Triangle Sum Theorem Notes and Worksheets - Lindsay Bowden - Free Printable

Let's solve each problem step by step using the Triangle Sum Theorem, which states:

> In a triangle, the 3 interior angles sum to 180°.

So:
$$
m\angle A + m\angle B + m\angle C = 180^\circ
$$

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Problem 1:

Given angles: $59^\circ$, $75^\circ$, and $x^\circ$

Use the triangle sum theorem:
$$
59 + 75 + x = 180
$$
$$
134 + x = 180
$$
$$
x = 180 - 134 = 46
$$

✔ Answer: $x = 46^\circ$
Missing angle: $46^\circ$

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Problem 2:

Given angles: $74^\circ$, $139^\circ$, and $x^\circ$

Wait — sum of two angles is already $74 + 139 = 213^\circ > 180^\circ$ — that’s impossible in a triangle.

But let's double-check: Is this possible?

No! The sum of all three angles must be exactly $180^\circ$. So if two angles are $74^\circ$ and $139^\circ$, their sum is $213^\circ$, which exceeds $180^\circ$. That can't happen.

But wait — maybe the $139^\circ$ is an exterior angle? But the diagram shows it inside the triangle.

Actually, looking at the drawing: It's a triangle with two labeled angles: $74^\circ$ and $139^\circ$. But since $74 + 139 = 213 > 180$, this is not possible.

Wait — perhaps one of them is not an interior angle?

But the problem says "interior angles" and the notation suggests both are interior.

This implies there may be a mistake in interpretation.

But let's suppose the triangle has angles $74^\circ$, $x^\circ$, and $139^\circ$. Then:

$$
74 + 139 + x = 180 \Rightarrow 213 + x = 180 \Rightarrow x = -33
$$

Impossible.

Therefore, one of these cannot be an interior angle.

But based on the diagram, it looks like both are interior angles. However, a triangle cannot have two angles greater than 90° unless it's obtuse, but even then, the total must not exceed 180°.

Wait — 139° is an interior angle, so the other two must add to $180 - 139 = 41^\circ$. But we're given $74^\circ$, which is already more than $41^\circ$. So contradiction.

✘ This triangle as drawn is impossible.

But likely, the 139° is meant to be an exterior angle, or there’s a typo.

Alternatively, maybe the triangle has angles $74^\circ$, $x^\circ$, and the third is $139^\circ$? But again, $74 + 139 = 213 > 180$, so no.

Wait — perhaps the $139^\circ$ is not an interior angle? Let's re-express.

Looking carefully: In the diagram, it's a triangle with one angle marked $74^\circ$, another marked $139^\circ$, and the third $x^\circ$. But since $74 + 139 = 213 > 180$, this is impossible.

So either:
- There's a typo.
- Or $139^\circ$ is an exterior angle.

But the problem says "solve for x", and "find missing angles", implying all are interior.

Wait — could it be that the $139^\circ$ is actually the supplement of an interior angle?

For example, if an exterior angle is $139^\circ$, then the interior angle is $180 - 139 = 41^\circ$.

That would make sense.

But the diagram doesn’t indicate that. It just labels $139^\circ$ inside the triangle.

Assuming the diagram is correct and all angles are interior, then this triangle is impossible.

But since this is a worksheet, likely a typo — perhaps it's supposed to be $59^\circ$ instead of $139^\circ$?

Alternatively, maybe the $139^\circ$ is not an interior angle — but the diagram shows it inside.

Wait — look again: In Problem 2, the triangle has two angles labeled: $74^\circ$ and $139^\circ$, and $x^\circ$.

But $74 + 139 = 213 > 180$, so impossible.

Unless... it's not a triangle? No, it's clearly labeled as a triangle.

So the only logical conclusion is that one of the angles is mislabeled, or perhaps $139^\circ$ is an exterior angle.

But the problem says “triangles may not be drawn to scale”, but doesn’t say anything about exterior angles.

Let’s assume that the 139° is an interior angle, and the $74^\circ$ is also interior. Then:

$$
74 + 139 + x = 180 \Rightarrow x = 180 - 213 = -33^\circ
$$

Not possible.

So either the problem has a typo, or we’re misunderstanding.

Wait — perhaps the $139^\circ$ is not an interior angle — maybe it's a typo and should be $39^\circ$?

Or maybe the triangle has angles $74^\circ$, $x^\circ$, and the third is $139^\circ$? Still same issue.

Alternatively, maybe the $139^\circ$ is an exterior angle, and we need to find the interior angle.

But the problem says “find all missing angles” — so likely all are interior.

Given the inconsistency, let's skip and come back.

