Find the missing alternate angles in each diagram.

To find the missing angles, we need to use two main rules about parallel lines cut by a transversal (the line crossing them):

1. Alternate Interior Angles are Equal: These are the angles on opposite sides of the crossing line and *between* the parallel lines. In these diagrams, $\angle 2$ and $\angle 4$ are alternate interior angles. So, $\angle 2 = \angle 4$.
2. Supplementary Angles add up to $180^\circ$: Angles that form a straight line add up to $180^\circ$. For example, $\angle 1 + \angle 2 = 180^\circ$ and $\angle 3 + \angle 4 = 180^\circ$. Also, vertical angles (angles opposite each other at an intersection) are equal, so $\angle 1 = \angle 3$ and $\angle 2 = \angle 4$.

Let's solve each problem step-by-step:

Problem 1)
* Given: $\angle 2 = 123.8^\circ$, $\angle 3 = 56.2^\circ$
* Find $\angle 4$: Since $\angle 3$ and $\angle 4$ are on a straight line, they add to $180^\circ$.
$\angle 4 = 180^\circ - 56.2^\circ = 123.8^\circ$.
*(Check: $\angle 4$ should equal $\angle 2$ because they are alternate interior angles. $123.8^\circ = 123.8^\circ$. Correct.)*
* Find $\angle 1$: $\angle 1$ and $\angle 2$ are on a straight line.
$\angle 1 = 180^\circ - 123.8^\circ = 56.2^\circ$.
*(Check: $\angle 1$ should equal $\angle 3$ because they are alternate exterior angles or corresponding logic. $56.2^\circ = 56.2^\circ$. Correct.)*
* Answer: $\angle 1 = 56.2^\circ$, $\angle 4 = 123.8^\circ$

Problem 2)
* Given: $\angle 3 = 123.9^\circ$, $\angle 4 = 56.1^\circ$
* Find $\angle 1$: $\angle 1$ and $\angle 4$ are not directly related by a simple name, but $\angle 1$ and $\angle 3$ are alternate exterior angles? No, let's look closer. $\angle 1$ and $\angle 4$ are consecutive interior? No.
Let's use vertical angles and linear pairs.
$\angle 1$ is vertically opposite to the angle adjacent to $\angle 4$? No, simpler: $\angle 1$ and $\angle 2$ are supplementary. $\angle 3$ and $\angle 4$ are supplementary ($123.9 + 56.1 = 180$).
$\angle 1$ corresponds to the angle vertically opposite $\angle 3$? No.
Let's use Alternate Exterior Angles: $\angle 1$ and $\angle 3$ are NOT alternate exterior. $\angle 1$ is bottom-right interior? No, looking at the arrows:
Line 1 (top) has angles 3 (left) and 4 (right).
Line 2 (bottom) has angles 2 (left) and 1 (right).
Actually, usually standard numbering is counter-clockwise or specific positions. Let's look at the positions in the diagram.
Diagram 2:
Top intersection: Left is $\angle 3$, Right is $\angle 4$.
Bottom intersection: Left is $\angle 2$, Right is $\angle 1$.
Therefore:
$\angle 3$ and $\angle 1$ are Alternate Exterior Angles? No, $\angle 3$ is top-left, $\angle 1$ is bottom-right. They are Alternate Exterior if they are outside the parallel lines. Here, $\angle 3$ is "inside" the left side? No, the parallel lines are horizontal. The transversal cuts them.
Let's assume standard positions relative to the parallel lines:
Top Line: $\angle 3$ is Top-Left? Or Interior Left? The arc for $\angle 3$ is between the transversal and the parallel line on the left. It looks like an interior angle. Wait, looking at Problem 1, $\angle 2$ and $\angle 4$ are clearly the "Z" shape angles (Alternate Interior).
So, $\angle 2$ (Bottom Left) and $\angle 4$ (Top Right) are Alternate Interior Angles. Thus $\angle 2 = \angle 4$.
And $\angle 3$ (Top Left) and $\angle 1$ (Bottom Right) are Alternate Interior? No.
Let's re-evaluate based on the "Z" shape.
In all diagrams:
$\angle 4$ is Top-Right (Interior).
$\angle 2$ is Bottom-Left (Interior).
So $\angle 2$ and $\angle 4$ are Alternate Interior Angles. $\angle 2 = \angle 4$.

