Find the value of x in each figure where two parallel lines are cut by a transversal.

Here are the step-by-step solutions to find the value of $x$ for each problem.

1)
The angle labeled $36^\circ$ and the angle labeled $x$ are alternate interior angles. When parallel lines are cut by a transversal, alternate interior angles are equal.
Therefore, $x = 36$.

2)
First, find the angle adjacent to the $131^\circ$ angle on the straight line. Angles on a straight line add up to $180^\circ$.
$180^\circ - 131^\circ = 49^\circ$.
This new $49^\circ$ angle and angle $x$ are alternate interior angles, so they are equal.
Therefore, $x = 49$.

3)
First, look at the top intersection. The angle vertically opposite to the $53^\circ$ angle is also $53^\circ$.
This $53^\circ$ angle and angle $x$ are consecutive interior angles (also called same-side interior angles). They are on the same side of the transversal and between the parallel lines. These angles add up to $180^\circ$.
$x + 53 = 180$
$x = 180 - 53$
$x = 127$

4)
First, find the angle adjacent to the $118^\circ$ angle on the straight line.
$180^\circ - 118^\circ = 62^\circ$.
This $62^\circ$ angle and angle $x$ are corresponding angles (they are in the same position at each intersection). Corresponding angles are equal.
Therefore, $x = 62$.

5)
Angle $x$ and the $55^\circ$ angle are consecutive interior angles. They add up to $180^\circ$.
$x + 55 = 180$
$x = 180 - 55$
$x = 125$

6)
First, find the angle adjacent to the $78^\circ$ angle on the straight line.
$180^\circ - 78^\circ = 102^\circ$.
This $102^\circ$ angle and angle $x$ are alternate interior angles. Alternate interior angles are equal.
Therefore, $x = 102$.

7)
Look at the triangle formed in the middle. The three angles inside any triangle must add up to $180^\circ$.
The angles are $151^\circ$? No, $151^\circ$ is an exterior angle. Let's look closer.
Actually, let's use parallel line rules directly.
The angle vertically opposite to the $29^\circ$ angle is inside the triangle. So one angle is $29^\circ$.
The angle supplementary to $151^\circ$ is $180^\circ - 151^\circ = 29^\circ$. This is another angle inside the "Z" shape or triangle context? Let's re-evaluate.
Let's use the property that the exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Alternatively, draw a line or use consecutive interior angles.
Let's look at the bottom left vertex. The angle inside the parallel lines is supplementary to $151^\circ$, which is $29^\circ$.
So we have a triangle with angles: $29^\circ$ (bottom left), $29^\circ$ (top right, vertical angle), and the third angle is adjacent to $x$.
Wait, simpler method:
The angle corresponding to the interior angle at the bottom left ($29^\circ$) is not helpful directly.
Let's use the "zig-zag" or triangle sum.
Inside the triangle formed by the transversal segments:
Top angle: Vertically opposite to $29^\circ$, so it is $29^\circ$.
Left angle: Supplementary to $151^\circ$, so $180 - 151 = 29^\circ$.
The third angle of this triangle is $180 - (29 + 29) = 180 - 58 = 122^\circ$.
Angle $x$ and this $122^\circ$ angle are supplementary (on a straight line)? No, $x$ is an exterior angle to the triangle at the bottom right vertex?
Looking at the diagram, $x$ and the interior angle of the triangle are on a straight line.
So, $x + 122 = 180 \rightarrow x = 58$.
Alternative Check: The exterior angle of a triangle equals the sum of the two opposite interior angles.
The exterior angle at the bottom right is vertically opposite to $x$? No.
Let's look at the relationship between the far left angle and far right angle.
Actually, there is a simpler rule for this "M" or "Sigma" shape pointing left: The sum of the angles pointing left equals the sum of the angles pointing right.
Angles pointing left: $151^\circ$ is obtuse, let's use the acute ones.
Let's stick to basic steps.
1. Angle adjacent to $151^\circ$ is $29^\circ$.
2. This $29^\circ$ angle and the top $29^\circ$ angle are alternate interior angles relative to a horizontal line? No.
Let's assume the standard triangle method.
Triangle vertices:
A (left): Interior angle is $180-151=29^\circ$.
B (top): Vertical angle to $29^\circ$ is $29^\circ$.
C (right): The angle inside the triangle is $y$.
$29 + 29 + y = 180 \Rightarrow y = 122^\circ$.
Angle $x$ and angle $y$ are supplementary on the straight line segment? The line goes through. Yes.
$x + 122 = 180 \Rightarrow x = 58$.

8)
The angle labeled $83^\circ$ and the angle labeled $97^\circ$ are on a straight line? $83+97=180$. Yes, they form a linear pair on the bottom parallel line.
Angle $x$ and the $97^\circ$ angle are consecutive interior angles? No.
Angle $x$ and the $83^\circ$ angle are alternate interior angles.
Therefore, $x = 83$.

