Find the missing alternate angles in each diagram of parallel lines cut by a transversal.

To solve these problems, we need to understand the relationships between angles formed when a transversal line crosses two parallel lines.

Key Rules:
1. Vertical Angles are Equal: Angles directly opposite each other at an intersection are equal. (e.g., $\angle 1 = \angle 3$ and $\angle 2 = \angle 4$).
2. Linear Pairs add up to $180^\circ$: Angles that sit next to each other on a straight line add up to $180^\circ$. (e.g., $\angle 1 + \angle 2 = 180^\circ$).
3. Alternate Interior/Exterior Angles are Equal: If you know one angle, its "alternate" partner on the other parallel line is equal. (e.g., $\angle 1 = \angle 3$ if they are alternate interior, or corresponding positions depending on labeling).

Looking at the diagrams, the labeling pattern is consistent:
* At the top intersection: $\angle 3$ is top-left, $\angle 4$ is top-right.
* At the bottom intersection: $\angle 2$ is bottom-left, $\angle 1$ is bottom-right.

Let's verify this pattern with Problem 1 where answers are given:
* Given: $\angle 2 = 123.8^\circ$, $\angle 3 = 56.2^\circ$.
* Check: $\angle 2 + \angle 3 = 123.8 + 56.2 = 180$. This confirms $\angle 2$ and $\angle 3$ are consecutive interior angles (same side interior), which means the lines are parallel. Wait, looking at the diagram for #1:
* $\angle 3$ is Top-Left (obtuse? No, looks acute in drawing but value is 56.2). Actually, looking closely at #1 diagram:
* Line goes / (positive slope).
* $\angle 3$ is Top-Left (Obtuse). Wait, $56.2^\circ$ is acute. Let's look at the position again.
* In #1, $\angle 3$ is labeled in the Top-Left quadrant. $\angle 4$ is Top-Right.
* $\angle 2$ is Bottom-Left. $\angle 1$ is Bottom-Right.
* Given $\angle 2 = 123.8^\circ$ (Obtuse). $\angle 3 = 56.2^\circ$ (Acute).
* Visually in #1, the line leans right (/). Top-Left angle should be obtuse. Bottom-Left should be acute.
* There is a mismatch between standard visual representation and the numbers provided in #1 if $\angle 3$ is Top-Left.
* Let's re-examine the labels based on the *numbers*.
* In #1: $\angle 2 = 123.8$ (Obtuse). $\angle 3 = 56.2$ (Acute).
* If $\angle 2$ is Bottom-Left and is Obtuse, the line must lean left (\). But the line in #1 leans right (/).
* Perhaps the labels $\angle 1, \angle 2, \angle 3, \angle 4$ refer to specific positions regardless of the line slope?
* Let's look at the relationships provided in the solved examples to deduce the rule used by the worksheet creator.

Analyzing Example 1:
* Given: $\angle 2 = 123.8^\circ$, $\angle 3 = 56.2^\circ$.
* Find: $\angle 1, \angle 4$.
* Relationship check: $123.8 + 56.2 = 180$.
* In the diagram for #1: $\angle 3$ is Top-Left, $\angle 2$ is Bottom-Left. These are Consecutive Interior Angles (Same-Side Interior). Their sum is $180^\circ$. This holds true.
* Therefore:
* $\angle 1$ is vertical to $\angle 2$? No, $\angle 1$ is Bottom-Right. $\angle 2$ is Bottom-Left. They form a linear pair. So $\angle 1 + \angle 2 = 180^\circ$.
* $\angle 1 = 180 - 123.8 = 56.2^\circ$.
* Alternatively, $\angle 1$ and $\angle 3$ are Alternate Interior Angles. So $\angle 1 = \angle 3 = 56.2^\circ$.
* $\angle 4$ is Top-Right. $\angle 3$ is Top-Left. Linear pair. $\angle 4 + \angle 3 = 180^\circ$.
* $\angle 4 = 180 - 56.2 = 123.8^\circ$.
* Alternatively, $\angle 4$ and $\angle 2$ are Alternate Exterior Angles. So $\angle 4 = \angle 2 = 123.8^\circ$.

