Triangle exterior angles worksheet with labeled diagrams for calculating angle measures.

Let’s solve each triangle problem one by one. We’ll use the Exterior Angle Theorem, which says:

> The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Also, remember:
→ The sum of all three interior angles in any triangle is always 180°.
→ If you see an exterior angle (outside the triangle), it forms a straight line with its adjacent interior angle → so they add up to 180°.

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Problem 1:

Triangle PQR. Exterior angle at Q is labeled as ∠OQR = ?
Interior angles given: ∠P = 64°, ∠R = 57°

We can find ∠Q first using triangle angle sum:
∠Q = 180° - 64° - 57° = 59°

Then, since ∠OQR is exterior and next to ∠Q, they form a straight line:
∠OQR = 180° - 59° = 121°

✔ OR — use Exterior Angle Theorem directly:
Exterior angle = sum of two opposite interior angles = 64° + 57° = 121°

Same answer!

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Problem 2:

Triangle UVW. Exterior angle at U is ∠XUV = ?
Interior angles given: ∠V = 33°, ∠W = 29°

Use Exterior Angle Theorem:
∠XUV = ∠V + ∠W = 33° + 29° = 62°

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Problem 3:

Triangle XYZ. Exterior angle at Y is ∠XYK = ?
Interior angles given: ∠X = 38°, ∠Z = 56°

Exterior Angle Theorem:
∠XYK = ∠X + ∠Z = 38° + 56° = 94°

Wait — let me double-check the diagram description. It says “m∠XYK” — that’s the exterior angle at Y? Actually, looking at standard labeling, if K is outside near Y, then yes — it should be the exterior angle formed by extending side XY or ZY.

But actually, in most diagrams like this, if it's labeled ∠XYK, and K is on the extension beyond Y, then yes — it’s the exterior angle at vertex Y.

So again: exterior angle = sum of remote interiors = 38° + 56° = 94°

BUT wait — hold on! Let me check again. In some cases, if the angle marked is NOT the exterior but something else... No, based on typical worksheet layout, and since it says “exterior angles” in title, we’re safe.

Actually — correction: Looking back at problem 3, the angle asked is m∠XYK — and from the sketch description, point K is likely on the extension of side YZ past Y, making ∠XYK the exterior angle at Y.

Yes — so remote interiors are X and Z → 38° + 56° = 94°

Wait — no! Wait a second — I think I made a mistake here.

If ∠XYK is the exterior angle at Y, then the two remote interior angles are ∠X and Z — yes.

But let me verify with another method.

Total interior angles: ∠X + Y + ∠Z = 180°
→ 38° + ∠Y + 56° = 180° → ∠Y = 86°

Then exterior angle at Y = 180° - 86° = 94° — same result.

Good.

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Problem 4:

Triangle JKL. Exterior angle at K is ∠JKL? Wait — no, it says m∠JKL — but that’s an interior angle?

Wait — look again: The label says “m∠JKL = ___”

In triangle JKL, points J, K, L. If ∠JKL is written, that’s the angle at K — which is INSIDE the triangle.

But the section is called “Exterior Angles”. Hmm.

Looking at the diagram description: There’s a point M outside, connected to K, forming angle ∠JKM or something?

Actually, re-reading: “m∠JKL” — but in the diagram, there’s probably an exterior angle shown.

Wait — perhaps it’s mislabeled? Or maybe ∠JKL is meant to be the exterior?

No — standard notation: ∠JKL means angle at K between points J-K-L — that’s interior.

But the problem says “find the measure of the indicated angle”, and in context of exterior angles chapter, likely it’s asking for the exterior angle.

Wait — looking at your original input: For #4, it says “m∠JKL = ___”

But in many worksheets, when they write ∠JKL and show an external ray, sometimes it’s still referring to the exterior.

Alternatively — perhaps it’s a typo and should be ∠JKM or similar.

But let’s assume based on common problems: In triangle JKL, with exterior angle at K, and interior angles given: ∠J = 47°, ∠L = 31°

Then exterior angle at K = ∠J + ∠L = 47° + 31° = 78°

And if the question is asking for the exterior angle (even if labeled ∠JKL by mistake), then answer is 78°.

Alternatively, if ∠JKL is truly the interior angle, then:

∠K = 180° - 47° - 31° = 102°, and exterior would be 78° — but the blank is labeled “m∠JKL”, which would be 102°.

This is confusing.

Wait — let me think differently. In the user’s image description, for #4, it might be that point M is drawn such that KM is an extension, and angle being asked is ∠JKM or ∠LKM.

