opposite? Wait, the vertices are \( S, T, U, V \), so the sides are \( ST, TU, UV, VS \). So the angles: \( \angle S \) is between \( VS \) and \( ST \), \( \angle T \) is between \( ST \) and \( TU \), \( \angle U \) is between \( TU \) and \( UV \), \( \angle V \) is between \( UV \) and \( VS \). So \( \angle S \) and \( \angle U \) are opposite, \( \angle T \) and \( \angle V \) are opposite? Wait no, that can't be. Wait, no, in a parallelogram, opposite angles are equal. So \( \angle S = \angle U \), \( \angle T = \angle V \). Wait, but the given angles are \( \angle S = 2z + 9 \) and \( \angle V = z \). Wait, that would mean \( \angle S \) and \( \angle V \) are not opposite. Wait, maybe I got the vertices wrong. Wait, the diagram: \( S \) is at the bottom left, \( T \) at the top, \( U \) at the top right, \( V \) at the bottom right? Wait, no, the diagram shows \( S \), \( T \), \( U \), \( V \) with \( S \) connected to \( T \) and \( V \), \( T \) connected to \( U \), \( U \) connected to \( V \). So \( STUV \) is a parallelogram, so \( ST \parallel UV \) and \( SV \parallel TU \). Therefore, \( \angle S \) and \( \angle T \) are consecutive (since \( ST \) is a side, so \( \angle S \) and \( \angle T \) are adjacent), so they should be supplementary? Wait no, consecutive angles in a parallelogram are supplementary. Wait, but also, \( \angle S \) and \( \angle V \): are they opposite? Wait, no, \( \angle S \) and \( \angle U \) are opposite, \( \angle T \) and \( \angle V \) are opposite. Wait, but the problem gives \( \angle S = 2z + 9 \) and \( \angle V = z \). Wait, maybe \( \angle S \) and \( \angle V \) are consecutive? Wait, no, let's look at the sides: \( S \) is connected to \( T \) and \( V \), \( T \) is connected to \( U \), \( U \) is connected to \( V \). So the sides are \( ST, TU, UV, VS \). So the angles: \( \angle S \) (between \( VS \) and \( ST \)), \( \angle T \) (between \( ST \) and \( TU \)), \( \angle U \) (between \( TU \) and \( UV \)), \( \angle V \) (between \( UV \) and \( VS \)). So \( \angle S \) and \( \angle T \) are adjacent (consecutive), \( \angle T \) and \( \angle U \) are adjacent, \( \angle U \) and \( \angle V \) are adjacent, \( \angle V \) and \( \angle S \) are adjacent? Wait, no, \( \angle V \) is between \( UV \) and \( VS \), and \( \angle S \) is between \( VS \) and \( ST \), so \( \angle V \) and \( \angle S \) are adjacent (consecutive). Therefore, consecutive angles are supplementary. Wait, but in a parallelogram, consecutive angles are supplementary, so \( \angle S + \angle T = 180^\circ \), \( \angle T + \angle U = 180^\circ \), etc. But also, opposite angles are equal: \( \angle S = \angle U \), \( \angle T = \angle V \). Wait, but the problem has \( \angle S = 2z + 9 \) and \( \angle V = z \). Wait, maybe \( \angle S \) and \( \angle V \) are not opposite, but \( \angle T \) and \( \angle V \) are equal? Wait, no, that would mean \( \angle T = \angle V = z \), and \( \angle S \) and \( \angle T \) are supplementary? Wait, no, let's re-express. Wait, maybe I made a mistake. Let's recall: in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So if \( STUV \) is a parallelogram, then \( \angle S = \angle U \) and \( \angle T = \angle V \), and \( \angle S + \angle T = 180^\circ \), \( \angle T + \angle U = 180^\circ \), etc. Wait, but the given angles are \( \angle S = 2z + 9 \) and \( \angle V = z \). Wait, \( \angle V \) is equal to \( \angle T \) (opposite angles), so \( \angle T = \angle V = z \). And \( \angle S \) and \( \angle T \) are consecutive, so they should be supplementary: \( \angle S + \angle T = 180^\circ \). So \( (2z + 9) + z = 180 \). Let's solve that.

Step2: Solve for \( z \)

So \( 2z + 9 + z = 180 \)
Combine like terms: \( 3z + 9 = 180 \)
Subtract 9 from both sides: \( 3z = 180 - 9 = 171 \)
Divide both sides by 3: \( z = \frac{171}{3} = 57 \)

Step3: Find \( m\angle T \)

Since \( \angle T = \angle V = z \) (opposite angles in parallelogram), so \( m\angle T = z = 57^\circ \)? Wait, no, wait: wait, \( \angle V \) is \( z \), and \( \angle T \) is equal to \( \angle V \)? Wait, no, maybe I got the opposite angles wrong. Wait, let's look at the diagram again. The angle at \( S \) is \( 2z + 9 \), angle at \( V \) is \( z \). So \( S \) and \( V \) are connected by a side, so \( \angle S \) and \( \angle V \) are consecutive? Wait, no, in the parallelogram, \( S \) to \( T \) to \( U \) to \( V \) to \( S \). So \( \angle S \) is between \( S V \) and \( S T \), \( \angle T \) is between \( S T \) and \( T U \), \( \angle U \) is between \( T U \) and \( U V \), \( \angle V \) is between \( U V \) and \( V S \). So \( \angle S \) and \( \angle T \) are consecutive (adjacent), \( \angle T \) and \( \angle U \) are consecutive, \( \angle U \) and \( \angle V \) are consecutive, \( \angle V \) and \( \angle S \) are consecutive. Wait, that would mean all consecutive angles are supplementary, so \( \angle S + \angle T = 180 \), \( \angle T + \angle U = 180 \), \( \angle U + \angle V = 180 \), \( \angle V + \angle S = 180 \). But also, opposite angles: \( \angle S = \angle U \), \( \angle T = \angle V \). So \( \angle S = \angle U \), \( \angle T = \angle V \). So if \( \angle S = 2z + 9 \), \( \angle V = z \), then \( \angle T = \angle V = z \), and \( \angle S + \angle T = 180 \), so \( 2z + 9 + z = 180 \), which we solved as \( z = 57 \). Then \( \angle T = z = 57 \)? Wait, but let's check: \( \angle S = 2*57 + 9 = 114 + 9 = 123 \) degrees. Then \( \angle S + \angle T = 123 + 57 = 180 \), which is supplementary, so that works. So \( \angle T = 57 \) degrees.

Answer:

\( 57 \)