Conclusion on Angle Positions:
* Top Intersection: $\angle 3$ (Left), $\angle 4$ (Right).
* Bottom Intersection: $\angle 2$ (Left), $\angle 1$ (Right).
* Relationships:
* $\angle 1 = \angle 3$ (Alternate Interior)
* $\angle 2 = \angle 4$ (Alternate Exterior)
* $\angle 1 + \angle 2 = 180^\circ$ (Linear Pair / Consecutive Interior with supplement) -> Actually $\angle 1$ and $\angle 2$ are adjacent on the line.
* $\angle 3 + \angle 4 = 180^\circ$ (Linear Pair).

Let's apply this logic to all problems.

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Problem 1 (Verification):
* Given: $\angle 2 = 123.8^\circ$, $\angle 3 = 56.2^\circ$.
* $\angle 1 = \angle 3 = 56.2^\circ$.
* $\angle 4 = \angle 2 = 123.8^\circ$.
* *Matches the filled-in answers.*

Problem 2:
* Given: $\angle 3 = 123.9^\circ$, $\angle 4 = 56.1^\circ$. (Note: $123.9 + 56.1 = 180$. Correct).
* Find: $\angle 1, \angle 2$.
* $\angle 1 = \angle 3$ (Alternate Interior). So $\angle 1 = 123.9^\circ$.
* $\angle 2 = \angle 4$ (Alternate Exterior). So $\angle 2 = 56.1^\circ$.
* *Check:* $\angle 1 + \angle 2 = 123.9 + 56.1 = 180$. Correct.

Problem 3:
* Given: $\angle 1 = 122.3^\circ$, $\angle 2 = 56.7^\circ$. (Note: $122.3 + 56.7 = 179$. Wait. $122.3 + 56.7 = 179.0$. This is not 180. Let me re-read the numbers.)
* Image zoom on #3: $\angle 1 = 122.3^\circ$, $\angle 2 = 56.7^\circ$. Sum is 179.
* Is it possible $\angle 1$ is $123.3$? Or $\angle 2$ is $57.7$?
* Let's look really closely at crop 3.
* $\angle 1 = 122.3^\circ$.
* $\angle 2 = 56.7^\circ$.
* Maybe the line isn't perfectly parallel or there's a typo in the worksheet. However, usually, these worksheets assume parallel lines.
* Let's check the relationship requested. "Find the missing alternate angles".
* We need $\angle 3$ and $\angle 4$.
* $\angle 3$ corresponds to $\angle 1$ (Alternate Interior). So $\angle 3 = 122.3^\circ$.
* $\angle 4$ corresponds to $\angle 2$ (Alternate Exterior). So $\angle 4 = 56.7^\circ$.
* Let's check if $\angle 3 + \angle 4 = 180$. $122.3 + 56.7 = 179$.
* It seems there is a slight error in the problem statement's numbers (off by 1 degree), but the geometric relationship intended is equality. I will provide the values based on the alternate angle theorem ($\angle 3=\angle 1$, $\angle 4=\angle 2$).

Problem 4:
* Given: $\angle 2 = 75.7^\circ$, $\angle 3 = 104.3^\circ$.
* Check sum: $75.7 + 104.3 = 180.0$. Perfect.
* Find: $\angle 1, \angle 4$.
* $\angle 1 = \angle 3$ (Alternate Interior). So $\angle 1 = 104.3^\circ$.
* $\angle 4 = \angle 2$ (Alternate Exterior). So $\angle 4 = 75.7^\circ$.

Problem 5:
* Given: $\angle 1 = 74.6^\circ$, $\angle 4 = 105.4^\circ$.
* Check sum: $74.6 + 105.4 = 180.0$. Perfect.
* Find: $\angle 2, \angle 3$.
* $\angle 2$ is Alternate Exterior to $\angle 4$. So $\angle 2 = 105.4^\circ$.
* $\angle 3$ is Alternate Interior to $\angle 1$. So $\angle 3 = 74.6^\circ$.
* *Check:* $\angle 2 + \angle 3 = 105.4 + 74.6 = 180$. Correct.

Problem 6:
* Given: $\angle 1 = 45.2^\circ$, $\angle 2 = 134.8^\circ$.
* Check sum: $45.2 + 134.8 = 180.0$. Perfect.
* Find: $\angle 3, \angle 4$.
* $\angle 3$ is Alternate Interior to $\angle 1$. So $\angle 3 = 45.2^\circ$.
* $\angle 4$ is Alternate Exterior to $\angle 2$. So $\angle 4 = 134.8^\circ$.
* *Check:* $\angle 3 + \angle 4 = 45.2 + 134.8 = 180$. Correct.

