Angle Relationships interactive worksheet - Free Printable

Let’s go step by step through each question.

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Part A: Fill in the correct angle.

We’re looking at a diagram with lines intersecting at point X. There are rays going to points A, B, C, D, E, F. We can see that:

- Line AD is horizontal (left-right).
- Line BE is vertical? Wait — actually, from the diagram, it looks like line BF and AE are diagonal, but there’s a right angle mark between AXB — so ∠AXB = 90°. That means ray XB is perpendicular to line AD.

Also, we can assume that lines EC and FD are straight lines passing through X? Actually, looking closely: Points E-X-C appear to be on one straight line, and F-X-D? Not exactly — let’s think based on standard vertical angles.

Vertical angles are opposite each other when two lines cross.

Supplementary angles add up to 180°.

Complementary angles add up to 90°.

Adjacent angles share a side and vertex.

Let’s tackle each:

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1) ∠AXE and ______ are vertical angles.

∠AXE is formed by rays XA and XE.

The vertical angle would be the angle directly opposite — which should be formed by the opposite rays: XD and XC? Wait — if E-X-C is a straight line, then yes.

Actually, if you look: Ray XA goes left, ray XE goes down-left. The opposite of that would be ray XD (right) and ray XC (up-right). So ∠DXC is vertical to ∠AXE.

Wait — let me double-check.

If two lines intersect, vertical angles are across from each other.

Assume line AC and line ED intersect at X? Hmm.

Actually, from the diagram, it seems that line EC passes through X, and line AD passes through X, and line BF also passes through X? But only some are straight.

Looking again: The right angle is at ∠AXB, meaning XB ⊥ AD.

So AD is a straight line (horizontal), XB is vertical upward.

Then, ray XC is going up-right, ray XF is going down-right, ray XE is going down-left.

Probably, line EC is straight — so E-X-C is a straight line.

Similarly, maybe F-X-B? Not sure.

But for vertical angles: When two lines intersect, they form two pairs of vertical angles.

So if line AD and line EC intersect at X, then:

∠AXE and ∠DXC are vertical angles.

Yes — because A and D are opposite on the horizontal line, E and C are opposite on the diagonal line.

So answer: ∠DXC

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2) ∠AXF and ______ are supplementary angles.

Supplementary means they add to 180°.

∠AXF is the angle from XA to XF.

Since AD is a straight line, any angle on one side plus the adjacent angle on the same line will be 180°.

So ∠AXF + ∠FXD = 180°, because together they make straight line AD.

Is that right? Point F is below, so from A to X to F, then F to X to D — yes, those two angles together make the straight angle along AD.

So ∠AXF and ∠FXD are supplementary.

But wait — is ∠FXD the same as ∠DXF? Yes.

So answer: ∠DXF or ∠FXD

But let’s check notation — usually we write the middle letter as vertex, so ∠FXD is fine.

Alternatively, since F-X-D might not be straight, but A-X-D is straight, so any angle starting at A and ending at F, plus the angle from F to D, should be 180°.

Yes.

Another possibility: ∠AXF and ∠FXB? No, that’s not necessarily 180.

Best answer: ∠FXD

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3) ∠DXC and ______ are complementary angles.

Complementary = add to 90°.

We know ∠AXB = 90° (right angle marked).

∠DXC is part of the upper right.

Notice that ∠DXC + ∠CXB = ∠DXB.

But ∠DXB is part of the right angle? Since XB is perpendicular to AD, then ∠BXD = 90°.

Because AD is straight, XB ⊥ AD, so ∠AXB = 90°, ∠BXD = 90°.

Now, ∠BXD is made of ∠BXC + ∠CXD.

So ∠CXD + ∠BXC = 90°.

Therefore, ∠DXC and ∠BXC are complementary.

Answer: ∠BXC

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4) ______ and ∠AXB are adjacent angles.

Adjacent angles share a common side and vertex, and don’t overlap.

∠AXB is between XA and XB.

What shares a side with it?

For example, ∠BXC shares side XB.

Or ∠AXE shares side XA.

Both are adjacent.

But probably they want the one next to it on the diagram.

Since ∠AXB is 90°, and above it is ∠BXC, sharing XB.

