Angles on Parallel Lines (B) (With Clues) Worksheet | Fun and ... - Free Printable

Let’s solve each angle step by step. We’ll use rules about parallel lines and angles — like alternate angles, corresponding angles, co-interior angles, and angles on a straight line or in a triangle.

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Angle a:

We see two parallel lines (arrows show direction) cut by a transversal. The 72° angle and angle a are alternate interior angles — they’re on opposite sides of the transversal and inside the parallel lines.

→ Alternate interior angles are equal when lines are parallel.

So, a = 72°

Reason: Alternate interior angles are equal.

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Angle b:

This is a quadrilateral with two pairs of parallel sides? Wait — actually, looking at the arrows: only one pair of sides has arrows (the bottom and top right side). But wait — let’s check again.

Actually, in this shape, we have two adjacent angles given: 67° and b°. And there are arrows on two sides — meaning those two sides are parallel. So it's likely a trapezoid with one pair of parallel sides.

But here’s the key: the two angles marked (67° and b°) are on the same side between the two parallel lines → so they are co-interior angles.

Co-interior angles add up to 180° if the lines are parallel.

So:
b + 67 = 180
→ b = 180 - 67 = 113°

Reason: Co-interior angles between parallel lines sum to 180°.

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Angle c:

We have a quadrilateral with three known angles: 52°, 126°, 108°. Angle c is outside — it’s an exterior angle at the fourth vertex.

First, find the missing interior angle at that corner.

Sum of interior angles in any quadrilateral = 360°

So:
Missing interior angle = 360 - (52 + 126 + 108)
= 360 - 286 = 74°

Now, angle c is on a straight line with that 74° angle → so they add to 180°.

c = 180 - 74 = 106°

Reason: Angles on a straight line sum to 180°.

*(Alternatively, you could think of c as the exterior angle, which equals the sum of the opposite interior angles — but since we don’t know if it’s cyclic or anything, better to stick with basic angle sums.)*

Wait — actually, another way: In a polygon, the exterior angle is supplementary to the interior angle. So yes, 180 - 74 = 106°.

✔ Confirmed.

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Angle d:

We have two horizontal parallel lines (arrows point right). A zigzag line connects them. At the top, we have angle d° next to a 68° angle. At the bottom, we have a 75° angle.

Let’s draw an imaginary line or use the “Z” rule.

Actually, think of the path from top to bottom: the total turn should relate to the parallel lines.

Another approach: Use the fact that the sum of angles around the "bend" can be found using parallel line properties.

Imagine extending the middle segment. Or better — use the idea that for two parallel lines cut by a broken transversal, the sum of the “inside” angles on one side equals the sum on the other? Not quite.

Better method: Draw a line through the bend point parallel to the top and bottom lines. Then split angle d into two parts.

But maybe simpler: Look at the triangle formed? No, not exactly.

Wait — here’s a trick: When you have two parallel lines and a zigzag between them, the sum of the angles on one side equals the sum on the other? Let me think.

Actually, consider the angles along the transversal path.

From top line down: angle d and 68° are adjacent at the top vertex. At the bottom, 75° is given.

If we consider the entire figure, the difference in direction from top to bottom must account for the parallel lines.

Alternative idea: The angle between the top line and the first segment is (180 - d) because d is shown as the reflex? Wait no — looking at diagram, d is the acute/obtuse angle inside the bend.

Actually, standard problem: For two parallel lines connected by a broken line with two bends, the sum of the left-facing angles equals the sum of the right-facing angles? Not sure.

Let me assign variables.

Call the top-left angle x = 180 - d (if d is measured inside the bend).

Wait — perhaps easier: Use the fact that consecutive interior angles.

Actually, let’s do this:

Draw a line through the middle vertex (where 68° and d meet) parallel to the top and bottom lines.

Then, the 68° angle splits into two parts: one part corresponds to the bottom 75° angle via alternate angles? Not directly.

Wait — here’s a reliable method:

The sum of the angles on the “left” side of the zigzag should equal the sum on the “right” side relative to the parallel lines.

In many textbooks, for such a Z-shaped or N-shaped path between parallel lines, the following holds:

Top angle + Bottom angle = Middle angle? No.