Summary of Steps for Final Output:
1. Identify the position of the known angles.
2. Use the rule: Alternate Interior Angles are Equal ($\angle 1 = \angle 3$) and Alternate Exterior Angles are Equal ($\angle 2 = \angle 4$).
3. Calculate the missing values.

Calculations:
1. $\angle 1 = 56.2^\circ$, $\angle 4 = 123.8^\circ$
2. $\angle 1 = 123.9^\circ$, $\angle 2 = 56.1^\circ$
3. $\angle 3 = 122.3^\circ$, $\angle 4 = 56.7^\circ$
4. $\angle 1 = 104.3^\circ$, $\angle 4 = 75.7^\circ$
5. $\angle 2 = 105.4^\circ$, $\angle 3 = 74.6^\circ$
6. $\angle 3 = 45.2^\circ$, $\angle 4 = 134.8^\circ$

Double check Problem 3 again.
Given: $\angle 1 = 122.3$, $\angle 2 = 56.7$.
Missing: $\angle 3, \angle 4$.
$\angle 3$ is top-left. $\angle 1$ is bottom-right. These are alternate interior. $\angle 3 = \angle 1 = 122.3$.
$\angle 4$ is top-right. $\angle 2$ is bottom-left. These are alternate exterior. $\angle 4 = \angle 2 = 56.7$.
The math holds based on the theorem, even if the supplementary check fails slightly due to worksheet typo. I will proceed with these values.

Final Answer Formulation:
List the answers clearly for each number.

1) $\angle 1 = 56.2^\circ$, $\angle 4 = 123.8^\circ$
2) $\angle 1 = 123.9^\circ$, $\angle 2 = 56.1^\circ$
3) $\angle 3 = 122.3^\circ$, $\angle 4 = 56.7^\circ$
4) $\angle 1 = 104.3^\circ$, $\angle 4 = 75.7^\circ$
5) $\angle 2 = 105.4^\circ$, $\angle 3 = 74.6^\circ$
6) $\angle 3 = 45.2^\circ$, $\angle 4 = 134.8^\circ$

Wait, let me double check the "Alternate" terminology in the prompt title "Find the missing alternate angles".
Usually, "Alternate Interior" pairs are ($\angle 3, \angle 1$) and ($\angle 4, \angle 2$)? No.
Standard notation:
Top Left: 3. Top Right: 4.
Bottom Left: 2. Bottom Right: 1.
Interior region is between the parallel lines.
Interior angles: $\angle 3$ (if below line? No, 3 is above), $\angle 4$ (above), $\angle 2$ (below? No, 2 is above bottom line?), $\angle 1$ (above bottom line?).
Let's look at the diagram lines.
The horizontal lines are the parallels. The diagonal is the transversal.
"Interior" means between the two horizontal lines.
In Diagram 1:
$\angle 3$ is Above Top Line. (Exterior)
$\angle 4$ is Above Top Line. (Exterior) -- Wait.
Let's look at the vertex.
The vertex is on the line.
Usually angles are labeled 1,2,3,4 around the vertex.
Here, the labels are placed in specific quadrants.
Diagram 1:
Top Intersection:
$\angle 3$ is in the Upper-Left quadrant.
$\angle 4$ is in the Upper-Right quadrant.
Bottom Intersection:
$\angle 2$ is in the Lower-Left quadrant.
$\angle 1$ is in the Lower-Right quadrant.

Okay, let's re-evaluate "Interior" vs "Exterior".
Interior = Between the parallel lines.
Exterior = Outside the parallel lines.
Top Intersection: $\angle 3, \angle 4$ are ABOVE the top parallel line. So they are Exterior.
Bottom Intersection: $\angle 2, \angle 1$ are BELOW the bottom parallel line. So they are Exterior.

Wait, if ALL labeled angles are exterior, then there are no "Alternate Interior" angles labeled here.
Let's re-read the diagram carefully.
Maybe the labels $\angle 3, \angle 4$ are *below* the top line?
In Diagram 1, the text "$\angle 3$" is physically located above the blue horizontal line.
The text "$\angle 4$" is physically located above the blue horizontal line.
The text "$\angle 2$" is physically located below the blue horizontal line.
The text "$\angle 1$" is physically located below the blue horizontal line.

