Question

find the value of the variables in the parallelogram shown below.
3x° (4x + 5)°
y°
x =

y =

Explanation:

Step1: Use property of parallelogram (adjacent angles supplementary)

In a parallelogram, adjacent angles are supplementary, so \( 3x + (4x + 5) = 180 \).
Simplify: \( 7x + 5 = 180 \).

Step2: Solve for \( x \)

Subtract 5 from both sides: \( 7x = 180 - 5 = 175 \).
Divide by 7: \( x = \frac{175}{7} = 25 \).

Step3: Find \( 3x \) to get \( y \) (opposite angles equal)

Calculate \( 3x = 3\times25 = 75 \).
In a parallelogram, opposite angles are equal, and also adjacent angles: \( y \) is supplementary to \( 3x \)? Wait, no—wait, \( y \) and \( (4x + 5) \) are opposite? Wait, no, looking at the diagram: \( 3x \) and \( y \) are adjacent? Wait, no, in the parallelogram, consecutive angles (adjacent) are supplementary, and opposite angles are equal. Wait, the angle \( 3x \) and \( (4x + 5) \) are adjacent (so supplementary), and \( y \) should be equal to \( (4x + 5) \) (opposite angles), or \( 3x \) and \( y \) are opposite? Wait, let's re - check.

Wait, in a parallelogram, opposite angles are equal, and adjacent angles are supplementary. So if one angle is \( 3x \), its adjacent angle is \( (4x + 5) \), so they add to 180. Then the angle opposite to \( 3x \) is \( y \)? Wait, no, maybe the diagram: the top left is \( 3x \), top right is \( (4x + 5) \), bottom left is \( y \). So in a parallelogram, top left and bottom right are equal, top right and bottom left are equal. Also, top left and top right are adjacent, so supplementary.

So \( 3x+(4x + 5)=180 \), we found \( x = 25 \), so \( 4x+5=4\times25 + 5=105 \). Then \( y \), which is bottom left, should be equal to top right (opposite angles), so \( y=4x + 5=105 \). Alternatively, since \( 3x = 75 \), and \( y \) is adjacent to \( 3x \)? Wait, no, if top left is \( 3x \), bottom left is \( y \), then they are adjacent? No, in a parallelogram, the sides are parallel, so the left side is common, so top left and bottom left are adjacent? Wait, no, the angles at the same vertex: no, in a parallelogram, angle at top left, top right, bottom right, bottom left. So top left and top right are adjacent (share the top side), top right and bottom right are adjacent (share the right side), bottom right and bottom left are adjacent (share the bottom side), bottom left and top left are adjacent (share the left side). Wait, so actually, all consecutive angles are adjacent and supplementary. So \( 3x \) (top left) and \( (4x + 5) \) (top right) are adjacent (supplementary), \( (4x + 5) \) (top right) and bottom right are adjacent (supplementary), bottom right and \( y \) (bottom left) are adjacent (supplementary), \( y \) (bottom left) and \( 3x \) (top left) are adjacent (supplementary). Wait, that can't be. Wait, no, opposite angles are equal. So top left (\( 3x \)) and bottom right are equal, top right (\( 4x + 5 \)) and bottom left (\( y \)) are equal. And adjacent angles (like top left and top right) are supplementary.

So we have \( 3x+(4x + 5)=180 \), solved \( x = 25 \), so \( 4x + 5=105 \), so \( y \) (bottom left) is equal to top right (\( 4x + 5 \)) because they are opposite angles. So \( y = 105 \).

Answer:

\( x = 25 \), \( y = 105 \)