SOLVED: Triangles Inequalities A Use the exterior Angle Inequality ... - Free Printable

We are given a triangle UVW with an exterior angle at vertex V. The interior angles are:

- ∠U = c°
- ∠W = b°
- ∠UVW = a° (interior angle at V)
- ∠d° is the exterior angle at V, formed by extending side VW beyond V.

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🔑 Key Theorems to Use:

1. Exterior Angle Inequality Theorem:
> The measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles.

So:
d > a is false — actually, d = a + b + c? Wait — no.

Actually, let’s clarify:

In triangle UVW, the exterior angle at V (angle d) is adjacent to interior angle a. Since they form a linear pair:

→ a + d = 180° (they are supplementary)

Also, by the Exterior Angle Theorem (not just inequality):

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

So:
→ d = b + c

✔ This is the key identity we’ll use for most parts.

And since a + d = 180°, we also have:
→ a = 180° - d

Also, from triangle angle sum:
→ a + b + c = 180°

But since d = b + c, then a + d = 180°, which checks out.

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Now let’s solve each problem:

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1. If b = 50° and c = 20°, then d > ______

By Exterior Angle Theorem:
d = b + c = 50° + 20° = 70°

But the question says “d > ______” — this is asking for the Exterior Angle Inequality Theorem, which says:

> The exterior angle is greater than each of the remote interior angles.

So d > b and d > c.

Since b = 50°, c = 20°, then d > 50° and d > 20°.

The question probably wants the larger of the two remote interior angles, since that gives the stricter inequality.

✔ Answer: d > 50°

*(Note: Technically, d > 20° is also true, but 50° is the more meaningful bound here.)*

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2. If a = 90° and c = 20°, then d = ______

We know:

- a + d = 180° → d = 180° - a = 180° - 90° = 90°

Alternatively, using triangle angle sum:

a + b + c = 180°
→ 90° + b + 20° = 180°
→ b = 70°

Then d = b + c = 70° + 20° = 90°

✔ Answer: 90°

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3. If b = 52° and c = 78°, then d = ______

By Exterior Angle Theorem:
d = b + c = 52° + 78° = 130°

✔ Answer: 130°

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4. If a = 90°, then b = ______ and c = ______

Wait — this is tricky. We are only given a = 90°, and asked to find b and c. But there are infinitely many possibilities unless more info is given.

However, note: the problem says “use the exterior angle inequality theorem and the given figure”.

But with only a = 90°, we can’t determine unique values for b and c — unless...

Wait — maybe it’s implying that the triangle is right-angled at V (since a = 90°), and perhaps we’re to assume something else?

Actually, looking back — in problem 2, when a=90° and c=20°, we found b=70°.

But here, no other values are given. So unless this is a trick or typo, we must consider:

From triangle angle sum:
a + b + c = 180°
→ 90° + b + c = 180°
→ b + c = 90°

And from Exterior Angle Theorem:
d = b + c = 90°

But still, without more info, we can’t assign specific values to b and c.

However, perhaps the problem expects us to realize that b and c can be any pair of positive numbers adding to 90°, but that doesn’t make sense for a fill-in-the-blank.

Wait — maybe it’s a misprint, and it meant to say “if a = 90° and d = 90°”, but that’s redundant.

Alternatively, perhaps the problem is testing if we realize that without additional constraints, b and c are not uniquely determined.

But that seems unlikely for this level.

Another possibility: Maybe in the original figure, it’s implied that the triangle is isosceles or something? But we don’t have that info.

Let me check problem 5 — it gives d = 105° and asks for c — again, not enough info unless we assume something.

Wait — perhaps all problems assume we are to express answers in terms of knowns, or maybe there’s a pattern.

Looking again at problem 4: “If a = 90°, then b = ______ and c = ______”

This is impossible to answer uniquely with current info.

Unless... perhaps the problem expects us to write “any values such that b + c = 90°”, but that’s not standard.

Alternatively, maybe it’s a trick, and since d = b + c and a = 90°, then d = 90°, so b + c = 90°, but again, no unique values.

Perhaps the problem intended to say: “If a = 90° and the triangle is isosceles with b = c”, then b = c = 45°.

That would make sense.

Given that this is a common setup, and since no other info is given, I think the intended answer is b = 45°, c = 45°, assuming isosceles right triangle.

✔ Answer: b = 45°, c = 45°

*(Note: This is an assumption, but likely what the problem expects.)*

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5. If d = 105°, then c = ______

Again, only d is given. We know:

d = b + c = 105°

But without knowing b, we can’t find c.

Same issue as problem 4.

Unless... perhaps the problem assumes that b and c are equal? Or maybe it's a continuation?

Wait — in problem 4, if we assumed b = c = 45°, then here maybe we assume something similar?

But 105° is odd for equal angles.

Alternatively, perhaps the problem expects us to leave it in terms of b? But that’s not a number.

Another thought: Maybe in the figure, angle b is labeled or implied? But we don’t have the figure.

Perhaps this is a mistake, and it should say “if d = 105° and b = 60°, then c = ?” — but it doesn’t.

Wait — let’s look back at the original image description. It says “use the exterior angle inequality theorem and the given figure”. But since we don’t have the figure, maybe in the figure, angle b is fixed?

Without more info, this is impossible.

But perhaps — in many textbook problems, when only d is given and they ask for c, they might imply that b is known from context or previous part.

Alternatively, maybe it’s expecting us to realize that c < d, which is true, but not a value.

This is problematic.

Wait — let’s consider problem 1 again. It said “d > ___” and we filled 50°, which is b.

In problem 5, if d = 105°, and we are to find c, perhaps they want the maximum possible value of c? But that would be less than d, so c < 105°, but again, not specific.

Alternatively, perhaps the problem is miswritten, and it should be “if d = 105° and a = 75°, then c = ?”

Because a + d = 180°, so if d = 105°, then a = 75°.

Then from triangle sum: a + b + c = 180° → 75° + b + c = 180° → b + c = 105°, which we already know.

Still stuck.

Perhaps the problem expects us to express c in terms of b, but that’s not a number.

I think there might be a typo in the problem.

But let’s try this: in some contexts, if no other info is given, and d is given, and they ask for c, perhaps they mean “what is c if b is minimized or something” — but that’s overcomplicating.

Another idea: Maybe in the figure, angle b is equal to angle c? Let’s assume that.

If b = c, and d = b + c = 2c = 105°, then c = 52.5°

That’s possible.

✔ Answer: 52.5° (assuming isosceles triangle with b = c)

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## 📝 Final Answers:

1. 50°
2. 90°
3. 130°
4. 45°, 45° *(assuming isosceles right triangle)*
5. 52.5° *(assuming b = c)*

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⚠️ Note: Problems 4 and 5 require assumptions due to insufficient information. In a real classroom setting, you might want to ask for clarification. But based on typical textbook problems, these assumptions are reasonable.

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✔ Boxed Answers:

1. $\boxed{50^\circ}$
2. $\boxed{90^\circ}$
3. $\boxed{130^\circ}$
4. $\boxed{45^\circ}$ and $\boxed{45^\circ}$
5. $\boxed{52.5^\circ}$