$\angle 3$ is Top-Left (Interior? Or Exterior?). Looking at the arc in #1, $\angle 3$ is adjacent to $\angle 4$ on the straight line. So $\angle 3 + \angle 4 = 180^\circ$.
$\angle 1$ is Bottom-Right (Interior? Or Exterior?). Looking at the arc in #1, $\angle 1$ is adjacent to $\angle 2$ on the straight line. So $\angle 1 + \angle 2 = 180^\circ$.

Let's re-read the diagram carefully.
Usually, 1,2,3,4 are quadrants.
But here, the labels are specific.
Top Intersection: $\angle 3$ is Left, $\angle 4$ is Right. They form a linear pair on the parallel line? No, they are on opposite sides of the transversal. They are adjacent angles on the straight parallel line. So $\angle 3 + \angle 4 = 180^\circ$.
Bottom Intersection: $\angle 2$ is Left, $\angle 1$ is Right. They form a linear pair on the parallel line. So $\angle 2 + \angle 1 = 180^\circ$.

Relationship between Top and Bottom:
$\angle 4$ (Top Right) and $\angle 1$ (Bottom Right) are Corresponding Angles. So $\angle 4 = \angle 1$.
$\angle 3$ (Top Left) and $\angle 2$ (Bottom Left) are Corresponding Angles. So $\angle 3 = \angle 2$.

Let's test this hypothesis on Problem 1:
Given $\angle 2 = 123.8$, $\angle 3 = 56.2$.
If $\angle 3 = \angle 2$, then $56.2$ should equal $123.8$. It does not.
So my assumption about which angle is which position is wrong.

Let's look at the "Z" again.
In Problem 1: $\angle 2 = 123.8^\circ$. $\angle 4$ is blank.
Visually, $\angle 2$ is obtuse. $\angle 4$ is obtuse.
$\angle 3 = 56.2^\circ$ (acute). $\angle 1$ is blank.
Visually, $\angle 1$ is acute.

Standard Alternate Interior Angles theorem: The angles inside the parallel lines on opposite sides of the transversal are equal.
In the diagram, the angles "inside" the parallel lines are $\angle 4$ (top, right of transversal) and $\angle 2$ (bottom, left of transversal)?
Or is it $\angle 3$ and $\angle 1$?
Let's look at the arcs.
In #1: Arc for $\angle 4$ is inside the parallel lines. Arc for $\angle 2$ is inside the parallel lines.
So $\angle 2$ and $\angle 4$ are Alternate Interior Angles.
Therefore, $\angle 2 = \angle 4$.

Consequently:
$\angle 3$ and $\angle 4$ are supplementary (linear pair on the top line). $\angle 3 + \angle 4 = 180^\circ$.
$\angle 1$ and $\angle 2$ are supplementary (linear pair on the bottom line). $\angle 1 + \angle 2 = 180^\circ$.
Also, $\angle 1$ and $\angle 3$ are Alternate Exterior Angles? No, $\angle 1$ is inside?
Let's look at $\angle 1$'s position. It is on the bottom line, right side of transversal. Is it inside or outside?
The arc for $\angle 1$ is between the parallel line and the transversal, on the "inside" region?
Actually, looking at #1, $\angle 1$ and $\angle 2$ are adjacent on the bottom line.
$\angle 3$ and $\angle 4$ are adjacent on the top line.

If $\angle 2$ and $\angle 4$ are Alternate Interior, they are equal.
Then $\angle 1$ (supplement of $\angle 2$) must equal $\angle 3$ (supplement of $\angle 4$).
Let's check #1:
$\angle 2 = 123.8$. So $\angle 4 = 123.8$.
$\angle 3 = 56.2$.
Check sum: $123.8 + 56.2 = 180$. This works perfectly.
So, $\angle 1$ is supplement of $\angle 2$. $\angle 1 = 180 - 123.8 = 56.2$.

Rule Established:
1. $\angle 2 = \angle 4$ (Alternate Interior Angles)
2. $\angle 1 = \angle 3$ (Alternate Interior/Exterior depending on definition, but effectively they are equal because they are supplements of equal angles).
3. $\angle 1 + \angle 2 = 180^\circ$
4. $\angle 3 + \angle 4 = 180^\circ$

Let's apply this to all problems.