Let’s calculate numerically.

Assume the direction change.

From top line going down-right, then down-left to bottom line.

The total deviation from original direction should be zero because lines are parallel.

The angle between top line and first segment: let’s say α = 180 - d (if d is the internal angle at top).

At the bend, the turn is 68°.

At the bottom, the angle between second segment and bottom line is 75°.

Since lines are parallel, the net rotation should be 0.

Going clockwise: from top line, turn by (180 - d) to go along first segment, then at bend turn by 68° (but direction?), then at bottom turn by 75° to align with bottom line.

This is getting messy.

Better approach: Use the formula for such configurations.

I recall that in such a setup with two parallel lines and a broken transversal making two angles on one side, the sum of the two outer angles equals the middle angle? Let’s test.

Suppose d + 75 = 68? That would give d negative — impossible.

Or d = 68 + 75? 143° — possible.

Check: If d = 143°, then the angle between top line and first segment is 180 - 143 = 37°.

Then at the bend, the angle inside is 68°, so the direction changes by 180 - 68 = 112°? Not helping.

Another idea: Consider the triangle formed if we connect endpoints — but not necessary.

Let’s look for similar problems.

Standard result: When two parallel lines are cut by a broken line with two segments, forming angles A, B, C as shown (A at top, B at bend, C at bottom), then A + C = B if the bends are on the same side.

In our case, d is at top, 68° at bend, 75° at bottom.

If d and 75° are on opposite sides of the transversal path, then d + 75 = 68? No, too small.

Perhaps d = 68 + 75 = 143°.

Let me verify with geometry.

Draw top line L1, bottom line L2, parallel.

Point P on L1, Q the bend point, R on L2.

Angle at P between L1 and PQ is, say, θ. Since d is shown as the angle inside the bend at P, probably d = 180 - θ.

Similarly, at R, angle between QR and L2 is 75°, so the angle inside might be 180 - 75 = 105°, but in diagram it's shown as 75° on the lower side.

At Q, the angle between PQ and QR is 68°.

Now, the direction from L1 to L2: the total turning angle should be consistent.

The vector from P to Q makes an angle φ with horizontal, then from Q to R makes angle ψ.

Since L1 and L2 are parallel, the net displacement vertical is fixed, but for angles, the sum of the deviations.

I found a better way: Use the fact that the sum of the interior angles on the same side.

Consider the polygon formed: points P, Q, R, and back along the lines — but it's not closed.

Add a line from P to R. Then we have triangle PQR.

In triangle PQR, we need angles.

At P: the angle between PQ and PR — but we don't know.

Perhaps overcomplicating.

Let me search my memory: In such diagrams, often d = 68 + 75 = 143°.

Why? Because the 68° is the "external" angle for the triangle formed, but let's assume that.

Calculate: if d = 143°, then the acute angle at top is 37°.

Then, the line PQ goes down at 37° below horizontal (assuming top line is horizontal).

At Q, it turns by 68° — if it turns towards the bottom line, the new direction might be 37° + 68° = 105° from horizontal, but then at R, it meets the bottom line at 75°, which would mean the angle between QR and horizontal is 75°, so 105° vs 75° — difference of 30°, not matching.

If at Q, it turns the other way: from 37° down, turn by 68° upwards, so new angle is 37° - 68° = -31°, i.e., 31° above horizontal, but bottom line is horizontal, so angle at R should be 31°, but it's given as 75° — not match.

Another try: Suppose the angle at P between the top line and PQ is x. Then d = 180 - x.

At Q, the angle between PQ and QR is 68°.

At R, the angle between QR and bottom line is y = 75°.

Since lines are parallel, the alternate interior angles should work if we consider the whole path.

The key insight: The sum of the angles on the "left" side of the transversal path equals the sum on the "right" side.

In this case, from top to bottom, the angles facing left are d and 75°, and the angle facing right is 68°? Or vice versa.

Standard rule for such a configuration: the middle angle equals the sum of the two outer angles if they are on the same side.

Here, d and 75° are both on the "inner" side, and 68° is on the other.

Perhaps d + 75 = 180 - 68? 180 - 68 = 112, so d = 112 - 75 = 37° — possible.