Wait — perhaps the triangle has angles $74^\circ$, $x^\circ$, and the third is $139^\circ$ — but that’s impossible.

Wait — unless the triangle is not a standard triangle? No.

I think there might be a typo. Let's assume it's $39^\circ$ instead of $139^\circ$. But we can't assume.

Alternatively, maybe the $139^\circ$ is an exterior angle, and the interior angle is $180 - 139 = 41^\circ$.

Then:

$$
74 + 41 + x = 180 \Rightarrow 115 + x = 180 \Rightarrow x = 65^\circ
$$

Then the missing interior angles are $41^\circ$ and $65^\circ$.

But the problem says “find all missing angles”, and labels $x^\circ$ as the unknown.

But if $139^\circ$ is an exterior angle, then the adjacent interior angle is $41^\circ$, and $x$ is the third interior angle.

But the diagram shows $139^\circ$ inside the triangle, so likely it's an interior angle.

This is confusing.

Let’s check the other problems — they seem consistent.

Perhaps it’s a mistake in the worksheet.

But let's try to proceed with the assumption that the triangle has interior angles 74°, x°, and 139°, but that gives negative angle.

So no solution exists unless we reinterpret.

Wait — perhaps the $139^\circ$ is not an interior angle — maybe it's a label for an angle outside.

But the diagram shows it inside.

Alternatively, maybe the triangle has angles $74^\circ$, $x^\circ$, and the third is $139^\circ$ — but again, impossible.

I think there's a typo. Likely, it's meant to be $39^\circ$ instead of $139^\circ$.

Let’s assume that.

Then:
$$
74 + 39 + x = 180 \Rightarrow 113 + x = 180 \Rightarrow x = 67^\circ
$$

But without confirmation, we can't assume.

Alternatively, perhaps the $139^\circ$ is an exterior angle, and the interior angle is $180 - 139 = 41^\circ$.

Then:
$$
74 + 41 + x = 180 \Rightarrow x = 65^\circ
$$

And the missing angles are $41^\circ$ and $65^\circ$.

But the problem asks to solve for $x$, and $x$ is labeled at the vertex opposite the $139^\circ$ angle.

If $139^\circ$ is exterior, then the interior angle is $41^\circ$, and $x$ is the third interior angle.

But the labeling is ambiguous.

Given that the problem says “triangles may not be drawn to scale”, but doesn’t mention exterior angles, and since $74 + 139 = 213 > 180$, I suspect a typo.

But let’s move on and return later.

---

Problem 3:

Angles: $(3x + 2)^\circ$, $113^\circ$, $(2x)^\circ$

Sum to $180^\circ$:
$$
(3x + 2) + 113 + (2x) = 180
$$
$$
5x + 115 = 180
$$
$$
5x = 65
$$
$$
x = 13
$$

Now find angles:
- $3x + 2 = 3(13) + 2 = 39 + 2 = 41^\circ$
- $2x = 2(13) = 26^\circ$
- Given: $113^\circ$

Check: $41 + 113 + 26 = 180^\circ$ ✔

✔ Answer: $x = 13$, angles are $41^\circ$, $113^\circ$, $26^\circ$

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Problem 4:

Right triangle (right angle symbol), so one angle is $90^\circ$

Other angles: $32^\circ$, $2x^\circ$

Sum:
$$
32 + 90 + 2x = 180
$$
$$
122 + 2x = 180
$$
$$
2x = 58
$$
$$
x = 29
$$

So $2x = 58^\circ$

Angles: $32^\circ$, $58^\circ$, $90^\circ$ → sum = $180^\circ$ ✔

✔ Answer: $x = 29$, angles: $32^\circ$, $58^\circ$, $90^\circ$

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Problem 5:

Angles: $(3x - 15)^\circ$, $(2x - 2)^\circ$, $(x + 41)^\circ$

Sum to $180^\circ$:
$$
(3x - 15) + (2x - 2) + (x + 41) = 180
$$
$$
(3x + 2x + x) + (-15 - 2 + 41) = 180
$$
$$
6x + 24 = 180
$$
$$
6x = 156
$$
$$
x = 26
$$

Now find angles:
- $3x - 15 = 3(26) - 15 = 78 - 15 = 63^\circ$
- $2x - 2 = 2(26) - 2 = 52 - 2 = 50^\circ$
- $x + 41 = 26 + 41 = 67^\circ$

Check: $63 + 50 + 67 = 180^\circ$ ✔

✔ Answer: $x = 26$, angles: $63^\circ$, $50^\circ$, $67^\circ$

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Problem 6:

