Question

find the measurement indicated in each parallelogram.

1. find sr
2. find m∠v
3. find m∠y
4. find cd

find the measure indicated.

1. find fe
2. find ed

Explanation:

Problem 1: Find \( SR \)

Step 1: Recall parallelogram property

In a parallelogram, opposite sides are equal. So \( RQ = ST \) and \( RS = QT \). Wait, actually, looking at the diagram, \( RS = 3x - 3 \) and \( QT = x + 7 \)? Wait, no, in parallelogram \( RQST \), \( RS \) and \( QT \) are opposite sides? Wait, maybe \( RS \) and \( QT \) are opposite? Wait, the sides: \( RQ \) and \( ST \) are opposite, \( RS \) and \( QT \) are opposite. Wait, the labels: \( R, Q, T, S \) – so \( RQ \) is adjacent to \( QT \), \( QT \) is adjacent to \( TS \), \( TS \) is adjacent to \( SR \), \( SR \) is adjacent to \( RQ \). So in a parallelogram, opposite sides are equal. So \( RS = QT \) and \( RQ = ST \)? Wait, the given sides: \( RS = 3x - 3 \), \( QT = x + 7 \). Wait, no, maybe \( RQ \) and \( ST \) are equal, and \( RS \) and \( QT \) are equal? Wait, the problem is to find \( SR \), so \( SR = 3x - 3 \), and \( QT = x + 7 \). Since in a parallelogram, opposite sides are equal, so \( RS = QT \)? Wait, no, maybe \( RQ \) and \( ST \) are equal, and \( RS \) and \( QT \) are equal. Wait, the diagram: \( R \) connected to \( Q \), \( Q \) connected to \( T \), \( T \) connected to \( S \), \( S \) connected to \( R \). So \( RQ \) and \( ST \) are opposite, \( RS \) and \( QT \) are opposite. So \( RS = QT \), so \( 3x - 3 = x + 7 \).

Step 2: Solve for \( x \)

\( 3x - 3 = x + 7 \)
Subtract \( x \) from both sides: \( 2x - 3 = 7 \)
Add 3 to both sides: \( 2x = 10 \)
Divide by 2: \( x = 5 \)

Step 3: Find \( SR \)

Substitute \( x = 5 \) into \( SR = 3x - 3 \):
\( SR = 3(5) - 3 = 15 - 3 = 12 \)

Step 1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary? Wait, no, opposite angles are equal, and consecutive angles are supplementary. Wait, the diagram: \( WXYZ \) (wait, the labels are \( W, X, V, U \)? Wait, the parallelogram is \( W, X, U, V \). So \( \angle W \) and \( \angle V \) – wait, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, the sides: \( WX \) and \( UV \), \( WU \) and \( XV \). Wait, the angles: \( \angle W \) and \( \angle V \) – are they opposite or consecutive? Wait, the given expressions: \( \angle W = 25x + 1 \), \( \angle U = 24x + 3 \). Wait, in a parallelogram, consecutive angles are supplementary? No, opposite angles are equal, and consecutive angles are supplementary. Wait, maybe \( \angle W \) and \( \angle U \) are consecutive? Wait, no, the problem is to find \( m\angle V \). Wait, maybe the parallelogram has \( \angle W \) and \( \angle V \) as opposite? Wait, no, let's check the sides. Wait, the user wrote some work: \( 2x - 2 = x + 7 \), \( -x + 2 -x + 2 \), \( x = 9 \) – maybe that's for another problem. Wait, let's re-examine. The parallelogram is \( W, X, U, V \), with \( \angle W = 25x + 1 \), and side \( WU = 24x + 3 \), \( WV \) – no, wait, maybe it's a parallelogram with \( \angle W \) and \( \angle U \) as angles? Wait, no, the problem is to find \( m\angle V \). Wait, maybe in a parallelogram, opposite angles are equal, so \( \angle W = \angle U \)? No, that doesn't make sense. Wait, maybe it's a parallelogram with \( \angle W \) and \( \angle V \) as consecutive angles? Wait, no, let's recall: in a parallelogram, opposite angles are equal, so \( \angle W = \angle V \) and \( \angle X = \angle U \)? Wait, no, that's not right. Wait, maybe the problem is that the sides are \( WX = 24x + 3 \) and \( WV = 25x + 1 \)? No, the labels are \( W, X, U, V \), so \( W \) to \( X \) to \( U \) to \( V \) to \( W \). So \( \angle W \) is at \( W \), between \( WX \) and \( WV \); \( \angle V \) is at \( V \), between \( VU \) and \( VW \). In a parallelogram, \( \angle W = \angle V \) (opposite angles) or \( \angle W + \angle V = 180^\circ \) (consecutive)? Wait, no, opposite angles are equal, consecutive are supplementary. Wait, maybe the given angles are \( \angle W = 25x + 1 \) and \( \angle U = 24x + 3 \), and we need to find \( \angle V \). Wait, maybe the user's work is for this problem: \( 25x + 1 = 24x + 3 \) (since in a parallelogram, opposite angles are equal? No, that's not correct. Wait, maybe it's a rhombus? No, the problem says parallelogram. Wait, maybe the angles at \( W \) and \( U \) are equal? Wait, no, let's solve \( 25x + 1 = 24x + 3 \):
\( 25x + 1 = 24x + 3 \)
Subtract \( 24x \): \( x + 1 = 3 \)
Subtract 1: \( x = 2 \)

Then \( \angle W = 25(2) + 1 = 51^\circ \). Since \( \angle W \) and \( \angle V \) are consecutive angles, they are supplementary? Wait, no, in a parallelogram, consecutive angles are supplementary. Wait, \( \angle W + \angle V = 180^\circ \)? No, opposite angles are equal. Wait, maybe \( \angle W \) and \( \angle V \) are opposite, so \( \angle V = \angle W = 51^\circ \)? No, that can't be. Wait, maybe I made a mistake. Let's check the user's work: they wrote \( 2x - 2 = x + 7 \), \( -x + 2 -x + 2 \), \( x = 9 \) – maybe that's for problem 4. Let's move to problem 3.

Problem 3: Find \( m\angle Y \)

Step 1: Recall rectangle property (since it's a rectangle, all angles are 90°, but wait, the diagram is a rectangle? \( Z, W, X, Y \) – with sides \( ZW = 4 + 16x \), \( YX = 15x + 10 \). In a rectangle, opposite sides are equal, so \( ZW = YX \).

Step 2: Set up equation

\( 4 + 16x = 15x + 10 \)
Subtract \( 15x \): \( 4 + x = 10 \)
Subtract 4: \( x = 6 \)

Step 3: Find angle \( Y \)

In a rectangle, all angles are \( 90^\circ \), so \( m\angle Y = 90^\circ \). Wait, but maybe it's a parallelogram? If it's a rectangle, then yes, all angles are 90°. So regardless of \( x \), \( \angle Y = 90^\circ \). Wait, but let's confirm. The sides: \( ZW = 4 + 16(6) = 4 + 96 = 100 \), \( YX = 15(6) + 10 = 90 + 10 = 100 \), so it's a rectangle. Thus, \( m\angle Y = 90^\circ \).

Answer:

\( 12 \)

Problem 2: Find \( m\angle V \)