Let me try d = 37°.

Then at P, the angle between top line and PQ is 180 - 37 = 143°? That seems large.

If d = 37°, and it's the angle inside the bend, then the actual angle between the line and the transversal is 37°.

Assume top line is horizontal. From P, the line PQ goes down to the right at an angle of d = 37° below horizontal? But in the diagram, d is shown as the angle at the top vertex between the top line and the first segment, and it's labeled as d°, and it looks obtuse in the diagram? Wait, in the image description, it's not specified, but typically in such problems, d is the angle inside the "corner".

Looking back at the user's image description: "d°" is at the top vertex, between the top parallel line and the first segment of the zigzag, and it's shown as the angle inside the bend, which for a typical Z-shape, if the zigzag is like / \ , then d would be the angle at the top of the / part.

Perhaps it's easier to use the following method:

Extend the middle segment or use parallel line properties with auxiliary lines.

I recall that for two parallel lines cut by a broken line with two segments, the following holds: the sum of the two angles on the same side of the transversal equals the angle at the bend if it's on the other side.

After thinking, I remember a standard result: in such a diagram, d = 68° + 75° = 143°.

Let me confirm with calculation.

Suppose we draw a line through the bend point Q parallel to the top and bottom lines.

Then, this line divides the 68° angle into two parts: let's say α and β, with α + β = 68°.

Now, the angle between the top line and PQ is equal to α (alternate interior angles), and since d is adjacent to that, d + α = 180°, so α = 180 - d.

Similarly, the angle between the bottom line and QR is equal to β (alternate interior angles), and that is given as 75°, so β = 75°.

Then, α + β = 68°
(180 - d) + 75 = 68
255 - d = 68
d = 255 - 68 = 187° — impossible, greater than 180.

Mistake: if the auxiliary line is drawn, and if the 68° is the angle at Q between PQ and QR, then when we draw the parallel line through Q, it may not split the 68° if the segments are on the same side.

Perhaps the 68° is the reflex angle? Unlikely.

Another possibility: the angle at Q is 68°, but it's the smaller angle, and the auxiliary line creates corresponding angles.

Let's define:

Let the auxiliary line through Q be parallel to top and bottom.

Then, the angle between PQ and the auxiliary line is equal to the angle between top line and PQ, which is, say, γ.

Similarly, the angle between QR and the auxiliary line is equal to the angle between bottom line and QR, which is 75°.

Now, depending on the configuration, these two angles γ and 75° could be on the same side or opposite sides of the auxiliary line.

In the diagram, since the zigzag is like a lightning bolt, likely γ and 75° are on opposite sides of the auxiliary line, so the total angle at Q between PQ and QR is |γ - 75°| or γ + 75°.

Given that the angle at Q is 68°, and assuming it's the smaller angle, then if γ and 75° are on opposite sides, the angle between PQ and QR is γ + 75° = 68°, which would make γ negative — impossible.

If they are on the same side, then |γ - 75°| = 68°.

So either γ - 75 = 68 or 75 - γ = 68.

First case: γ = 68 + 75 = 143°

Second case: γ = 75 - 68 = 7°

Now, γ is the angle between PQ and the auxiliary line, which is equal to the angle between top line and PQ (since auxiliary || top line).

In the diagram, at the top, the angle d is shown as the angle between the top line and PQ, and it's likely the larger angle or the one inside.

If γ = 143°, then the angle between top line and PQ is 143°, so d = 143° (if d is that angle).

If γ = 7°, then d = 7°.

Which one makes sense? In the diagram, d is probably obtuse, so 143° is more likely.

Also, if d = 143°, then the acute angle is 37°, and at the bottom, 75°, and at bend 68°, and 37° + 75° = 112°, not 68°, but earlier calculation showed that if γ = 143°, and β = 75°, and they are on the same side, then the angle between PQ and QR is |143 - 75| = 68°, yes! Perfect.

So, if the auxiliary line is drawn, and γ = 143° (angle between PQ and aux line), β = 75° (angle between QR and aux line), and since both are measured from the aux line in the same direction (say, both below or both above), then the angle between PQ and QR is |143 - 75| = 68°, which matches.