But since the text says “m∠JKL”, and in standard geometry, that’s interior.

However, given the section title is “Exterior Angles”, and other problems ask for exterior angles, it’s very likely that for #4, even though it says ∠JKL, it’s meant to be the exterior angle — or perhaps it’s a different labeling.

Another possibility: Maybe ∠JKL is the exterior angle because L is on the extension? Unlikely.

Let me check online or recall standard problems.

Actually — I found a better way: Let’s calculate both ways.

If interior ∠K = 180 - 47 - 31 = 102°, then exterior = 78°.

Now, if the problem intends to ask for the exterior angle, answer is 78°.

Given that all other problems are clearly exterior angles, and this is under “Exterior Angles” section, I will assume that despite the label “∠JKL”, it’s meant to be the exterior angle at K.

Perhaps it’s a misprint, and it should be ∠JKM or ∠MKL.

To resolve: Let’s look at the value. 47+31=78 — nice number. 102 is also possible, but less likely for exterior angle section.

I think safest bet: Since the section is exterior angles, and the angle indicated is probably the exterior one, I'll go with 78°.

But to be precise — let’s note: If ∠JKL is interior, answer is 102°; if exterior, 78°. Given context, I choose 78°.

Wait — actually, in the original problem statement from user: “Find the measure of the indicated angle in each triangle.” and for #4, it’s “m∠JKL”

In many textbooks, when they draw an exterior angle, they might still call it by the vertex and one side, but specify.

Perhaps I should calculate the interior first.

Let me do this: Assume ∠JKL is the interior angle at K.

Then ∠K = 180 - 47 - 31 = 102°

But then why is this in "exterior angles" section? Unless they want us to find the exterior, but labeled wrong.

Another idea: Perhaps "∠JKL" is a typo and it's "∠JKM" where M is outside.

Given the ambiguity, and since 47+31=78 is clean, and matches exterior angle theorem, I'll go with 78° for the exterior angle.

Final decision: 78°

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Problem 5:

Triangle MNO. Exterior angle at N is ∠MNX = ?
Interior angles given: ∠M = 41°, ∠O = 68°

Exterior Angle Theorem:
∠MNX = ∠M + ∠O = 41° + 68° = 109°

Check: Interior ∠N = 180 - 41 - 68 = 71°, so exterior = 180 - 71 = 109° — correct.

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Problem 6:

Triangle STU. Exterior angle at T is ∠STX = ?
Interior angles given: ∠S = 62°, ∠U = 55°

Exterior Angle Theorem:
∠STX = ∠S + ∠U = 62° + 55° = 117°

Check: Interior ∠T = 180 - 62 - 55 = 63°, exterior = 180 - 63 = 117° — good.

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Problem 7:

Triangle CDE. Exterior angle at D is ∠CDE? Wait — again, label issue.

It says “m∠CDE = ___”

In triangle CDE, ∠CDE is angle at D — interior.

But section is exterior angles. Likely, it’s meant to be the exterior angle.

Interior angles given: ∠C = 36°, ∠E = 45°

If ∠CDE is interior, then ∠D = 180 - 36 - 45 = 99°, exterior = 81°

But if they want exterior, it should be 81°.

Again, conflict in labeling.

Given pattern, I think they mean the exterior angle.

So exterior at D = ∠C + E = 36° + 45° = 81°

I’ll go with that.

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Problem 8:

Triangle BCD. Exterior angle at C is ∠BCX = ?
Interior angles given: ∠B = 64°, ∠D = 37°

Exterior Angle Theorem:
∠BCX = ∠B + D = 64° + 37° = 101°

Check: Interior ∠C = 180 - 64 - 37 = 79°, exterior = 180 - 79 = 101° — correct.

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Now, compiling answers:

1. 121°
2. 62°
3. 94°
4. 78° *(assuming exterior angle despite label)*
5. 109°
6. 117°
7. 81° *(assuming exterior angle)*
8. 101°

For problems 4 and 7, if the label strictly means interior angle, answers would be 102° and 99° respectively, but given the section title and consistency, I believe exterior is intended.

To confirm, let’s see if there’s any clue in the numbering or diagram — but since we don’t have visual, we rely on context.

In most such worksheets, when they say “indicated angle” under “exterior angles”, and show a small arc outside, it’s exterior.

So final answers:

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Final Answer:
1) 121°
2) 62°
3) 94°
4) 78°
5) 109°
6) 117°
7) 81°
8) 101°