Angles on Parallel Lines (A) (With Clues) Worksheet | Cazoom Maths ... - Free Printable

Let’s solve each problem step by step. We’ll use the clues given at the top to find missing angles and give reasons.

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Section A

Problem 1: Angle a°
We see two parallel lines with a zigzag line (like a Z) crossing them. The angle marked 48° is on the top line, and angle a° is on the bottom line — they are on opposite sides of the diagonal line and inside the parallel lines.

This is an alternate angle. Alternate angles are equal when lines are parallel.

→ So, Angle a = 48°
Reason: It is an alternate angle and therefore equal.

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Problem 2: Angle b°
The 52° angle is above the top parallel line, and angle b° is below the bottom parallel line — both on the same side of the diagonal line.

These are corresponding angles. Corresponding angles are equal when lines are parallel.

→ So, Angle b = 52°
Reason: It is a corresponding angle and therefore equal.

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Problem 3: Angle c°
We have two parallel lines with a transversal making a “C” shape. The 65° angle and angle c° are both inside the parallel lines and on the same side of the transversal.

These are interior angles. Interior angles add up to 180°.

So, c° = 180° - 65° = 115°

→ Angle c = 115°
Reason: It is an interior angle and therefore adds to 180°.

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Problem 4: Angle d°
The 71° angle is between the parallel lines, and angle d° is outside — but notice that angle d° is vertically opposite to the angle that corresponds to 71°.

Wait — let’s look again. Actually, angle d° is directly across from the angle that would be corresponding to 71°. But even simpler: angle d° and the 71° angle are vertically opposite? No — not exactly.

Actually, if you extend the lines, angle d° is in the same position as the 71° angle relative to the parallel lines — so they are corresponding angles.

But wait — looking at the diagram: the 71° is inside the top parallel line, and angle d° is outside the bottom one — same side → yes, corresponding.

Alternatively, maybe it's easier: angle d° is vertically opposite to the angle that is alternate to 71°? Let’s think differently.

Actually, the simplest way: angle d° and the 71° angle are corresponding angles because they are in matching corners relative to the transversal and parallel lines.

→ So, Angle d = 71°
Reason: It is a corresponding angle and therefore equal.

*(Note: Some might argue it’s vertically opposite to an alternate angle — but since we’re told to choose from the list, “corresponding” fits best here.)*

Wait — actually, looking again: the 71° is on the right side of the transversal, above the top line. Angle d° is on the right side of the transversal, below the bottom line — that’s corresponding! Yes.

✔ Confirmed: d = 71°, reason: corresponding angle.

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Section B

Problem 1: Angle a°
We have two parallel lines cut by a transversal. One angle is 108°, and angle a° is next to it on the straight line.

Angles on a straight line add to 180°.

So, a° = 180° - 108° = 72°

→ Angle a = 72°
Reason: Angles on a straight line add up to 180°.

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Problem 2: Angle b°
There’s a 43° angle above the top parallel line. Angle b° is below the bottom parallel line, on the same side of the transversal.

That makes them corresponding angles.

→ So, Angle b = 43°
Reason: It is a corresponding angle and therefore equal.

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Problem 3: Angle c°
We have 114° and angle c°. They are on opposite sides of the transversal, and both are outside the parallel lines? Wait — actually, 114° is above the top line, and c° is below the bottom line — but on opposite sides.

Actually, these are alternate exterior angles — but our clue list doesn’t say that. However, note that angle c° is vertically opposite to the angle that is alternate to 114°.

Better approach: The angle adjacent to 114° on the straight line is 180° - 114° = 66°. That 66° angle and angle c° are corresponding angles — so c° = 66°.

But wait — let’s check the diagram again. Actually, 114° and c° are on opposite sides of the transversal and both are outside — so they are alternate exterior angles, which are equal. But since our clues don’t include that, perhaps we should use another path.

Alternative: The angle vertically opposite to 114° is also 114°, and that angle and c° are interior on the same side? No.

Wait — actually, looking at standard positions: 114° and c° are alternate angles if we consider the full setup. But 114° is obtuse, and c° looks acute — so probably not.

Let me recalculate:

If 114° is on the top left, then the angle directly below it (on the bottom line, same side) would be corresponding — but that’s not labeled.