And since the auxiliary line is parallel to top line, the angle between top line and PQ is also 143°, so d = 143°.

Yes!

So d = 143°

Reason: By drawing an auxiliary line parallel to the given parallel lines through the bend point, the angle d corresponds to the angle between the top line and the first segment, which equals the angle between the auxiliary line and the first segment. The difference between this angle and the angle at the bottom (75°) gives the bend angle (68°), leading to d = 75° + 68° = 143°.

More precisely: d - 75° = 68°, so d = 143°.

Because the two angles on the same side of the auxiliary line differ by the bend angle.

✔ Confirmed.

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Angle e and f:

We have two parallel lines (arrows on them). A triangle is formed with angles 39°, f°, and e° is at the intersection.

Specifically, there's a triangle with vertices: one at the bottom left with 39°, one at the top with f° and 81° adjacent, and one at the crossing with e°.

Note that e° is vertically opposite to the angle inside the triangle at that vertex.

Also, the two lines with arrows are parallel.

First, look at the top vertex: there's an angle of 81° shown, and f° is adjacent to it, and together they form a straight line? Or are they part of the triangle?

In the diagram, at the top vertex, the 81° is outside, and f° is inside the triangle.

Actually, the 81° and f° are adjacent angles that form a straight line with the parallel line.

Since the top line is straight, and f° and 81° are on it, so f + 81 = 180°? Is that correct?

Looking at the description: "f°" is at the top vertex of the triangle, and "81°" is the angle between the top parallel line and the side of the triangle.

Typically, in such diagrams, the 81° is the angle between the parallel line and the transversal, and f° is the angle inside the triangle at that vertex.

Moreover, since the two lines are parallel, we can use alternate angles.

Let me denote:

Let the triangle be ABC, with A at bottom left, B at top, C at the crossing point.

At A: angle is 39°.

At B: angle is f°.

At C: angle is, say, g°, but e° is vertically opposite to g°, so e = g.

Also, at B, there is an additional angle of 81° between the top parallel line and side BC.

Since the top line is straight, and side BA is another line, but actually, the 81° is likely the angle between the top parallel line and the side of the triangle that is not BA.

Assume that from point B, one side goes to A (down-left), and another side goes to C (down-right), and the top parallel line is horizontal.

Then, the angle between the top line and BC is 81°, and since the top line is straight, the angle between the top line and BA would be something else.

But f° is the angle inside the triangle at B, between BA and BC.

So, if the top line is straight, and BC makes 81° with it, then the angle between BA and the top line can be found if we know the direction.

Since the two lines are parallel, and we have a transversal, we can find alternate angles.

Notice that the side from B to C crosses the lower parallel line, and at the crossing, we have angle e°.

Also, the side from A to B is another transversal.

At point A, angle is 39°, which is between the lower parallel line and side AB.

Since the lines are parallel, the alternate interior angle to the 39° would be at the top, but on the other side.

Specifically, the angle between the top parallel line and side AB should be equal to 39° if it's alternate interior.

Is that correct? Side AB connects lower line to upper line, so yes, it's a transversal.

So, the angle that AB makes with the lower line at A is 39°, so the alternate interior angle at B between AB and the top line should also be 39°.

Therefore, at point B, the angle between AB and the top line is 39°.

But we also have that the angle between BC and the top line is 81°.

Now, depending on whether AB and BC are on the same side or opposite sides of the perpendicular.

In the diagram, likely AB is going down-left, BC down-right, so from the top line, AB is 39° below on the left, BC is 81° below on the right.

Then, the angle between AB and BC, which is f°, would be the sum of these two angles: 39° + 81° = 120°.

Because they are on opposite sides of the vertical.

So f = 39 + 81 = 120°.

Then, in the triangle, angles sum to 180°.

We have angle at A: 39°, angle at B: f = 120°, so angle at C: 180 - 39 - 120 = 21°.

Now, angle e° is vertically opposite to this angle at C, so e = 21°.

Vertically opposite angles are equal.

So e = 21°.

Confirm: yes.

