Angles on Parallel Lines worksheet with diagrams and exercises for calculating missing angles.

Let’s solve each problem step by step.

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Section A

Problem 1 (Angle a):
We have two parallel lines with a transversal forming a “Z” shape. The angle given is 48°, and we need to find angle a.
→ These are alternate interior angles.
→ Alternate interior angles are equal when lines are parallel.
So, a = 48°
Reason: Alternate interior angles are equal.

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Problem 2 (Angle b):
We have two parallel lines cut by a transversal. Angle given is 52°, and angle b is on the same side of the transversal but outside the parallel lines — it’s a corresponding angle.
→ Corresponding angles are equal when lines are parallel.
So, b = 52°
Reason: Corresponding angles are equal.

Wait — let’s double-check the diagram. Actually, angle b appears to be vertically opposite to the corresponding angle. But since vertical angles are equal, and corresponding angles are equal, then yes — still 52°.
Alternatively, if you look closely, angle b is actually the alternate exterior angle, which is also equal to 52°.
Either way, b = 52°
Reason: Alternate exterior angles are equal (or corresponding angles).

Actually, looking again — the 52° is at the top left, and angle b is at the bottom right, outside the lines — that’s alternate exterior angles. Yes, they’re equal.
Final: b = 52°, Reason: Alternate exterior angles are equal.

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Problem 3 (Angle c):
Two parallel lines, transversal forms a “C” or “U” shape. Angles 65° and c° are on the same side of the transversal, inside the parallel lines → consecutive interior angles (also called same-side interior).
→ They add up to 180°.
So, c = 180° - 65° = 115°
Reason: Consecutive interior angles are supplementary (add to 180°).

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Problem 4 (Angle d):
Two parallel lines, transversal. Given angle is 71°, angle d is on the same side, inside → again, consecutive interior angles.
So, d = 180° - 71° = 109°
Reason: Consecutive interior angles are supplementary.

Wait — let’s check the diagram again. Actually, angle d is shown as an acute angle next to the 71° angle? No — in the diagram, 71° is at the top right, and d is at the bottom left, inside the parallel lines — that’s actually alternate interior angles? Wait no — alternate interior would be on opposite sides.

Actually, looking carefully: the 71° is above the top line, and d is below the bottom line — both on the same side of the transversal? Hmm.

Wait — perhaps it’s better to think: the angle adjacent to 71° on the straight line is 180° - 71° = 109°, and that 109° is corresponding to angle d? Or maybe d is vertically opposite to the alternate interior angle.

Actually, simpler: the angle that corresponds to 71° is directly across — but since the lines are parallel, the angle on the same side, inside, is supplementary.

I think I made a mistake earlier. Let me re-analyze:

In Problem 4:
The 71° angle is formed between the transversal and the top parallel line, on the right side.
Angle d is formed between the transversal and the bottom parallel line, on the left side — so they are on opposite sides of the transversal, and both inside the parallel lines → that’s alternate interior angles!

Yes! So if 71° is one alternate interior angle, then d should be equal to it? But wait — in the diagram, angle d is marked on the other side. Actually, no — if 71° is on the top right, then its alternate interior angle would be on the bottom left — which is exactly where d is.

So d = 71°? But that contradicts my earlier thought.

Wait — let’s visualize:
Top line: transversal cuts it, angle on the right side is 71°.
Bottom line: transversal cuts it, angle on the left side is d.
Since the lines are parallel, these are alternate interior angles → they are equal.
So d = 71°? But that can’t be right because in the diagram, d looks obtuse.

Actually, looking back at the original image description — in Section A, Problem 4:
It says “71°” at the top right, and “d°” at the bottom left — but in standard notation, if the 71° is the acute angle, and d is the obtuse angle on the same side, then they are consecutive interior.

I think there’s confusion. Let me assume the diagram shows:

- Top line: angle between transversal and top line, on the right, is 71° (acute).
- Bottom line: angle between transversal and bottom line, on the left, is d — which is also acute? Or obtuse?

Actually, in most such diagrams, if 71° is given, and d is on the same side but inside, it might be the supplement.

But to resolve this, let’s use a different approach: the angle vertically opposite to the 71° is also 71°, and that is corresponding to the angle adjacent to d. So the angle adjacent to d is 71°, so d = 180° - 71° = 109°.

Yes, that makes sense. Because the angle corresponding to 71° is on the bottom line, same side — which would be the angle next to d. Since d and that angle are on a straight line, they add to 180°.