9)
First, find the angle adjacent to $130^\circ$ on the straight vertical line.
$180^\circ - 130^\circ = 50^\circ$.
This $50^\circ$ angle and the $50^\circ$ angle at the bottom are alternate interior angles?
Let's check the positions.
The $50^\circ$ we just found is "inside" the parallel lines on the left.
The given $50^\circ$ is "inside" on the left? No, it's below the transversal.
Let's look at the triangle formed.
Or simpler:
The angle vertically opposite to the bottom $50^\circ$ is $50^\circ$ (inside the parallel strip).
The angle supplementary to $130^\circ$ is $50^\circ$ (inside the parallel strip).
These two $50^\circ$ angles are on the same side of the transversal?
Let's trace the transversal.
Top intersection: Angle inside is $50^\circ$.
Bottom intersection: Angle inside is $50^\circ$ (vertical to the labeled 50).
Since these alternate interior angles are equal ($50=50$), the lines are parallel (which we know).
Now, where is $x$?
$x$ is an exterior angle at the top right.
The angle vertically opposite to the interior $50^\circ$ (calculated from 130) is also $50^\circ$.
$x$ and this $50^\circ$ angle are supplementary? No, they are vertically opposite to... wait.
Let's look at the top intersection again.
We have a vertical line and a transversal.
Angle $130^\circ$ is bottom-left.
Angle vertically opposite is top-right. Let's call it $v$. $v = 130^\circ$.
Angle $x$ is adjacent to $v$ on the straight vertical line? Or on the transversal?
Usually, $x$ is marked as the angle between the transversal and the parallel line.
If $x$ is the angle shown in the top right quadrant between the vertical line and transversal:
The angle adjacent to $130^\circ$ on the straight vertical line is $50^\circ$ (top-left).
The angle vertically opposite to that $50^\circ$ is the bottom-right angle.
Angle $x$ is in the top-right position.
Angles on a straight line (the vertical one) add to 180.
The angle next to $x$ (top-left) is $50^\circ$.
So $x + 50 = 180 \Rightarrow x = 130$.
Let's double check.
Corresponding angle to $x$ would be the top-right angle at the bottom intersection.
At the bottom intersection, we have a $50^\circ$ angle labeled. It looks like it's in the bottom-left position relative to the intersection point? Or bottom-right?
The arc is between the downward vertical line and the transversal going down-left. That is the bottom-left angle.
So bottom-left is $50^\circ$.
Then top-right (corresponding to bottom-left? No, corresponding is same position).
Top-right corresponds to Bottom-right.
Bottom-left ($50^\circ$) and Top-right ($x$) are alternate exterior angles.
Alternate exterior angles are equal.
Therefore, $x = 50$?
Let's re-read the diagram carefully.
Top part: $130^\circ$ is the angle between the upward vertical and the transversal going down-left. That is the Top-Left angle.
So Top-Left = $130^\circ$.
$x$ is the angle between the upward vertical and the transversal going up-right. That is the Top-Right angle.
Top-Left and Top-Right are supplementary on the straight vertical line.
$x + 130 = 180$.
$x = 50$.

Let's verify with the bottom part.
Bottom part: $50^\circ$ is the angle between the downward vertical and transversal going down-left. That is the Bottom-Left angle.
If Top-Left is $130^\circ$, then Bottom-Left should be $180-130=50^\circ$ if they were consecutive interior? No.
Top-Left ($130$) and Bottom-Left ($50$) are consecutive interior? No, they are on the same side of the transversal but one is exterior?
Let's use Corresponding Angles.
Top-Left angle is $130^\circ$.
The corresponding angle at the bottom is the Bottom-Left angle.
So Bottom-Left should be $130^\circ$.
But the diagram labels Bottom-Left as $50^\circ$.
Contradiction?
Ah, look at the transversal direction.
Top: Transversal goes from Top-Right to Bottom-Left.
Bottom: Transversal goes from Top-Right to Bottom-Left.
Okay.
Top Intersection:
Angle marked $130^\circ$ is between the Vertical Line (Up) and Transversal (Down-Left). This is the Top-Left angle.
Angle marked $x$ is between the Vertical Line (Up) and Transversal (Up-Right). This is the Top-Right angle.
Since the vertical line is a straight line, Top-Left + Top-Right = $180^\circ$.
$130 + x = 180$.
$x = 50$.