Given:
- $m\angle ABC = (15x - 4)^\circ$
- $m\angle BCA = (4x + 5)^\circ$
- $m\angle BAC = 65^\circ$

Sum to $180^\circ$:
$$
(15x - 4) + (4x + 5) + 65 = 180
$$
$$
19x + 66 = 180
$$
$$
19x = 114
$$
$$
x = 6
$$

Now find angles:
- $\angle ABC = 15(6) - 4 = 90 - 4 = 86^\circ$
- $\angle BCA = 4(6) + 5 = 24 + 5 = 29^\circ$
- $\angle BAC = 65^\circ$

Check: $86 + 29 + 65 = 180^\circ$ ✔

✔ Answer: $x = 6$, angles: $86^\circ$, $29^\circ$, $65^\circ$

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Back to Problem 2:

We now have a conflict.

But let's reconsider: Perhaps the $139^\circ$ is not an interior angle, but an exterior angle?

Suppose the triangle has angles: $74^\circ$, $x^\circ$, and the third interior angle is $y^\circ$. Then:

Exterior angle at one vertex is $139^\circ$, so the interior angle is $180 - 139 = 41^\circ$

Then:
$$
74 + x + 41 = 180 \Rightarrow x = 65^\circ
$$

So the missing angles are $41^\circ$ and $65^\circ$

But the problem labels $x^\circ$ as the unknown angle, and $139^\circ$ is shown inside.

But if $139^\circ$ is outside, then it's an exterior angle.

But the diagram shows it inside the triangle.

However, in some diagrams, the angle might be labeled incorrectly.

Alternatively, maybe the triangle has angles $74^\circ$, $x^\circ$, and $139^\circ$, but that's impossible.

Another possibility: Maybe the $139^\circ$ is not part of the triangle — but it's labeled as such.

Wait — perhaps the triangle is not a triangle? No.

I think the most plausible explanation is that the $139^\circ$ is a typo, and it should be $39^\circ$.

Then:
$$
74 + 39 + x = 180 \Rightarrow x = 67^\circ
$$

But since we can't change the problem, let's consider the only way this makes sense: the $139^\circ$ is an exterior angle.

Then the interior angle is $180 - 139 = 41^\circ$

Then:
$$
74 + 41 + x = 180 \Rightarrow x = 65^\circ
$$

So the missing angles are $41^\circ$ and $65^\circ$

Even though the diagram shows $139^\circ$ inside, perhaps it's a labeling error.

Alternatively, maybe the triangle has angles $74^\circ$, $x^\circ$, and $139^\circ$, but that's impossible.

So I conclude: Either the problem has a typo, or $139^\circ$ is an exterior angle.

But since the problem says “find all missing angles”, and labels $x^\circ$ as the unknown, and the other two angles are $74^\circ$ and $139^\circ$, but sum exceeds 180, no valid solution exists.

But to resolve, let’s assume that the $139^\circ$ is an exterior angle, so the interior angle is $41^\circ$, and solve.

Then:
$$
74 + 41 + x = 180 \Rightarrow x = 65^\circ
$$

✔ Answer (assuming 139° is exterior): $x = 65^\circ$, interior angles: $74^\circ$, $41^\circ$, $65^\circ$

But the problem says “triangles may not be drawn to scale”, but doesn’t mention exterior angles.

Alternatively, perhaps the $139^\circ$ is not an angle of the triangle — but it’s labeled as such.

Given the ambiguity, and since all other problems are solvable, I’ll go with the most likely intended solution: that $139^\circ$ is an exterior angle, so interior angle is $41^\circ$, and $x = 65^\circ$

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Final Answers:

#### 1.
$x = 46^\circ$
Angles: $59^\circ$, $75^\circ$, $46^\circ$

#### 2.
Assume $139^\circ$ is exterior, so interior angle is $41^\circ$
Then: $74 + 41 + x = 180 \Rightarrow x = 65^\circ$
Angles: $74^\circ$, $41^\circ$, $65^\circ$

#### 3.
$x = 13$
Angles: $41^\circ$, $113^\circ$, $26^\circ$

#### 4.
$x = 29$
Angles: $32^\circ$, $58^\circ$, $90^\circ$

#### 5.
$x = 26$
Angles: $63^\circ$, $50^\circ$, $67^\circ$

#### 6.
$x = 6$
Angles: $86^\circ$, $29^\circ$, $65^\circ$

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Fill-in-the-blank:

Triangle Sum Theorem – In a triangle, the 3 _interior_ angles sum to _180°_.

✔ Answer: interior, 180°

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Let me know if you'd like the worksheet filled out neatly!