Actually, angle c° is on the bottom right, and 114° is on the top left — so they are alternate exterior angles — which are equal. But again, not in our clue list.

Wait — perhaps the intended solution is:

The angle adjacent to 114° on the straight line is 66°. That 66° angle and angle c° are corresponding angles — so c° = 66°.

Yes, that works with our clues.

→ Angle c = 66°
Reason: First, angles on a straight line add to 180°, so the angle next to 114° is 66°. Then, that 66° angle and angle c° are corresponding angles, so c° = 66°.

But the question asks for one reason. Since the clue list includes “angles on a straight line”, and we need to get to c°, perhaps we can say:

Actually, angle c° is vertically opposite to the angle that is alternate to the supplement of 114° — too complicated.

Let’s simplify: In many textbooks, this configuration is treated as alternate exterior angles being equal — but since our clues don’t have that, and we must choose from the list, the best fit is:

Notice that angle c° and the 114° angle are not directly related by one rule — but if we take the angle vertically opposite to 114°, it’s still 114°, and then the angle adjacent to that on the straight line is 66°, and that 66° is corresponding to c°.

But the worksheet likely expects us to recognize that c° is equal to the alternate angle of the supplement.

Wait — I think I made it harder than needed.

Looking at the diagram: 114° is on the top, and c° is on the bottom, on the opposite side — so they are alternate angles? But 114° is outside, c° is outside — alternate exterior.

Since our clues don’t specify exterior, but “alternate angle” is listed, and in some contexts, alternate includes exterior, perhaps we can say:

Actually, no — let’s calculate numerically.

Assume the lines are parallel. The angle corresponding to 114° on the bottom line would be 114° on the same side. Then angle c° is vertically opposite to the angle that is supplementary to that — messy.

Better: The angle adjacent to 114° is 66° (straight line). That 66° angle and angle c° are corresponding angles — so c° = 66°.

And the reason for c° being 66° is because it’s corresponding to the 66° angle, which came from 180° - 114°.

But the question says "give a reason for your answer" — implying one reason from the list.

Perhaps the intended answer is that c° and 114° are interior angles on the same side? No, they are not both interior.

I think there’s a mistake in my initial assumption.

Let me re-express:

In Section B, Problem 3: The 114° angle and angle c° are on opposite sides of the transversal, and both are outside the parallel lines — so they are alternate exterior angles, which are equal. But since "alternate angle" is in the clue list, and it doesn't specify interior, perhaps we can use that.

However, typically "alternate angles" refer to interior alternate angles. To avoid confusion, let's use the straight line method.

Final decision:

Step 1: The angle next to 114° on the straight line is 180° - 114° = 66°.

Step 2: That 66° angle and angle c° are corresponding angles (same relative position), so c° = 66°.

But the reason for c° is "it is a corresponding angle", assuming we've already calculated the 66°.

Since the worksheet allows using the clues, and "angles on a straight line" is a clue, we can combine, but the format asks for one reason per answer.

Looking at similar problems, often for this configuration, they expect:

Angle c° = 114° because they are alternate exterior angles — but since that's not in the clues, perhaps it's 66°.

I recall that in some curricula, alternate angles include exterior, but to be safe, let's go with 66° and reason: it is a corresponding angle to the angle that is supplementary to 114°.

But that's two steps.

Perhaps the diagram shows that c° is vertically opposite to an angle that is alternate to 114°'s supplement.

I think I found it: the angle vertically opposite to 114° is 114°, and then the angle adjacent to that on the bottom line is 66°, and c° is vertically opposite to that 66° angle — so c° = 66°.

Then the reason could be "vertically opposite angle" for the last step, but it's indirect.

To resolve this, let's assume the intended answer is c° = 66°, with reason: "It is a corresponding angle" — referring to the 66° angle derived from the straight line.

But since the instruction is to give one reason, and the clue list has "angles on a straight line", perhaps for this problem, the reason is "angles on a straight line add up to 180°" — but that gives the adjacent angle, not c° directly.

I think there's a better way: in the diagram, angle c° and the 114° angle are actually alternate angles if we consider the full picture — but 114° is obtuse, c° is acute, so unlikely.

Upon second thought, in many worksheets, for this exact diagram, c° is taken as equal to 114° because they are alternate exterior angles, and "alternate angle" is accepted.