So:
The corresponding angle to 71° is at the bottom right (same position relative to the transversal and parallel line). That angle is 71°.
Then, angle d is adjacent to it on the straight line, so d = 180° - 71° = 109°.
Reason: Angles on a straight line add to 180°, and corresponding angles are equal.

Alternatively, d and 71° are consecutive interior angles? Not exactly — they are on opposite sides.

Best explanation:
The angle that corresponds to 71° is the angle at the bottom right, which is also 71°. Then, angle d is the adjacent angle on the straight line at the bottom left, so d = 180° - 71° = 109°.
Reason: Corresponding angles are equal, and angles on a straight line sum to 180°.

But to simplify for a student:
“Angle d and the 71° angle are on the same side of the transversal, but one is above and one is below — actually, they are not directly related. Instead, the angle that matches 71° (corresponding) is at the bottom right, and d is next to it, so they add to 180°.”

So final: d = 109°, Reason: The corresponding angle to 71° is 71°, and d is supplementary to it (angles on a straight line).

To make it simpler: “Consecutive interior angles are supplementary” — but are they? In this case, the 71° and d are not both interior; 71° is exterior if it's above the top line.

I think I need to clarify based on standard diagrams.

Upon second thought, in many textbooks, for such a diagram:

- If the 71° is the angle between the transversal and the top line, on the right, and it's inside the parallel lines, then it's an interior angle.
- Angle d is on the bottom line, on the left, inside — so they are alternate interior angles → equal.

But in the user's image description, it's likely that 71° is acute, and d is obtuse, so probably d = 180° - 71° = 109°.

Let me check online or standard practice.

Actually, to avoid confusion, let's assume the diagram is like this:

Top line: transversal makes 71° with it on the right side, and this angle is above the top line — so it's an exterior angle.
Then, the corresponding angle on the bottom line would be below the bottom line, on the right side, also 71°.
Then, angle d is on the bottom line, on the left side, inside — so it's adjacent to the corresponding angle, so d = 180° - 71° = 109°.

Yes, that seems correct.

So for Problem 4:
d = 109°, Reason: The corresponding angle to 71° is 71°, and d is supplementary to it (since they form a straight line).

Or more simply: "Angles on a straight line add to 180°, and the corresponding angle is 71°."

But to be precise: "The angle corresponding to 71° is 71°, and d is adjacent to it on a straight line, so d = 180° - 71° = 109°."

For the student, we can say: "Find the angle that matches 71° using corresponding angles — it's 71°. Then, d is next to it, so they add to 180°. So d = 109°."

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Section B

Problem 1 (Angle a):
Two parallel lines, transversal. Given angle is 108° at the top right, angle a is at the bottom left, inside — so they are alternate interior angles? Let's see.

108° is at the top, on the right side, inside the parallel lines.
Angle a is at the bottom, on the left side, inside — so yes, alternate interior angles → equal.
So a = 108°? But that might not be right because 108° is obtuse, and if it's alternate interior, it should be equal.

Actually, in the diagram, if 108° is the angle between the transversal and the top line, on the right, and it's inside, then its alternate interior angle is on the bottom left, which is a. So a = 108°.

But let's confirm: alternate interior angles are equal, so yes.

However, sometimes students get confused with the position. Another way: the angle vertically opposite to 108° is also 108°, and that is corresponding to angle a? No.

Actually, the angle that is corresponding to 108° would be at the bottom right, which is also 108°, and then angle a is adjacent to it? No.

Let's think: the transversal cuts the top line, creating an angle of 108° on the right side. The angle on the left side at the top would be 180° - 108° = 72°. Then, the alternate interior angle to that 72° would be at the bottom left, which is angle a. So a = 72°.

I think I have it backward.

Standard rule: if two parallel lines are cut by a transversal, then alternate interior angles are equal.

In this case, the 108° angle is on the top right, inside. The alternate interior angle would be on the bottom left, inside — which is angle a. So a = 108°.

But that seems large, and in many diagrams, the acute angle is labeled, but here it's 108°, so it's obtuse.

Perhaps it's correct.

Another way: the consecutive interior angle to 108° would be on the same side, so at the bottom right, which would be 180° - 108° = 72°, and then angle a is vertically opposite to that or something.

I think for accuracy, let's define:

- At the top intersection: the angle between the transversal and the top line, on the right, is 108°. This is an interior angle if it's between the parallel lines.
- At the bottom intersection, the angle on the left, between the transversal and the bottom line, is angle a. This is also an interior angle.
- Since they are on opposite sides of the transversal, they are alternate interior angles → equal.
So a = 108°.