Does this match the bottom info?
Bottom Intersection:
Angle marked $50^\circ$ is between Vertical Line (Down) and Transversal (Down-Left). This is the Bottom-Left angle.
Are the lines parallel?
If Top-Left is $130$, then Bottom-Left (Consecutive Interior? No. Same-Side Exterior? No.)
Let's check Alternate Interior.
Top-Right is $x=50$.
Bottom-Left is $50$.
These are Alternate Interior Angles?
Top-Right is "Interior"? No, it's above the parallel line? Wait, the arrows are on the vertical lines. The horizontal-ish lines are the transversals?
NO. The arrows are on the vertical lines. This means the vertical lines are parallel.
The diagonal line is the transversal.
Okay, this changes everything.
Parallel Lines: Vertical.
Transversal: Diagonal.

Re-evaluating Problem 9 with Vertical Parallel Lines:
1. Identify Parallel Lines: The two vertical lines with arrows are parallel.
2. Identify Transversal: The diagonal line cutting them.
3. Analyze Top Intersection:
* Angle $130^\circ$ is between the parallel line (up) and transversal. It's in the "North-West" position relative to the intersection.
* Angle $x$ is between the parallel line (up) and transversal. It's in the "North-East" position.
* These two angles form a linear pair along the parallel line? No, they are on opposite sides of the transversal but share the vertical ray? No, they share the vertex. They are adjacent angles on the straight line defined by the transversal? No.
* They are adjacent angles on the straight line defined by the parallel line? No, the vertical line is straight. The angle $130$ and angle $x$ are adjacent and their non-common sides form the transversal? No.
* Let's look at the rays.
* Ray Up (Parallel)
* Ray Down-Left (Transversal) -> Angle is 130.
* Ray Up-Right (Transversal) -> Angle is x.
* Wait, the transversal is a single straight line. It goes from Top-Right to Bottom-Left.
* So at the top intersection, we have the Vertical Line and the Diagonal Line.
* Angle $130^\circ$ is bounded by Vertical-Up and Diagonal-Down-Left.
* Angle $x$ is bounded by Vertical-Up and Diagonal-Up-Right.
* The Diagonal-Down-Left and Diagonal-Up-Right form a straight line (the transversal).
* Therefore, Angle $130^\circ$ and Angle $x$ are supplementary because they lie on the straight line of the transversal? No, they lie on the straight line of the transversal only if the vertical line was the one splitting them.
* Actually, Angle($130$) and Angle($x$) are adjacent. Their outer rays are Diagonal-Down-Left and Diagonal-Up-Right. These form a straight line.
* Therefore, $130 + x = 180$.
* $x = 50$.

* Let's check consistency with the bottom angle.
* Bottom Intersection:
* Angle $50^\circ$ is bounded by Vertical-Down and Diagonal-Down-Left.
* We found $x = 50^\circ$ (Top-Right, bounded by Vertical-Up and Diagonal-Up-Right).
* Are these related?
* Angle $x$ (Vertical-Up, Diag-Up-Right) and Angle $50$ (Vertical-Down, Diag-Down-Left).
* Vertical-Up and Vertical-Down are opposite rays (parallel lines direction).
* Diag-Up-Right and Diag-Down-Left are opposite rays (transversal).
* These are Alternate Exterior Angles?
* Let's check positions relative to the parallel vertical lines.
* $x$ is to the Right of the Left Parallel Line. (Interior?) No, it's between the parallels?
* Let's assume the region between the vertical lines is "Interior".
* $x$ is inside the parallel strip? Yes, it points towards the other line.
* The bottom $50^\circ$ angle: Bounded by Vertical-Down and Diag-Down-Left. This is to the Left of the Right Parallel Line? No, it's the Left Parallel Line?
* There are two vertical lines.
* Top intersection is on the Left Line? Or Right?
* Usually, standard orientation implies Left and Right.
* Let's assume Top Intersection is on the Left vertical line.
* $130^\circ$ is Top-Left (Exterior).
* $x$ is Top-Right (Interior).
* They are supplementary on the straight transversal line? No, on the straight vertical line?
* Angle 130 is between Vertical-Up and Transversal-Down-Left.
* Angle x is between Vertical-Up and Transversal-Up-Right.
* Since Transversal-Down-Left and Transversal-Up-Right are opposite rays, the angles $130$ and $x$ are supplementary.
* $130 + x = 180 \rightarrow x = 50$.
* Now check Bottom Intersection on the Right vertical line.
* Angle $50^\circ$ is between Vertical-Down and Transversal-Down-Left.
* This angle is Bottom-Left relative to the intersection.
* Is it Interior or Exterior? It is to the Left of the Right Parallel Line, so it is Interior.
* So we have:
* Top Interior Angle ($x$) = $50^\circ$. (Position: Top-Right of Left Line intersection).
* Bottom Interior Angle = $50^\circ$. (Position: Bottom-Left of Right Line intersection).
* These are Alternate Interior Angles.
* Alternate Interior Angles are equal.
* $50 = 50$.
* The geometry is consistent.

* Conclusion for 9: $x = 50$.

Final Answer:
1) 36
2) 49
3) 127
4) 62
5) 125
6) 102
7) 58
8) 83
9) 50