So:

f = 120°

Reason: The angle between AB and the top parallel line is 39° (alternate interior angles with the 39° at A), and the angle between BC and the top line is 81°, and since they are on opposite sides, the angle between AB and BC is 39° + 81° = 120°.

e = 21°

Reason: Vertically opposite angles are equal; the angle in the triangle at the crossing point is 180° - 39° - 120° = 21°, and e is vertically opposite to it.

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Angle g:

We have a triangle with angles 74°, 21°, and g° is at the top, but g° is shown as the exterior angle or something.

Diagram: triangle with angles 74° at bottom left, 21° at top right? And g° is at the top vertex, but outside.

Specifically, there is a triangle, and at the top vertex, there is an angle g° which is adjacent to the triangle's angle.

Also, there are arrows indicating parallel lines? In this diagram, only one pair of sides has arrows? Looking back: "two arrows on two sides" — probably the two non-base sides have arrows, meaning they are parallel? But in a triangle, sides can't be parallel unless degenerate.

Re-read: in the last diagram, it says "two arrows" on two sides, but in a triangle, if two sides have arrows in the same direction, it might indicate they are parallel, but that's impossible for a triangle.

Perhaps the arrows indicate that those sides are part of parallel lines, but it's a single triangle.

Looking at the description: "g°" is at the top vertex, and there is a line extending out, and arrows on two sides of the triangle.

Probably, the two sides with arrows are meant to be parallel to some other lines, but in this context, likely the arrows indicate that the two sides are parallel to each other? But that can't be.

Another interpretation: perhaps the arrows show that the two sides are cut by a transversal, but it's confusing.

In many such problems, when a triangle has two sides with arrows, it might mean that those sides are parallel to corresponding sides in another figure, but here it's standalone.

Perhaps the arrows indicate direction for angle purposes, but I think in this case, the key is that g° is an exterior angle.

Look at the triangle: angles given are 74° and 21°, so the third angle is 180 - 74 - 21 = 85°.

Then, g° is adjacent to this 85° angle, and they form a straight line, so g + 85 = 180, thus g = 95°.

But why are there arrows? Perhaps to indicate that the lines are straight, or for parallel lines elsewhere, but in this diagram, no other lines.

In the user's description: "two arrows on two sides" — probably indicating that those two sides are parallel to each other, but in a triangle, that's impossible unless it's degenerate.

Unless... perhaps it's not a triangle alone; maybe there is another line.

Re-examining: in the last diagram, it might be that the two sides with arrows are part of parallel lines, but the figure is a triangle with an extension.

Standard interpretation: g° is the exterior angle at the top vertex.

In a triangle, the exterior angle is equal to the sum of the two opposite interior angles.

So, if the two given angles are 74° and 21°, and they are not adjacent to g, then g = 74 + 21 = 95°.

Yes, that makes sense.

And the arrows might be red herrings or to indicate the sides are straight, but in this case, since no other parallel lines are involved, probably g is simply the exterior angle.

To confirm: if the interior angle at the top is 180 - 74 - 21 = 85°, then the adjacent exterior angle g = 180 - 85 = 95°, or directly g = 74 + 21 = 95° by exterior angle theorem.

So g = 95°

Reason: Exterior angle of a triangle is equal to the sum of the two opposite interior angles.

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Now, summarize all answers:

- Angle a: 72°
- Angle b: 113°
- Angle c: 106°
- Angle d: 143°
- Angle e: 21°
- Angle f: 120°
- Angle g: 95°

With reasons as above.

Final Answer:

Angle a: 72°
Reason: Alternate interior angles are equal.

Angle b: 113°
Reason: Co-interior angles between parallel lines sum to 180°.

Angle c: 106°
Reason: Angles on a straight line sum to 180° (after finding the interior angle of the quadrilateral).

Angle d: 143°
Reason: Using an auxiliary parallel line, the angle d equals the sum of the bottom angle and the bend angle (75° + 68°).

Angle e: 21°
Reason: Vertically opposite angles are equal; calculated from triangle angle sum.

Angle f: 120°
Reason: Sum of alternate interior angle (39°) and given angle (81°) at the vertex.

Angle g: 95°
Reason: Exterior angle of a triangle equals sum of two opposite interior angles (74° + 21°).