Problem 1)
Given: $\angle 2 = 123.8^\circ$, $\angle 3 = 56.2^\circ$
$\angle 4 = \angle 2 = \mathbf{123.8^\circ}$
$\angle 1 = \angle 3 = \mathbf{56.2^\circ}$ (or $180 - 123.8$)

Problem 2)
Given: $\angle 3 = 123.9^\circ$, $\angle 4 = 56.1^\circ$
Check sum: $123.9 + 56.1 = 180$. Good.
$\angle 1 = \angle 3 = \mathbf{123.9^\circ}$
$\angle 2 = \angle 4 = \mathbf{56.1^\circ}$

Problem 3)
Given: $\angle 1 = 122.3^\circ$, $\angle 2 = 56.7^\circ$
Check sum: $122.3 + 56.7 = 179.0$? Wait.
$122.3 + 56.7 = 179.0$. This is not 180. There might be a typo in the problem or my reading.
Let me re-read the numbers in image 3.
$\angle 1 = 122.3^\circ$? Or $123.3$? It looks like $122.3$.
$\angle 2 = 56.7^\circ$?
$122.3 + 56.7 = 179$.
Let's look at $\angle 1$ and $\angle 2$. They are a linear pair. They MUST add to 180.
Maybe $\angle 1$ is $123.3$? $123.3 + 56.7 = 180$. That makes sense.
Or maybe $\angle 2$ is $57.7$? $122.3 + 57.7 = 180$.
Looking closely at crop 3... The number for L1 is `122.3`. The number for L2 is `56.7`.
This is a contradiction in the problem statement itself ($122.3+56.7 \neq 180$).
However, often in these worksheets, you just follow the equality rules.
Rule: $\angle 3 = \angle 1$ and $\angle 4 = \angle 2$.
So $\angle 3 = 122.3^\circ$ and $\angle 4 = 56.7^\circ$.
Let's check if $\angle 3 + \angle 4 = 180$. $122.3 + 56.7 = 179$.
I will provide the answers based on the direct equality properties ($\angle 1=\angle 3, \angle 2=\angle 4$), noting the slight arithmetic error in the question's source data if necessary, but for a student, just applying the rule is best.
$\angle 3 = \angle 1 = \mathbf{122.3^\circ}$
$\angle 4 = \angle 2 = \mathbf{56.7^\circ}$

Problem 4)
Given: $\angle 2 = 75.7^\circ$, $\angle 3 = 104.3^\circ$
Check sum for top line: We don't have $\angle 4$ yet.
We know $\angle 2 = \angle 4$. So $\angle 4 = \mathbf{75.7^\circ}$.
We know $\angle 1 = \angle 3$. So $\angle 1 = \mathbf{104.3^\circ}$.
Check consistency: Does $\angle 3 + \angle 4 = 180$?
$104.3 + 75.7 = 180.0$. Yes.
Does $\angle 1 + \angle 2 = 180$?
$104.3 + 75.7 = 180.0$. Yes.
Answers: $\angle 1 = 104.3^\circ$, $\angle 4 = 75.7^\circ$.

Problem 5)
Given: $\angle 1 = 74.6^\circ$, $\angle 4 = 105.4^\circ$
We know $\angle 3 = \angle 1$. So $\angle 3 = \mathbf{74.6^\circ}$.
We know $\angle 2 = \angle 4$. So $\angle 2 = \mathbf{105.4^\circ}$.
Check consistency:
$\angle 1 + \angle 2 = 74.6 + 105.4 = 180.0$. Yes.
$\angle 3 + \angle 4 = 74.6 + 105.4 = 180.0$. Yes.
Answers: $\angle 2 = 105.4^\circ$, $\angle 3 = 74.6^\circ$.

Problem 6)
Given: $\angle 1 = 45.2^\circ$, $\angle 2 = 134.8^\circ$
We know $\angle 3 = \angle 1$. So $\angle 3 = \mathbf{45.2^\circ}$.
We know $\angle 4 = \angle 2$. So $\angle 4 = \mathbf{134.8^\circ}$.
Check consistency:
$\angle 1 + \angle 2 = 45.2 + 134.8 = 180.0$. Yes.
Answers: $\angle 3 = 45.2^\circ$, $\angle 4 = 134.8^\circ$.

Final Answer:

1)
$\angle 1 = 56.2^\circ$
$\angle 4 = 123.8^\circ$

2)
$\angle 1 = 123.9^\circ$
$\angle 2 = 56.1^\circ$

3)
$\angle 3 = 122.3^\circ$
$\angle 4 = 56.7^\circ$
*(Note: The given angles $122.3^\circ$ and $56.7^\circ$ add up to $179^\circ$ instead of $180^\circ$, but using the angle correspondence rules gives these values.)*

4)
$\angle 1 = 104.3^\circ$
$\angle 4 = 75.7^\circ$

5)
$\angle 2 = 105.4^\circ$
$\angle 3 = 74.6^\circ$

6)
$\angle 3 = 45.2^\circ$
$\angle 4 = 134.8^\circ$