But to match the clues strictly, let's calculate:

Suppose we call the angle adjacent to 114° as x. x = 180° - 114° = 66°.

Then, x and c° are corresponding angles, so c° = 66°.

The reason for c° is "it is a corresponding angle".

Yes, that works.

→ Angle c = 66°
Reason: It is a corresponding angle and therefore equal. (referring to the 66° angle)

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Problem 4: Angle d°
We have 79° and angle d°. They are on the same side of the transversal, and both are between the parallel lines? Let's see.

79° is on the bottom, d° is on the top — same side of the transversal, and both inside the parallel lines — so they are interior angles on the same side.

Interior angles add to 180°.

So, d° = 180° - 79° = 101°

→ Angle d = 101°
Reason: It is an interior angle and therefore adds to 180°.

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Now, compiling all answers:

Section A:
- Angle a: 48°, Reason: It is an alternate angle and therefore equal.
- Angle b: 52°, Reason: It is a corresponding angle and therefore equal.
- Angle c: 115°, Reason: It is an interior angle and therefore adds to 180°.
- Angle d: 71°, Reason: It is a corresponding angle and therefore equal.

Section B:
- Angle a: 72°, Reason: Angles on a straight line add up to 180°.
- Angle b: 43°, Reason: It is a corresponding angle and therefore equal.
- Angle c: 66°, Reason: It is a corresponding angle and therefore equal. (after finding the supplementary angle)
- Angle d: 101°, Reason: It is an interior angle and therefore adds to 180°.

For Section B, Problem 3, to be precise, the reason "it is a corresponding angle" assumes we know the other angle is 66°, which comes from the straight line. But since the clue list includes "angles on a straight line", and we might need to use it, but the format asks for one reason per answer, I'll stick with "corresponding angle" for c°, as the direct reason for its value once the setup is understood.

Alternatively, some might argue that c° and 114° are vertically opposite to angles that are alternate, but I think 66° is correct.

Let me double-check with a different approach: in parallel lines, alternate exterior angles are equal. If 114° is an exterior angle, then the alternate exterior angle would be c°, so c° = 114°. But that contradicts the diagram where c° looks acute.

Looking back at the user's image description (though I shouldn't describe it), in standard problems, if 114° is given and c° is on the opposite side outside, it should be equal if alternate exterior.

But in the clue list, "alternate angle" is listed, and it doesn't specify interior, so perhaps c° = 114°.

I think I made a mistake earlier.

Let me clarify with a reliable method:

When two parallel lines are cut by a transversal:
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Consecutive interior angles sum to 180°.

In Section B, Problem 3: 114° is an exterior angle on the top left. Angle c° is an exterior angle on the bottom right — so they are alternate exterior angles, hence equal.

Therefore, c° = 114°.

And the reason is "It is an alternate angle and therefore equal." — since "alternate angle" can encompass alternate exterior in some contexts, or the clue might intend it broadly.

Given that, and to match common textbook solutions, I'll go with c° = 114°.

Similarly, in many online resources, for such diagrams, alternate exterior angles are considered under "alternate angles".

So, correction:

→ Angle c = 114°
Reason: It is an alternate angle and therefore equal.

This makes sense because the diagram likely shows them as alternate exterior.

For consistency with the clues, and since "alternate angle" is listed without specification, this is acceptable.

So, final answers:

Section A:
- a: 48°, alternate angle
- b: 52°, corresponding angle
- c: 115°, interior angle
- d: 71°, corresponding angle

Section B:
- a: 72°, angles on a straight line
- b: 43°, corresponding angle
- c: 114°, alternate angle
- d: 101°, interior angle

Yes, this seems consistent.

Final Answer:
Section A:
- Angle a: 48°, Reason: It is an alternate angle and therefore equal.
- Angle b: 52°, Reason: It is a corresponding angle and therefore equal.
- Angle c: 115°, Reason: It is an interior angle and therefore adds to 180°.
- Angle d: 71°, Reason: It is a corresponding angle and therefore equal.

Section B:
- Angle a: 72°, Reason: Angles on a straight line add up to 180°.
- Angle b: 43°, Reason: It is a corresponding angle and therefore equal.
- Angle c: 114°, Reason: It is an alternate angle and therefore equal.
- Angle d: 101°, Reason: It is an interior angle and therefore adds to 180°.