Below it is ∠AXE, sharing XA.

Either could work, but likely they expect ∠BXC or ∠AXE.

Looking at the diagram order, perhaps ∠BXC is more natural.

But let’s see — adjacent just needs to share a side.

I’ll go with ∠BXC — it shares ray XB.

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5) ______ and ∠CXD are supplementary angles.

Supplementary = 180°.

∠CXD is in the upper right.

What adds to 180° with it?

If we consider line EC (assuming E-X-C is straight), then ∠CXD + ∠DXE = 180°? Not necessarily.

Better: Since AD is straight, and C is above, then ∠CXD + ∠CXA = 180°? Let's see.

Points: A-X-D straight.

Ray XC is going up-right.

So angle between XC and XD is ∠CXD.

Angle between XC and XA is ∠CXA.

And since A-X-D is straight, ∠CXA + ∠CXD = 180°.

Yes! Because they form a linear pair on line AD.

So answer: ∠CXA

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6) ______ and ∠AXC are vertical angles.

∠AXC is from XA to XC.

Vertical angle would be opposite: from XD to XE.

Because if line AD and line EC intersect at X, then ∠AXC and ∠DXE are vertical angles.

Yes.

So answer: ∠DXE

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Part B: Fill in the correct angle measurement.

7) Complement of 11°

Complement = 90° - given angle

90 - 11 = 79°

Answer: 79°

8) Supplement of 92°

Supplement = 180° - given angle

180 - 92 = 88°

Answer: 88°

9) Complement of 56°

90 - 56 = 34°

Answer: 34°

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For #10–12, use the diagram to the right.

Diagram shows two lines crossing: one horizontal, one diagonal.

Angles labeled: 41°, then 2, 3, 4 around the intersection.

Standard setup: when two lines intersect, vertical angles are equal, adjacent angles are supplementary.

Given: one angle is 41°.

Assuming it’s angle 1 (not labeled, but implied).

In the diagram, it says “41°” near angle between left and top-left ray.

Then angle 2 is adjacent to it on the top.

Angle 3 is below, angle 4 is on the right.

Typically:

- Angle 1 = 41°

- Angle 2 is adjacent → 180 - 41 = 139°

- Angle 3 is vertical to angle 1 → 41°

- Angle 4 is vertical to angle 2 → 139°

But let’s confirm labeling.

The diagram has:

Left ray, right ray (horizontal line)

Top-left ray, bottom-right ray (diagonal line)

Angle between left ray and top-left ray is 41° — that’s angle 1.

Then angle 2 is between top-left ray and right ray.

Angle 3 is between right ray and bottom-right ray? No.

Actually, standard labeling: going clockwise or counterclockwise.

Usually, angle 1 and 3 are vertical, 2 and 4 are vertical.

If angle 1 is 41°, then angle 3 = 41° (vertical)

Angle 2 = 180 - 41 = 139° (supplementary to angle 1)

Angle 4 = 139° (vertical to angle 2)

In the problem, it labels:

m∠2 = ?

m∠3 = ?

m∠4 = ?

And the 41° is shown where angle 1 would be.

So:

10) m∠2 = 180 - 41 = 139°

11) m∠3 = 41° (vertical to 41° angle)

12) m∠4 = 139° (vertical to ∠2)

Yes.

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Final Answers:

1) ∠DXC
2) ∠FXD
3) ∠BXC
4) ∠BXC (or ∠AXE, but ∠BXC is better)
Wait — for #4, I said ∠BXC, but let me confirm.

∠AXB is between XA and XB.

Adjacent angles: could be ∠BXC (shares XB) or ∠AXE (shares XA).

But in many textbooks, they might expect the one continuing the rotation.

Perhaps both are acceptable, but I think ∠BXC is fine.

Actually, looking back, in the diagram, after ∠AXB, going clockwise is ∠BXC, so likely that’s intended.

But to be safe, let’s see if there’s a standard.

I think it’s okay.

For #2, I said ∠FXD, but is that correct?

∠AXF and ∠FXD: do they form a straight line?

Point F is below, so from A to X to F, then F to X to D — yes, since A-X-D is straight, and F is another ray, so ∠AXF + ∠FXD = angle from A to D via F, which is 180° only if F is on the line, but it’s not.