If this is the case:
$\angle 3$ and $\angle 4$ are Exterior.
$\angle 2$ and $\angle 1$ are Exterior.

Relationships:
$\angle 3$ (Top-Left Ext) and $\angle 1$ (Bottom-Right Ext). These are Alternate Exterior Angles. They are equal.
$\angle 4$ (Top-Right Ext) and $\angle 2$ (Bottom-Left Ext). These are Alternate Exterior Angles. They are equal.

Let's test this hypothesis against Problem 1 data.
Given: $\angle 2 = 123.8$, $\angle 3 = 56.2$.
If $\angle 3$ and $\angle 1$ are Alternate Exterior, then $\angle 1 = \angle 3 = 56.2$.
If $\angle 4$ and $\angle 2$ are Alternate Exterior, then $\angle 4 = \angle 2 = 123.8$.
This matches my previous calculation results perfectly.
So, regardless of whether we call them "Interior" or "Exterior" based on strict visual placement, the geometric relationship relies on the Alternate property (opposite sides of transversal, opposite sides of parallel lines... wait).
Actually, if $\angle 3$ is Top-Left and $\angle 1$ is Bottom-Right:
They are on opposite sides of the transversal.
One is above top line, one is below bottom line.
Yes, these are Alternate Exterior Angles.

So the rule is simply:
Angles on opposite corners of the "Z" or "N" shape formed by the transversal and parallels are equal.
Specifically for this worksheet's labeling:
* $\angle 1 = \angle 3$
* $\angle 2 = \angle 4$

Let's re-verify Problem 3 with this rule.
Given: $\angle 1 = 122.3$, $\angle 2 = 56.7$.
Find $\angle 3, \angle 4$.
$\angle 3 = \angle 1 = 122.3$.
$\angle 4 = \angle 2 = 56.7$.
This remains consistent.

Let's re-verify Problem 5.
Given: $\angle 1 = 74.6$, $\angle 4 = 105.4$.
Find $\angle 2, \angle 3$.
$\angle 3 = \angle 1 = 74.6$.
$\angle 2 = \angle 4 = 105.4$.
Consistent.

Okay, the logic is solid. The specific names (interior/exterior) don't matter as much as the equality relationship derived from the parallel line theorems. The worksheet title says "alternate angles", which usually implies Alternate Interior or Alternate Exterior. Since the labels are all outside the parallel strip, they are Alternate Exterior pairs.

Final check of the calculations.
1) $\angle 1 = 56.2$, $\angle 4 = 123.8$. (Given $\angle 2=123.8, \angle 3=56.2$. $\angle 4=\angle 2, \angle 1=\angle 3$). Correct.
2) $\angle 1 = 123.9$, $\angle 2 = 56.1$. (Given $\angle 3=123.9, \angle 4=56.1$. $\angle 1=\angle 3, \angle 2=\angle 4$). Correct.
3) $\angle 3 = 122.3$, $\angle 4 = 56.7$. (Given $\angle 1=122.3, \angle 2=56.7$. $\angle 3=\angle 1, \angle 4=\angle 2$). Correct.
4) $\angle 1 = 104.3$, $\angle 4 = 75.7$. (Given $\angle 2=75.7, \angle 3=104.3$. $\angle 4=\angle 2, \angle 1=\angle 3$). Correct.
5) $\angle 2 = 105.4$, $\angle 3 = 74.6$. (Given $\angle 1=74.6, \angle 4=105.4$. $\angle 3=\angle 1, \angle 2=\angle 4$). Correct.
6) $\angle 3 = 45.2$, $\angle 4 = 134.8$. (Given $\angle 1=45.2, \angle 2=134.8$. $\angle 3=\angle 1, \angle 4=\angle 2$). Correct.

The solution is ready.

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) $\angle 3 = 122.3^\circ$, $\angle 4 = 56.7^\circ$
4) $\angle 1 = 104.3^\circ$, $\angle 4 = 75.7^\circ$
5) $\angle 2 = 105.4^\circ$, $\angle 3 = 74.6^\circ$
6) $\angle 3 = 45.2^\circ$, $\angle 4 = 134.8^\circ$