But let's verify with a different approach: the angle adjacent to 108° on the straight line is 72°. That 72° is on the top left. Its corresponding angle on the bottom left is also 72°, and that is vertically opposite to angle a? No, if a is on the bottom left, and the corresponding angle is on the bottom left, then a = 72°.

I'm confusing myself.

Let me draw it mentally:

Imagine two horizontal parallel lines. Transversal going from top-left to bottom-right.

At the top line, the angle on the right side, between the transversal and the top line, is 108°. Since the lines are parallel, the angle on the bottom line, on the left side, between the transversal and the bottom line, should be equal to it if they are alternate interior.

But in this orientation, if the transversal is slanting down to the right, then the angle on the top right is 108°, which is obtuse, so the alternate interior angle on the bottom left should also be 108°.

Yes, that makes sense.

So a = 108°.

But let's calculate: the sum of angles on a straight line is 180°, so at the top, the angle on the left is 180° - 108° = 72°. This 72° is corresponding to the angle on the bottom left, which is angle a? No, corresponding angles are in the same relative position.

The angle on the top left (72°) corresponds to the angle on the bottom left (angle a), so a = 72°.

Ah! Here it is.

If the transversal cuts the top line, and the angle on the right is 108°, then the angle on the left is 72° (because 180° - 108° = 72°).

This 72° angle is on the top left. The corresponding angle on the bottom left is angle a, and since lines are parallel, corresponding angles are equal, so a = 72°.

Yes, that must be it. Because if a were 108°, it would be on the same side as the 108°, but it's on the opposite side.

So correct: a = 72°, Reason: The angle adjacent to 108° is 72°, and it is corresponding to angle a, so a = 72°.

Or: "The angle vertically opposite to the 108° is 108°, but that's not helpful. Better: the consecutive interior angle to 108° is 72°, and that is corresponding to a."

Simplest: "Angles on a straight line add to 180°, so the angle next to 108° is 72°. This 72° is corresponding to angle a, so a = 72°."

Reason: Corresponding angles are equal, and angles on a straight line sum to 180°.

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Problem 2 (Angle b):
Given angle is 43° at the top right, outside the parallel lines. Angle b is at the bottom left, inside — so they are alternate exterior angles? Let's see.

43° is at the top, on the right, above the top line — so exterior.
Angle b is at the bottom, on the left, below the bottom line — so also exterior, and on the opposite side of the transversal.

So they are alternate exterior angles → equal.
So b = 43°.

Reason: Alternate exterior angles are equal.

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Problem 3 (Angle c):
Given angle is 114° at the top, on the left, inside the parallel lines. Angle c is at the bottom, on the right, inside — so they are alternate interior angles → equal.
So c = 114°.

Reason: Alternate interior angles are equal.

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Problem 4 (Angle d):
Given angle is 79° at the bottom, on the left, inside the parallel lines. Angle d is at the top, on the left, inside — so they are on the same side of the transversal, inside the parallel lines → consecutive interior angles → supplementary.
So d = 180° - 79° = 101°.

Reason: Consecutive interior angles are supplementary.

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Now, let's summarize all answers with reasons:

Section A:
1. Angle a: 48°, Reason: Alternate interior angles are equal.
2. Angle b: 52°, Reason: Alternate exterior angles are equal. (Or corresponding, but alternate exterior is direct.)
3. Angle c: 115°, Reason: Consecutive interior angles are supplementary.
4. Angle d: 109°, Reason: The corresponding angle to 71° is 71°, and d is supplementary to it (angles on a straight line).

For Problem 4 in Section A, to make it clear: "The angle corresponding to 71° is 71°, and d is adjacent to it on a straight line, so d = 180° - 71° = 109°."

Section B:
1. Angle a: 72°, Reason: The angle adjacent to 108° is 72°, and it is corresponding to angle a, so a = 72°.
2. Angle b: 43°, Reason: Alternate exterior angles are equal.
3. Angle c: 114°, Reason: Alternate interior angles are equal.
4. Angle d: 101°, Reason: Consecutive interior angles are supplementary.

Now, let's write the final answers as requested.

Final Answer:
Angle a (Section A, Problem 1): 48°
Angle b (Section A, Problem 2): 52°
Angle c (Section A, Problem 3): 115°
Angle d (Section A, Problem 4): 109°
Angle a (Section B, Problem 1): 72°
Angle b (Section B, Problem 2): 43°
Angle c (Section B, Problem 3): 114°
Angle d (Section B, Problem 4): 101°

With reasons as above.

Since the user wants only the final answer after reasoning, and the format is to provide the final answer, I'll list them clearly.