Mistake here!

If A-X-D is straight, and F is not on that line, then ∠AXF and ∠FXD are not necessarily supplementary unless F is on the line, which it’s not.

Oh no!

Let me rethink.

Supplementary angles don't have to be adjacent; they just need to sum to 180°.

But in this context, probably they mean adjacent supplementary, i.e., linear pair.

But the question says "are supplementary angles", not necessarily adjacent.

However, in geometry worksheets, often they imply adjacent when saying "supplementary" in such contexts.

But technically, any two angles that add to 180° are supplementary.

But for ∠AXF, what angle is supplementary to it?

It could be many, but likely they want the one that forms a linear pair.

Since A-X-D is straight, the angle adjacent to ∠AXF on the other side of XF would be... wait.

Actually, the ray XF divides the plane.

The angle ∠AXF is on one side.

The supplement would be the angle that completes to 180° with it.

But without more info, perhaps we need to find an angle that is clearly supplementary.

Note that ∠AXF and ∠FXB might not be.

Another idea: since XB is perpendicular, ∠AXB = 90°.

∠AXF includes ∠AXB and ∠BXF? Depending on position.

From the diagram, F is below, so from XA to XF, it goes through XB? Probably not.

Assume the order of rays around X.

Starting from XA (left), then going clockwise: XA, then XE (down-left), then XF (down-right), then XD (right), then XC (up-right), then XB (up), back to XA.

But XB is up, and there's a right angle at AXB, so from XA to XB is 90°.

Then from XB to XD is another 90°.

Now, ray XC is between XB and XD, ray XF is between XE and XD, etc.

For ∠AXF: from XA to XF.

This angle spans from left to down-right.

The total around point X is 360°.

But to find a supplementary angle, we need one that adds to 180° with ∠AXF.

Perhaps ∠FXC or something.

This is messy.

Alternative approach: in many such diagrams, when they say "supplementary", and it's on a straight line, they mean the adjacent angle on the straight line.

But for ∠AXF, it's not on the straight line AD alone.

Unless... perhaps ∠AXF and ∠FXD are not supplementary, but ∠AXF and the angle on the other side.

Let's calculate roughly.

Suppose ∠AXB = 90°.

Assume that ∠BXC = x, then ∠CXD = 90° - x, since ∠BXD = 90°.

Similarly, on the bottom, ∠AXE = y, ∠EXF = z, etc.

But we don't have values.

For question 2, perhaps they intend ∠FXD, but as I said, it may not be correct.

Another thought: perhaps "supplementary" here means they form a linear pair, so share a side and their non-common sides form a straight line.

For ∠AXF, the non-common side with what? If I take ∠FXD, then common side is XF, and non-common sides are XA and XD, which are on a straight line! Yes!

XA and XD are opposite rays, forming straight line AD.

So ∠AXF and ∠FXD share the ray XF, and their other rays XA and XD form a straight line.

Therefore, they are adjacent and supplementary — a linear pair.

So yes, ∠AXF and ∠FXD are supplementary.

My initial answer was correct.

Similarly for others.

So I'll stick with that.

For #4, adjacent to ∠AXB: could be ∠BXC or ∠AXE.

But in the diagram, since ∠AXB is at the top-left, and ∠BXC is next to it clockwise, probably ∠BXC is expected.

To confirm, let's see if there's only one answer.

I think it's fine.

So final answers:

1) ∠DXC
2) ∠FXD
3) ∠BXC
4) ∠BXC
5) ∠CXA
6) ∠DXE
7) 79°
8) 88°
9) 34°
10) 139°
11) 41°
12) 139°

For #4, is ∠BXC the only choice? What about ∠AXE? It also shares a side.

But perhaps the worksheet expects ∠BXC.

Maybe I should write both, but the blank is for one.

Looking at the diagram, after ∠AXB, the next angle is ∠BXC, so likely that's it.

I think it's acceptable.

So I'll proceed.

Final Answer:
1) ∠DXC
2) ∠FXD
3) ∠BXC
4) ∠BXC
5) ∠CXA
6) ∠DXE
7) 79°
8) 88°
9) 34°
10) 139°
11) 41°
12) 139°